Essentials of Symbolic Logic R. L. Simpson
Start Citation[PU]Broadview of Press[/PU][DP]1999[/DP]End of Citation
© 1999 R. L. Simpson All rights reserved. The use of any part of this publication reproduced, transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, or stored in a retrieval system, without prior written consent of the publisher - or in the case of photocopying, a licence from CANCOPY (Canadian Copyright Licensing Agency) 6 Adelaide St. E., Suite 900, Toronto, Ontario M5C 1H6- is an infringement of the copyright law. Canadian Cataloguing in Publication Data Simpson, R.L. Essentials of symbolic logic revised edition Includes index. ISBN 1-55111-250-7 1. Logic, symbolic and mathematical. I Title. BC135.S556 1999 160 C98-932753-1 N.B. This title was originally published by Routledge in 1989 (ISBNs 0415-01870-6 hc, 0-415-01871-4 pb); Essentials of Symbolic Logic is now available worldwide exclusively in this new Broadview Press edition, which incorporates substantial revisions. Broadview Press Ltd. is an independent, international publishing house, incorporated in 1985. North America: PO Box 1243, Peterborough, Ontario, Canada K9J 7H5 3576 California Road, Orchard Park, NY, USA 14127 phone, fax and e-mail addresses for North America: (705) 743-8990 (phone); (705) 743-8353 (fax);
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Acknowledgements This book owes much to more people than can be listed here. Over the years, many of my students have pointed out errors and omissions in various versions of the text. Among the former students who have been particularly helpful are Sandy Bannikoff, John Davison and Dan Rubin. Various teachers of logic have given advice and encouragement. Norman Swartz in particular spent an extraordinary amount of time and effort when the enterprise of writing a new text seemed overwhelming. I have also received help and encouragement from Skip Bassford, Bob Goldman, Michael Picard and James Young. I am indebted to the readers at Broadview Press. They caught many minor errors and more than one major one. The readers made suggestions and criticisms, some accepted and some rejected. Many suggestions were ignored on the grounds that a short book could not deal with the suggested changes or additions. A number of readers had serious criticisms of a few claims made in the text. In several cases I rejected them on the basis that I did not agree. All the criticisms, rejected or not, were helpful. Where I disagreed, I tried to reword the text so that contentious issues are not presented in a way that suggests that opposing views are not worth considering. My family, of course, has had to put up with my preoccupation with the book. In the future, I hope it will be possible to eat a meal in a restaurant without asking someone for a slip of paper to write down a new idea. Start of Citation[PU]Broadview Press[/PU][DP]1999[/DP]End of Citation
Page i
Contents Chapter One §1.1: The Aims of this Book
1
§1.2: Primarily for the Student
3
§1.3: Primarily for the Instructor
4
Chapter Two §2.1: The Fundamentals
7
§2.2: Truth Functions
8
§2.3: Sentence Letters and Symbols
9
§2.4: Metalanguage
11
§2.5: Well-Formed Formulas
12
§2.6: Exercises
13
§2.7: Simple Translation: Negation
14
§2.8: Simple Translation: Conjunction
15
§2.9: Simple Translation: Disjunction
17
§2.10: Simple Translation: The Horseshoe
18
§2.11: Simple Translation: The Triplebar
20
§2.12: Translating Complicated Sentences
21
§2.13: Exercises
24
§2.14: Defining the Operators
26
§2.15: Truth Tables
29
§2.16: Exercises
32
§2.17: Types of Formulas
32
§2.18: Truth-Functional Equivalences
35
§2.19: Exercises
39
§2.20: Arguments
39
§2.21: Exercises
43
§2.22: Translating Arguments
45
§2.23: Exercises
50
§2.24: Summary of Topics
51
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Page ii
Chapter Three §3.1: What a Derivation Is
55
§3.2: Conjunction Introduction
57
§3.3: Conjunction Elimination
59
§3.4: Negation Elimination
61
§3.5: Disjunction Introduction
62
§3.6: Horseshoe Elimination
64
§3.7: Triplebar Elimination
66
§3.8: Exercises
67
§3.9: Constructing Derivations
68
§3.10: Exercises
79
§3.11: Assumptions
80
§3.12: Reiteration
82
§3.13: Horseshoe Introduction
84
§3.14: Triplebar Introduction
88
§3.15: Negation Introduction
91
§3.16: Disjunction Elimination
95
§3.17: Exercises
97
§3.18: More Derivation Construction
99
§3.19: Exercises
112
§3.20: Indirect Proof
113
§3.21: Exercises
121
§3.22: Categorical Derivations
123
§3.23: Exercises
126
§3.24: Summary of Topics
127
Chapter Four §4.1: Individuals and Predicates
133
§4.2: One-Place and Multi-Place Predicates
135
§4.3: Choices in Translation
137
§4.4: Exercises
138
§4.5: Quantifiers
139
§4.6: The Existential Quantifier
140
§4.7: The Universal Quantifier
142
§4.8: Using Both Quantifiers
144
§4.9: Exercises
147
§4.10: People and Things
148
§4.11: Exercises
149
§4.12: Identity
150
§4.13: Translation Problems and Domains
154
§4.14: Exercises
159
§4.15: Arguments in Predicate Logic
160
§4.16: Exercises
163
§4.17: Simple Interpretations
165
§4.18: Exercises
171
§4.19: Summary of Topics
174
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Chapter Five §5.1: Predicate Logic Derivations
177
§5.2: Existential Quantifier Introduction
178
§5.3: Universal Quantifier Elimination
181
§5.4: Existential Quantifier Elimination
183
§5.5: Universal Quantifier Introduction
186
§5.6: Identity Introduction
190
§5.7: Identity Elimination
191
§5.8: Exercises
192
§5.9: Derivation Strategies
194
§5.10: Exercises
203
§5.11: More Derivation Strategies
204
§5.12: Exercises
215
§5.13: Categorical Derivations in Predicate Logic
217
§5.14: Exercises
220
§5.15: Summary of Topics
220
Chapter Six §6.1: The Scope of the Text
223
§6.2: Existence and Translation Problems
224
§6.3: The Names of the Rules
224
§6.4: Unfamiliar Terminology and Symbols
225
§6.5: Additional Derivation Rules
228
§6.6: Replacement Rules
229
§6.7: Avoiding Reiteration
230
§6.8: Joint Denial and the Sheffer Stroke
231
§6.9: Consistency and Truth Trees
233
Index Technical Terms
239 Start of Citation[PU]Broadview Press[/PU][DP]1999[/DP]End of Citation
Page iv
Symbols The symbols used in the text are introduced at the indicated pages. Various symbols mentioned only in Chapter 6 are not included. A v ttt V } XIX { } ( )
Greek Letters Several Greek letters are used in the text. They are listed below with their names and a rough guide to pronunciation. Upper-Case Letters: D phi rhymes with 'pie'- 'ph' as in 'PHone' V psi rhymes with 'pie' - 'ps' as in 'tiPSy' Q omega oh MEga or oh MAY ga Lower-Case Letters: x chi rhymes with 'pie'- 'ch' as in 'loCH' t tau rhymes with 'cow' Start of Citation[PU]Broadview Press[/PU][DP]1999[/DP]End of Citation
CHAPTER ONE
INTRODUCTION
SI. 1: THE AIMS OF THIS BOOK ... God has not been so sparing to men to make them barely twolegged c reatures, and left it to Aristode to make them ra lional.. . God has been more boumiful to mankind than so. He has given them a mind that can reason, withom being instructed ... John Loc ke, An Essay Concerning Human Understanding. Bk. IV, Ch. XVII . This quotation reflects Locke's belief that onc can reason logically without having studied the subj ect called 'logic'. I sha re Locke's view. T he srudy o f logic, of course, is far different now from what it was in Locke's time; the student of today is likely to spend much mo re time on symbolic logic tha n on the logic o f Aristode. Notwithstanding the great diffe rences between Aristotelian logic and contemporary symbolic logic, the view that one should learn logic in order to reason properl y is Slill preva lent. I do not deny that the study of logic can - and ohen does - sharpen one's ability to reason; what I deny is that this is the only, or even (he prima ry, reason for studying the subject. A common approach to the teaching of logic begins with the claim or at least the strong suggestion that the scudent can neither evaluate arguments
2
S1. 1:
THE AIMS OF THIS BOOK
nor construct decent arguments without some formal training in logic. Coupled w ith the equally dubious claim or suggestion that knowledge of logically respectable argumentation is useful in producing persuasive arguments, this approach [Q logic leads - by a decided ly shaky inference co the claim that every student should study logic , preferably symbolic logic with a banery of esoteric symbols to impress the uninitiated. There are at least three reasons why it is wrong to promote the study of logic in this way. First, people of any intelligence, from the time of Sextus Empiricus to the present, have realized that logicians do not have a monopoly on rational ity. The suggestion that one needs to study logic in o rder to deal with eve ryday arguments is manifestly false. T he very enterprise of studyi ng logic requires (he ability to reason, and students recognize this. To begin the study of logic with an attempt to establish a conclusion by means of an obviously unsound argument is hardly a way lO encourage people to assess their beliefs on some rational basis. Second, the claim that logic should be studied only because it is 'useful' belittles the subject. Proving the obvious is a sophisticated activity; pretending that the obv ious is not obvi ous until it is demonstrated is both dishoncst and pedagogically unhelpful. The study o f symbolic logic is a worthwhile activity in itself; there is no need lO invent spuri ous justifications for learning something about an intel1ectually exciring activity, Third, the all toO common practice of attempting to express every philosophical claim and argument in symbolic terms frequently ignores subtleties which are comparatively clear in English and completely lost in the symbolic language. Although the use of symbols may not, as some claim, be the last resort of me philosophical scoundrel, students who complain about 'philosophical squid' hiding in an inky cloud of symbols often have a legitimate complaint. T he study o f symbolic logic has, of course, practical benefits, but these benefits are not primarily skills in applying a testing procedure to a rguments. Over the years I have dealt with hundreds of logic students. M y experience has been that most students gain twO things from an introductory course in symbolic logic. First, translating sta tements and arguments from English into a symbolic language makes the student read and write carefully. For the most part, post-secondary students do not need to learn how to read a nd write carefully; they already know how. Nevertheless. translating from English into a symbolic language forces students to usc the knowledge they have, and in many cases habits of dose reading and disciplined writing whic h are helpful lO the students in all their studies are developed . Second, producing derivations in a natural deduction system is an exercise in problem-solving. The problem of getting a fo rmula in the right place using only a pa rticular set of rules is challenging. T he simi la rities between
S1.2:
PRIMARILY FOR THE STUDENT
3
problems of this son and problems facing students in computer science will be obvious to anyone with any experience in programming computers. Many students have told me that their wo rk with derivations has helped them in their computer science courses. Whateve r the student goes on to study, the habit of taking a systematic approach to problems is worth developing. The study of logic, like the study of any intellectually respectable subject, tends to inculcate this. Given this perception of what in fact is learnt by most students in an introductory course, the choice of a text is not easy. There are many excell ent texts available. My primary objection to most o f them is that they cover much more than would be taught in an introductory course, with the result that the students have to pay for material which they do not use and are intimidated by a text of forbidding length and complexity. Afte r teaching seve ral courses using no tes prepared by myself and ot hers, I decided to write the material in JUSt the way I want to present it in class. The result is Es5ef1tiais of Symbolic Logic. In producing this book, my concern is to produce a text which is short and affordable. Many topics are simply ignored, but the experience o f teaching ove r the years has convinced me that there is enough materia l here for a respectable course.
§ 1.2: PRIMARILY FOR THE STUDENT Many students take a course in symbolic logic only because it is a required course for a degree o r diploma . Others [3ke such a course because they have come across various strange symbols and want to kn ow what people are doing with them. Whatever one's reasons fo r studying symbolic logic, the study can be interesting and fruitful. Essentials of Symbolic Logic was written for neither fools nor illiterates. It was nOt written to teach anyone 'how to think properly' - whatever that may mean. It was written to supply a short text which would make the student familiar with a standard set of logical symbols and reasonably adept at using those symbols. The majo r part of the book is concerned with solving derivation problems, the problems discussed in Chapter 3 and Chapter 5. These problems can often be baffling, but considerable care has been taken to provide the student with a way o f approaching problems that minimizes frustration. Chapter 2 and Chapter 3 deal with sentence logic, the rdatio nship between sentences and clauses that ca n stand alone as meaningful sentences. Cha pter 4 and Chapter 5 dea l with predicate logic , which is concern ed with the internal structure of sentences. An understanding of Chapter 2 is necessary to understand anythi ng that follows. Howeve r, if one wants to
~ 1.3: PRIMARILY FOR THE INSTRUcrOR
4
skip the material on derivations in Chapter 3, one can go straight from
Cha pter 2 to Chapter 4. Chapter 5, the discussion of derivations in predicate logic, relies heavily on (he previous u eatment o f derivations, and should not be approached until onc has mastered the mater ia l in all of the earlie r chapte rs. Logic is a serious discipline and ca nnot be learm without se rious intellectual effon. However, learning some elementary symbolic logic is far easier than it might appea r at fi rst glance; the symbols which initially seem
so confusing we re developed to ma ke matters simpler, not more difficult . In working through [he [ext, students should become clearer in th ei r understanding of the logical structure underlying English sentences and develop problem-solving ski!ls as they work at producing derivations. In the process, many students actually enioy themselves.
S1.3:
PRIMARILY FOR THE INSTRUCTOR
Various versions of Essentials of Symbolic l.ogic have been used fo r some dozen years with students of varying abilities and intc res(S. While no text ca n anticipate all the problems whi(;h studel1ts migh t hOlve, (;OIre hOls been taken to add ress recu rrent diffic ulties. Metatheory is nO( considered for three reasons. First, it would add to the length and cost of the book. Second, many students arc nm concerned with metatheo retical Questions; their interem and career plans lead them in Quite different di rections. T hi rd, most students need to be ve ry familiar with so me system of logic before they can address the problems of meta theory seriously. There is not very much time spent on instructions for uanslating English sentences and arguments, although these are important skills. M ost students al ready have an understanding of English which is sufficient to cope with Quite complicated semences and arguments; they do not read and write with care because they have not been forced ta do so. Ave rage studems are Quite ca pable of distinguish ing between 'if' and 'only if' without being instructed. Similarly, any student who can read this text can identify the premisses and conclusions of Quite compl ex arguments. It is not necessary to spend much time on teaching such things; simply demanding ca reful work (and grading exerc ises accordingly) fo rces students to use the knowledge and skills they already have. This text has been written with the recognition that many stude nts will go on to study more logic from other texts. Since there are different cOllvemions current, it is impossible to avoid so me inconsistencies. Th ere are also a couple o f eccent ricities which require explanation. First, the way in which the colum ns of truth tables are numbered may strike some
S1.3:
PRIMARILY FOR THE INSTRUCTOR
5
instructors as being potentially confusing to the students. I have marked thousands of truth tables, and getting the students to number the columns in the way I suggest is the only way I have found which forces them to complete every column on the truth table and to indicate clearly what they take to be the main operator in eac h formula. Second, I have deliberately avoided the common practice of writing variables in italics. As far as pOssible, I have.: [(jed to set up formulas, tables, argumCIlIS and derivations in a way that any student can duplicate with no great calli ~raph ic skills. Neither of these departures from the more usual con ventions sho uld cause any difficulty for the student wh o moves on ro anOther texI. I use the caret symbol, '/\', rather than the more common ampersand or dot. DOts produced by students are often shapeless blobs or virtually invisible, and many students have great difficulty in writing a legible ampersand. Students have better things to do with their time than to Struggle with such difficulties. In accordance with almost universal usage in logic, I rder to me '-' as a 'tilde' although strictly spea king it is ca lled a 'swing dash'. (Typographers will insist that the tilde is a diacritical mark which appears over a character, as in 'canon'.) I refer to parentheses as 'brackets'. This usage is formally correct in some countri es and colloquiall y common wherever English is spoken. Students will nOt be confused by the use of either word. The material can, of course, be supplemented by whatever other material the instructor sees fit. In the past, I have adjusted the course to the interests and capabilities of the students by introducing a discussion of set theory and Venn diagrams in one case, and some work on electrical analogues of the truth-functional operators in anOther. The common procedure of using incomplc:te truth tabl es is not discussed. Capable students can see when a truth table need not be completed, and incomplete truth tables can be discussed in class. My own practice in teaching has been to be sure above all else that the basics are thoroughly understOod. Additions, such as those above or a bit of meta theory, can be added as time permits. As far as possible, I have kept the technical vocabulary to an absolute minimum, Students already have to learn several unfamiliar terms, and learning unnecessary vocabulary distracts students from understanding logical concepts and dealing with logical problems. To keep the book short and ine xpensive, there are not many exercises. The exercises cover all me major wpics, and it will not be onerous fo r instructors to create their own exercises. Solutions to exercises in texts tend to be circulated after a text has been used for a while, and there is no substilute for produc ing fresh exercises for e.:a~h new dass. Students are not given solutions for any of the exercises in the book. The problem with providing solutions to exercises is that students almost
6
S 1.3;
PRIMARILY FOR THE INSTRUcrOR
invaria bly look at the solution when they tun into any difficulty, realize that there is no insu rmountable problem, and think they have learnt something whe n in fac t the y have not. A brief discussion of seve ral topics not discussed in the main text is included as Chapte r 6. A very short discussion of truth trees is included in [hat chapter. Additional rules such as modus tallens and various replacement rules are also mentioned ill [his chapter. Understand ing the basic set of fU IeS is essential to understand ing symbolic logic. While the additional rules make shorter derivations possible, they add link of theoretical interest. Some instructors have suggested that the insistence on reiteration makes derivations unnecessarily long. In Chapter 6, I mention t he possibil ity of avoiding reiterations, but this is not a practice r would recommend to students. It is easy to see how the rdteration steps could have been skipped in a completed derivation. However, in the process o f const ructing der ivatiolliO, pa rticularly ill predicate logic, reiterating fo rlllula s onto tile scope line on which one is working makes the problems more manageable and helps to avoid se rious errors.
CHAPTER TWO
SENTENCE LOGIC
§2.1: THE FUNDAMENTALS The fir st part of this chapter will dea l with a simple artificial language and the relationship between English and that language. The second part of the chapter wilt use (he anificiallanguagc to deal with some matters which 3rc most easily dealt with by using a non-natural language. It is not claimed tha t the artificial language which will be developed is bener than a natural language such as English; the claim is merely that for some purpoSeS the artificial language is morc convenient than a natural language. The basic unit to be consi dered is a declarative sentence, the SOrt of thing which can be tfue or false. For example, 'Rome is an ancient city' is a sentence having a truth value, which is JUSt another way of saying that it can be true o r false. By contrast, 'Is Rome an ancient city?' is neither true no r false; it does nm have a truth value. Similarly, a n~Ques[ like 'Please shut (he door' cannm be said to be either true or false. The only sentences which will be considered in t his book are sentences having truth values. The sentence 'The equator is to the south' is tfue when uttered in England and false when uttered in Australia. Similarly, the sentence 'My !lame is Bill' is true when uttered by Bill and false when uttered by John. The fact that th e truth value of a sentence ca n depend on the occasi on of its uttera nce has led
7
8
\2 .2 : TRUTH FUNCfIOSS
some logicia ns to make a distinction between a statement or proposition which strictly speaking has a truth value and a sentence which is a meaningful seque nce of words. T his distinction is not wi thout its difficulti es, but there is no need to consider them here. For present purposes, the distinction can simply be ignored. In this book the word 'semencc' is to be unde rstood as referring to the sort of thing that can be tfue or fa lse, and w ill be used according! y. Two points about truth and falsity should be noted here. First. dete rm ining the t ruth or fa lsity of a se ntence such as 'The Chi nese invented gunpowde r' is not necessa ry for prescnt pu rposes; one can sim pl)' assume tha t it is true or assume t hat it is false. In the terminology of logic, to make an assumption about the truth or fa lsity of a sentence is to assign a truth value to the sen tence. Second, the re is no need to complicate matters by conside ring the poss ibility of differing degrees of truth or falsity. Va rious logicians have attempted to develop logics in which one can deal with this sort of thing, but this book will deal with only two truth values. Thus, the sentence ' Wine helps the digestion' will have to be regarded simply as true or fa lse, and not partly true or [rue fo r some people. It should be noted that this places no restriction on what one can say. If one wants to say that wine drunk in moderation helps t he diges tion of some people, one ca n say just that by usi ng the sentence 'Wine drunk in moderation helps the digestion of some people .'
5 2. 2: "!RUTH FUN CTIO NS A sentence such as 'Jack is a lawyer and Joh n is an accounta nt ' is usually called a compound sentence. It is formed by combi ning the tWO simple sente nces, 'Jack is a lawyer' and 'John is an acco untant.' In o rder to tell whether or not the compound se ntence is true, one need only know w hethe r or not 'Jack is a lawyer' is t rue and whether or nOt 'John is an accountant' is tr ue. The tr uth value of 'Jack is a lawyer and Jo hn is an accountant' is a function of the truth value of 'Jac k is a lawyer' a nd the truth value of 'John is an accounta nt.' Such functions a re tfuth functions . T he word 'and' in t he compound sentence ser ves as a truth·funcri onal operatof be tween the twO simple sentences. It is easy to find other examples of truth-functional operalOrs in everyday English. T he word 'or' in the se ntence 'Either Bill will go to th e weddi ng or Jo hn wi ll go to the wedding' serves as a truth-functional operator. The truth value of the compound sentence can be de ter mi ned from the trut h values of the simple sentences. For example, if one knows that Bill w ill go to the wedding, o ne knows that {he compound sentence is {rue. Si milarly, if one
52.3:
Sn..'TENCE LEITERS AND SYMBOLS
9
knows that Bill will not go to the wedding and one knows that Jo hn will not go to the wedding, then one knows that the compound sentence is false. Altho ugh th e phrase 'It is false that .. .' does not connect twO simple sentences, if one knows the truth value of 'Mary is married to Brian', one knows the truth value of 'It is false that Mary is married to Brian.' The truth value of 'It is false that Mary is married to Brian' is a function o f the truth value o f 'Mary is ma rried to Brian.' Accordingly, the phrase 'It is false that .. .' serves a role which is analogous to (he role served by 'and' and 'or' in the sentences JUSt discussed, and it is a truth-functional o perator. Truthfuncti onal operators arc o ften called 'truth-fun ctional connectives', and in much writing in logic (he wo rd 'connective' is frequ ently used when 'operator' is used in this book. It is because 'connectives' often do not in any straightfo rwa rd sense do any connecting that 'o perator' is lhe prefe rred vocabulary in this book. Man y sentences are formed by words or phrases which plainly do nm serve as truth-fun ctional operators. Fo r example, the truth value of '1·le is irritable because he lost the game' cannot be determined from the truth value o f 'He is irritable' and the truth value of 'H e lost the game '; one can know that he is irritable and know that he lost the game without knowing that his irr ita tion is the result of his losing the game. Simila rl y, the truth va lue o f '] went home before I went to the play' cannot be determined from (he (rmh values of ' I went home' and ' I went 10 the play.' Phrases such as 'It is obvious that .. .' a nd 'It is necessarily the case tha t .. .' are logically different from phrases such as 'It is false that .. .' This is shown by the fact that one can nm determine the truth value of 'It is obvious that Smith is guilty of (he cri me' from the truth value of 'Smith is guilty of the crime.' These examples of words and phrases which do not serve as truth-functional ope rators are mentioned in order ( 0 help clarify the limits of semence logic. Sentence logic is concerned with truth-functional operators. In the next fe w sections, fiv e different truth-functional operators will be discussed.
S2.3: SENTENCE LETfERS AND SYMBOLS When the nOli on of a truth function was introd uced in 52.2, simple sentences such as 'Mary is married to Brian' were picked arbitrarily. It should be clear that th e pa rticular sentences involved did nOt matter. For purposes of doing logic, ra ther than writing out arbitrarily picked sentences, one can use leners to stand for sentences. The symbolic language to be developed in this chapter will use any upper-case Roman letter to represent a sentence, so such express ions as 'It is fal se (hat C' and ' R o r S' can be used to discuss truth functions. Alth ough one is restricted to twenty-six leners, there is no theoretical limit to the numbe r o f distinct sentences which can be
!O
S2.3:
SENTENCE LEHERS AND SYMBOLS
represented by sentence letters. The need will never arise , but it is possible to extend the srock of sentence leners by using subscripts. Thus, one can write ' As and B8 ' or 'It is false that GJ2.' Use of subscripts for this purpose should be avoided because it is unnecessary. Th e possibility of using more than twenty-six sentence letters is mentioned onl y to make the theoretica l poim that the symbolic language docs not restrict o nc lO a limited number of sente nces. There a rc advantages in using s(Otc nce leners rathe r than senten ces. As a maner of simple convenie nce. it is JUSt quicker to write a tener than to write a sentence . More imporlant, in considering the way logical operators work, one should be careful nOI 10 leI the content of Ihe sentences confuse Ihe issue. Using sentence letters allows one to see clearly the logical connections bet weell various simple alld cOlllpoulld SClltI:m::es withuut juugillg isSues of truth and falsit y on the basis of the COntent o f particular sentences. Similar considerations ma ke it desirable to use symbols instead of Engl ish words or phrases for the various truth-functi onal operators. This will become increasingly appare nt as the symbols and their approximate English equiva lents a re disc ussed. Five sym bols will be defined and the re lationship betwten thuSt symbols ailu vari ous English wurds alii..! phrases will be considered. A very crude account of the wa y t hese symbols are used fo llows:
'- A' is the rough equivalent of 'It is false that A.' The '-' is the tilde. 2 '( A 1\ H)' is the rough eq uivalent of ' A and B.' Th e '1\' is the caret. 3 '(A v B)' is the rough equivalent of ' A o r B.' Th e 'v' is the vel sometimes called the ' wedge'. 4 '( A:;) B)' is the rough equivalent of 'If A, then B.' T he ':;)' is the horseshoe some times called the 'hook'. 5 '( A ;;;: H)' is the rough equivalent of 'A. if and only if H.' T he ' ",,' is the triplebar some times called 'tribar'. All [he elements of the symbolic language whic h will be developed for sentence logic have now been exhibited. T hey arc very si mple, and consiS[ of three kinds of items: A stock o f twenty-six sentence letters, in principl e su pplemented by subscripted leIters. 2 Five symbols for creating new sentences from existing sentences. 3 A left bracket '(' and a right bracket punctuation marks.
'I'
which se rve as the only
S2.4: METAlANCUAGE
II
52.4: METALANGUAGE Now that a very sketchy outline of an artificial language has been given, it is time to consider a reasonable approach to discussing th is language. Most speakers of English who learn French are ta ught about the foreign language in thei r native language. Thus, a beginning student of French may learn some French by hearing or reading the English sentence, 'The French word "et" has the same meaning as the English word "and".' The language being discussed, French in this case, is the object language. The language uSed ( 0 discuss the object language, English in this case, is the metalanguage. The symbolic language of logic which is discussed in this book is the object language. It will be talked abo ut in a meta language, which is English supplemented by a few symbols borrowed from the object language and by a handful of Gree k letters. English speakers, of course, do not study only fo reign languages; they also study their own language. When they do, the ob ject language and the metalanguage are one and the same. In practice, this works quite well. Given some grasp of basic English, one can understand a grammar text written in English. T he re are, however, theoretical problems about using a single language as both object la nguage and metalanguage, and the sym bolic language in rhis text will never be used to talk about itself. The difficulty in using the same language as bOl h the objec t language and the metalanguage can easily be seen by considering the problems o ne wou ld ha ve in trying (Q show in written English the use o f quotation ma rks in English. It would be hard to avoid writing something which does nOl presuppose an understa nding of ho w English quotati on marks are used. The following sentence, for example, looks deci dedly strange and it is hard to see how anyone could learn anything by reading it: In English, the convention is to begin quotations with""''' and end quotations with""". By comrast, the following sentence does not look so strange and it is easy to see how someone could learn something by reading it: In French, the conven ti on is to begin qu mations with "oc" and end quo tations with .... ". For purposes of illustrati on, double quotati on marks are used in both examples. The same problems wo uld arise if single quota tion marks were used o r a combination of single and double quotation ma rks. Akin to the distinction between an object language and a metalanguage is the distinction between what can be shown within a logica l system and what can be shown about that logical system . Within the logica l system to be developed in this book, several rules will be given for doing such things as deriving one sente nce in the symbol ic language from another sentence.
12
~2.5: WELL-FoRMED FORMULAS
Th ese rules will simply be presented with the implied claim that tbey are logically respectable, but no auempt will be made to give a clear account of what that involves. Such an account is a question in metatheory; it is a maner of discussing the logic of the system rather than using it. Metatheory embraces this and many other topics, but none of them will be considered in this book. Unlike the rLiles of English spelling and grammar, the rules of the symbolic language arc extremely simple and completely consistent. Questions o f spelling do nO[ arise, and the only punctuation marks are the ldl and right brackets. In this book, the rules of the object language will be followed scrupulously - with onc exception. The onc permissible departure from the rules is discussed in S2.5. Many of thc quirks o f everyday English, however, will remain in the metalanguage.
51.5: WELL-FORMED FORMULAS JUSt as English has rules of grammar, so the symbolic language has rules. Some combinations of symbols conform to (he rules. These are well-formed fonnulas . A well-formed formula is a WFF. The usual prollUlICiatiull u f 'WFF' is 'whi ff' or ' woof' . The terms ' well-formed form ula' and 'WFF' will be used imerchangeably in this book. The rules for creating well· formed formulas will be explained as the language is developed. Other combinations do not conform to the rules. When these are shown for purposes o f illustration, [hey will be flagged with (hree daggers ('ttt') to emphasize that what has just been written is mistaken in some way. (Throughout this book, the three daggers will be used to mark errors.) A sentence letter standing alone is an atomic formula . Thus, 'A' is an atomic formu la, and '- A' is not. Any atomic formula is a WFF. Thus, 'A' is a well-formed formula, as are 'B', 'C' and 'Y'. Putting a tilde in front of any WFf results in another WFF, the negation of the o rigina l WFF. Thus, '- A', '- G' and '- M ' are all WFFs. Since '- A' is a WFF, '- - A' is a WFF, as are '- - - A', '- - - - A' and '- - - - - A'. Using the Greek letter 'ell' to stand for any well-formed formula whatever, '- ell' is a well-formed formula. (It should be noted that 'ell' is not a WFF in the symbolic language. It is part o f the metalanguage used to discuss the symbolic language.) Using the '¢I' and ''V' to stand for an y well -formed formulas, '(41 " \jI)' is a WFF, '( Ill v 1jI)' is a WFF, '(IP ::J 'V )' is a WFF and '( IP !!! 1jI)' is a WFF. Strictly speaking, the brackets are essential; '4> " 'P', 'Ill v \II', 'IP ::J \II' and '41 == 'V' are not WFFs, although the lack of brackets causes no confusi on. Throughout the rest of this book, when dropping brackets causes no ambiguity, they will usually be dropped.
13
S2.6: EXERCISES
For va rious purposes, diffe rent conventions have been established to disambiguate arithmetical expressions. However, for a normal reader, an expression sueh as '6 + 4 + 2' is ambiguous. Is one to add 6 and 4 and then divide the result by 2, or is one to divide 4 by 2 and then add 6 to the resu lt? Adding brackets in the appropriate places ma ke it clear whether it is the first or second which is intended: plain ly, '(6 + 4) + 2' means something different from '6 + (4 ... 2)'. In just the same way, ' A v B ::l C ttt is not a WFF because it is ambiguous between ' A v (B :::. C)' and '( A v B) :::. C. The former should be read as 'Either A, or if B then C, while the lan er should be read as 'If either A or B, then C. Strictly speaking, ne ither ' A v (B ::l C)' nor '(A II B) :::> C is a WFF, but the absence of the outer brackets does not create any ambiguity. In fa ct, the absence of the outer brackets makes formulas easier to read, and there is no reason to add them every time one uses 'A', 'v', ':::>' or '='. However, it is important to add brackets as mo re complex formu las arc created. For example, the two formulas'S' and 'P := Q ' can be used in various more complex form ulas such as 'S v (P := Q)', '( P := Q ) " S' and 'S :J (P ::= Q)'. In eac h case, brackets must be added to ' P ;;; Q ' in o rder to avoid sueh ambiguous expressions as'S v P E Q ' ttt , 'P s: Q " S' ttt 'nd'S => p. Q' ttt . Every formu la which is not an atomic formula has a main operalOr. In the formula'S v (P ;;: Q)', 'v' is the main ope rator; in the formula '(P :; Q) " S', ',,' is the main operator; and in the formula'S :::> (P :; Q )', '::')' is the ma in operator. The main operatOr of an y formula in sentence logic can be determined by considering the way in which the WFF is built up either from anothe r formula in the case of the tilde or fro m other fo rmulas in the case of the other o pe rators. The last operatOr wh ic h is added is the ma in operator. If there is no way of determining which operatOr is the main ope rator o f a fo rmula, then t hat formu la is not a WFE
51.6: EXERCISES Which of the following are strictly speaking well -formed formulas? 1 - - - 8 2 ((A) => 8)
3 (F
A
(G v H I)
4 ((F
A
G) v H )
5 ((F
A
G v H I)
6 - ( - M => ((R v G ) • SI) 7 - (- M => ((( R v G ) " S))
14
S2.7: SIMPLE TRANSLATION: NEGATIOl'
8 (F v G)
1\
(F v G))
9 R v 5
10 If
::>
Ig v s))
51.7: SIMPLE lRANSLATION: NEGATION It is now possible to do a linle translation from English inro the symbolic language. For example, one can translate 'Jack is greedy' 3S 'G', Then, if one wants to assert that Jack is not greedy, that it is fals e rhat Jack is greedy, onc can write ' - G'. Similarly, if the sentence Icncr 'R' is used to represent the English sentence 'Robert is my brother', the denial of the statement that Robert is my brother is represented by '- R'. As a gene ral rule, in translating English sentences inm the symbolic language, one should try to use atomic formulas and oper:Jtors to reflect the logica l Structure of the English sentence. Using 'M' for 'Martin went to the party' and translating 'It is false that Martin went to the party' as '- M ' is an example of a good translation. Using ' M ' for 'It is fa lse that Manin went to the party' is an example of a poor translation. 'il is false thal Martin wem to the party' is no t, of cou rse, the most colloquial way of saying that he did not go. The 'It is false that ... ' locution has been used until now beca use the English negation is before the sentence it negates, just as the ti lde is before the formula it negates. In ordinary English, however, there are ve ry many ways of expressing negation, few of them involving a negating clause at the beginning of a sentence. A knowledge of basic English is all tha t is requi red to recognize them, but a few are listed here:
It is not true that Sam is old. 0: Sam is old. -0
John is not coming. C: John is coming. - C It is false that she is abnormal. N: She is normal.
--N This is illegal. I.: This is legal. - L It is not true that this is not illegal. L: This is Icgal. - - - L
S2.8: SIMPLE TRANSLATION: COSJUNCn01\
15
They are dissatisfied. S: They are satisfied. - 5 He is incompetent. C: He is com petent.
-c It is irregular.
R: It is regular. - R One must always use care in translation and pay serious anemion to the meaning of the English semence. For example, to translate 'He is infamous' as 'It is false that he is famous' is plainly wrong; to say that someone is infamous is not to deny that th e person in question is famous. Simila rly, to translate 'Everyone is dissatisfied' as 'It is false that everyone is satisfied' is to change the meaning of the original sentence.
S2.8: SIMPLE 1RANSLATION: CONJUNCTION A formula such as 'A " S' is a conjunction, and the two fo rmulas flanking the caret are conjunclS. Using 'M' to represent ' Mary studies biology' and 'K' to represent ' Kate studies philosophy', one can translate ' Mary studies biology and Kate studies philosophy' as ' M " K'. Adding the sentence lener T to represent 'Jane studies history', one can translate 'Mary studies biology, Kate studies philosophy and Jane studies history' as '(M " K) " J'. ' M 1\ (K 1\ J)' would also be a correct translati on, but 'M " K A J' ttt would nol. This is because the formula is not well formed; it could not be produced by placing '1\' between twO WFFs. Even with the indusion o f outer brackets, '(M " K 1\ J )' ttt is ambiguous between '(M 1\ K) 1\ J' and 'M " (K 1\ j)' and therefore not a WFF. The fact [hat in this case the ambiguity is of no consequence does nm make the formula well formed . There are many ways in which conjunctive sentences are wrinen in English. As with negation, recognizing these involves no more than a little care and knowledge of English. Some of the more common locutions are given below: Bill is a denti$[ and John is a lawyer. B: Bill is a dentist. J: John is a lawyer. B A J
16
S2.g: SIMPLE TRA NSLATIOK COSJUNCTION
Bill and John are dentistS. B: Bill is a dentist. J: John is a dentist.
B
A
J
She was poor but she was honest. P: She was poor. H: She was honest.
P
A
H
Although cigarenc smoking is a common habit, it is dangerous . C: Cigarette smoking is a common habit. D: Cigarette smoking is dangerous. CAD Carl plays tennis and jogs. T: Carl pla ys tennis. J; Carl iogs.
T
A
J
Paul is successful; his brother, however, is nOl. P: Paul is successful. B: Paul's brothe r is successful.
r
1\
-
B
In translating English into symbols, pronouns should be replaced whenever possi ble by the nouns or noun phrases to which they refer. Someone reading the translation of one sentence should be able to understand it without having to refer to another se ntence. In translating conjunctive sentences, one needs to pay careful attemion to the sense of the English. It would be ludicrous to translate 'No one is a Prmestant and a Muslim' as the conjunction of 'No onc is a Prmestant' and 'No one is a Muslim', for exa mple. One reason for care in translating the word 'and' is chac ie is ohen used in English as something mher than a simple conjunction. When 'and' is used in such a way, it cannot be treated as a conjunction without some loss o f meaning. For example, 'We had dinner and went to [he theatre' is naturally taken to mean that we had dinner before we went to the theatre. 'H e caught pneumonia and died' is usually taken to mean that he died as a result of the pneumonia. In mOSt comcx {s, 'Sa m and Janet are married' should be ullderstood as fJltal~illg that they are married to taCh Other. Where something of consequence other than simple conjunction is expressed, an English semence cannQl be translated into twO formulas flanking a ca ret.
Sl.9:
SIMPLE TRANSLATION: DISJUNCfION
17
51.9: SIMPLE 1RANSLATION: DISJUNCfION A formu la such as ' M v P' is a disjunction , and the twO formu las flan king the vel are disjuncts. ' F v G', 'F v - G' and ' F v (G " H )' are all disjunctive fo rmulas. Using ' P' to represent 'Lana is in Paris', 'R' to represent 'Lana is in Rome' and ' B' to represent 'Lana is in Brussels', one can translate 'lana is in Paris, Rome or Brussels' in severa l ways. 'P v (R v B)' and '(P v R) v B' are both correct translations . Although there is nothing to choose between these two translations, '( P v R v B)' ttt is not a good translation because it is not a WFF. In English, [he word 'or' is sometimes taken to mean 'either but not both ' and sometimes taken to mean 'ei ther and perhaps both '. The former is the exclusive sense of 'or' and the latter is the inclusive sense of 'or'. In most cases, the context makes it perfectly clear which sense is intended, bue one often sees expressions such as 'and/or' and 'one or the other but not bach ' when these are needed to make the meaning perfectly clear. The symbol 'v' expresses {he inclusive se nse of 'or'. (In Latin, the exclusive sense o f 'or' is 'aut' while the incl usive se nse of 'ur' is 'vd' . This expla ins du: symbul and its name. ) Th ere are not so many ways o f expressing disjunction in English as there are ways of expressing cun junctio n. A few typical exampl es follow: Bob or Charlie will win a schola rshi p. S: Bob will win a scholarship. C: Cha rlie will win a scholarship. S v C Bob or Cha rl ie, but not both, will wi n a scholarship. S: Bob will win a scholarship. C: Charli e will win a schola rship. (8 v C) " - (8 A C) Either Jane or Mary will come to the party. J: j ane will come to the pa rry. M : Mary will come 10 the party. Jv M Either Jack will co me to the picnic or Sam won't. j: j ack will come to the picnic. S: Sam will come [Q the picnic. Jv - 5 Ma ry will go, or j ill and Sally will go. M: Mary will go. J: Jill will go. S: Sally will go.
M
v
IJ
A
51
18
S2.10: SIMPLE TRANSLATION: THE H ORSESHOE
Mary o r Jill will go, and Sally will go. M: Mary will go.
J:
Jill will go. S: Sally will go. (M v
1>
A
S
Either Peter's mother will be upset or he is doing his homework. M: Peter's mother will be upset. H : Peter is doing his homework. M v H Peter's mother will be upset unless he is doing his homework.. M: Peter's mothe r will be upset. H: Peter is doing his homework. M v H Translating both 'Ei ther Peter's mother will be upset or he is doing his
homework' and 'Peter's mOl her will he upset unless he is doing his homework' as disju nctions may strike some readers as odd. Treating ' unless' as a disjunction produces the most economical translation, but the word can he translated in other ways. The claim that Peter's mother will be upser unless he is doing his homework can be thought o f as claiming that Peter 's mother will be upset if he is not doing his homework, and translating the sentence in (his way is perf..-cdy correct. T his will b..- mad..- clea rer in S2.1 0 and in )2.IS where truth-functional equivale nces are discussed. Translating disjunctive sen tences is straightforward if one is ca reful not to distort the meaning o f the English sentences. It requires no logica l soph istication to recognize that 'Every number is odd or even', for example, can not be translated as the disjunction of 'Every number is odd' a nd ' Every number is even. '
§2.10: SIMPLE lRANSLATION: THE HORSESHOE Using 'P' to represent 'Peter will go to France' and 'J' to represent 'Jim will go to France', the sentence 'If Jim goes to Franc..-, Peter will go' ca n be translated as ') ~ P'. The fo rmula CO the left of the horsesho..-, the 'jf' parr, is the antecedent. The formula to the right of the horsesh~, the 'then' part, is the consequent. Both antecedents and consequents can be atomic formulas or complex fo rmulas. 'If ... then' sentences are ofte n called 'conditional' sentences.
)2.10: SIMPLE TRANSLATION: THE HORSESHOE
19
There are many ways o f expressing conditional semences in English. Some of the more common ones fo llow: If it is su nny, I will go.
S: It is sunny. G: I wilt go.
S ::> G I will go if it is sunny. S: It is sunny. G: I will go. S ::> G
1£ Peter is not doing his ho mework, his mother will be upset. M: Peter's mother will be upset. H: Peter is doing his homework. - H => M
Pete r's mother will be upset unless he is doing his homework. M : Peter's mother will be upset. H: Peter is doing his homework. - H => M Assuming that I don't lose my job, I'll buy a new car this year.
L: I will lose my job. B: I will buy a new car this yea r. - L => B
If you are an American citizen by birth, you can be President, provided that you are at least thirty-five yea rs old. C: You afe an American citizen by birth . P: You can be President. 0: You are at least thirty-five years old.
e
=> (0 => P)
You cannot castle when you have mo ved you r king.
C: You can castle. K: You have moved your king. K => - C James will get the scholarship only if he learns a little German. J: James will get the sc holarship. L: James learns a little German. J ::> L There is fir e only if there is oxygen. F: There is fi re. 0 : T here is oxygen. F ::> 0
20
S2.ll: SIMPLE l'RAKSLATION: THE TRIPlEBAR
The everyday expression 'only if' confuses many people when they are translating. 'There is fire only if there is oxygen' means [hat oxygen is necessary for fire - if one sees fir e, one knows that oxygen is present. It does not mean [hat if oxygen is present then fire is present. The distinction between 'if' and 'only if' is clearly shown in [he expressi on 'if and only if' which is discussed in S 2.11. III ITlOSt Cases, translating English 'if ... theil' SentenCeS iuto symbuls ilti not difficult. As always, one needs to be sure that the meaning of the English sentence is not changed. For example, 'If something is a cow, then it is a mammal' anainiy cannot be translated as ' If something is a CO w , thell something is a mammal. ' The former semence assens that all cows are mamma ls, while the latter is far less informative. There are some special complexities concerning the horseshoe which will become apparent when this symbol is formally defined. These complex ides will be discussed later in this chapter.
§2.ll, SIMPLE "TRANSLATION, THE "TRIPLEBAR Somt: t:xamplt:s of formulas whi(;h cOlHain 1:1 lriplebar are 'K -= L' , '- J -= - M' and ' S -= (V ::> H)'. Translating 'I will apply for the job' as 'A'
and translating 'Peter resigns' as 'P', one can translate the semence ' I will apply for the job if and only if Peter resigns' as 'A -= P'. A semence using 'if and only if' is often called a 'biconditional' sentence. Expressions such as 'if and only if' are relatively rare in colloquial English . When people use such expressions they are usually speaking carefully, which tends to make the task of translation straightforward. Some of the more common ways of expressing 'i f and only if' follow: You are eligible if and only if you are a student. E: You are eligible. S: You are a studem.
E.S You can go when and on ly when the light is green.
G; You can go. l;
The light is green.
G.L Saying that it is an acid is equivalem to saying that it is a prOton donor. A: It is an acid . P; It is a proton donor. A a p
S2.12: TRANSLAnNG COMPLICATED SF.l\.TE~ C ES
21
Provided, but only provided, that [here are no objections from the major shareholder, the merger will take place. S: There are obj ections from the major shareholder. M: The me rger will take place.
- S. M The equipment will be used juSt in case there is a fire. E: The equipment will be used. F: There is a fire.
E·F John will take the course iff Kate takes it. J: John will take the course. K: Kate will take the course.
J•
K
'Iff' in the final example is not a misspelling; it is logicians' shorthand for 'if and only if', It is pronoun,ed 'if-if' and the triple bar can always be used in translating it. Translating English sentences into formulas which contain a triplebar is usually quite straightforward, but of course onc muSt nOt translate blindl y with no anention w [he meaning of the sentence under considerati on. Fo r example, 'I am tak ing my umbrella JUSt in case it rains' is nO( to be unde rstood as '\ am taking my umbrella if and only if it rains.' Simi larly, 'Every animal has a heart if and only if it has a kidney' must not be read as 'Every anima l has a heart if and only if every an imal has a kidney. '
§l.12: 1RANSLATING COMPLICATED SENTENCES A knowledge of how the truth-functional operators work in simple cases, wgether with an ordinary grasp o f the English language, is all that is required to deal with very complicated sentences. One need on ly pay careful attention to the meani ng of the sentence bei ng translated. A few examples should show this. The sentence 'If Sam goes to the chess club or visits his sister, he will be late for the play' can be broken up into three simple sentences which are joined by truth-functi onal operators. One should begin the translation by assigning sentence letters to the three simple sentences. The choice o f sentence letters is arbitrary, but one might as well usc letters which help one to remember which letter represents wh ich sentence. 'C' can represent 'Sam goes to the chess club', 'S' can represeO[ 'Sam visits his sister' and 'P' can represent 'Sam will be late for tht play.' The senttnce asserts that if Sam does either of two things he will be late for the play; if ht either goes to the chess club or visits his sister, then he will be late fur the play. Thus, a correcr translatio n of tht original sentence is '(C v S) ~ P'. (There are mher
22
S2.1 2:
TRANSLATING COMPLICATED S£~TENCES
correct translations of this sentence. The various possibilities will be discussed later in this chapter.} One can now write the original and the translation in a clear way: If Sam goes to the chess club or visits his sister, he wilt be late fo r the play. C: Sam goes to the chess clu b. S: Sam visits his sister. 1': Sam will be late for the play, (C v 5) ~ P
Four points about this translation should be noted. Fi rst, the reader is given a clear indication of what is being translated, which sentence lette r represents which simple sentence, and what the translatio n o f rhe o riginal sentence is. Second, eac h sentence lener is assigned to a complete English sentence something which has a truth value . 'If Sam goes [Q the chess club' and 'or visits his sister' are nOl, of course, grammatical se ntences. It is often necessary to do a little rewording in order to have a grammatical si mple sentence to which one can assign a sente nce letter. Third , wherever possible, pronouns are replaced. As far as possible, each sentence letter shou ld be assigned to a se ntence which does not require rde rence to some other sentence for an understanding of its meaning. Fourth, the posit ion of the brackets is extremely important. Unlike '(C v S) =::. P', 'C v (S =::. P)' says that either Sam goes to the chess club o r he will be late for the play if he visits his sister. Pla inly, this is not the meaning o f the original sentence. 'If Sam goes to (he chess club or visits his sister, he will be late for the play' is a conditional sentence, while 'Either Sam goes to the chess club or he will be late fo r the play if he visits his sister' is a disjunctive sentence. The twO sentences have diffe rent mai n operators. In translating a sentence from English into the symbolic language, the brackets should reflect the structure of the English sentence. It would be hard to misread the sentence about Sam which was JUSt d iscussed. The presence of a comma in 'If Sam goes to the chess club or visits his siste r, he will be lare for the play' makes it completely cl ear that this is a conditiona l sen rence. However, one cannot rely on the presence of pu nctuation ma rks. One must often determine the intended mea ning of a sentence by keeping in mind what someone might be expected to mean by that sentence. This means that one frequentl y has to depend on the context to determine how a sente nce is to be understood. The position of the brackets does not always affect the meaning of the sentence. It does not matter, fOf example, in translating 'Jones, Smith o r the butler did it.' 'U v 5) v B' and 'J v (S v B), are both perfectly good
23
)2.12: TRANSLATIl\G CO:-.1PI.!CATED SE:\"T£:\"CF};
translations. '0 v S v BY ttt. however, is not a good translation because there is no wa y of tell ing which vel is the main operator. In everyday English, mose sentences are quite short. Occasionally, however, one comes across a long sentence which ca n cause some confusion. In such cases one should break up the sentence in stages. Th e follow ing sentence is grammatically correct, although far [00 lengthy from a stylistic poi nt of view: If j ack is neither on {he track nor in the pits, {hen his car has broke n down or he has crashed and a caution fla g is out; but if a caution flag is out, aU the cars will slow down and everyone will know it.
This is a con junctive sente nce, with ' buc' se rving as che main operator. The first part o f the sentence is a complex 'if ... then' clause and the second part o f the se ntence is a less complex 'if ... then ' clause. The fi rst thing to do is to identify the simple sentences. The re arc seven . If Jack is neither on the track nor in the pits, then his car has broken down or he has crashed and a caution fJag is ou c; but if a cau tion flag is out, all the cars will slow down and everyone will know if. T: Jack is on the track. P: Jack is in the pin. B: j ack's car has broken down. C: Jack has crashed. F: A caution flag is out. S: All the ca rs will slow down. K: Everyone will know that a caution flag is out.
Since it is the simp lest, one can tackle the second 'if ... then ' clause fi rst. Thi s is easily translated as 'F :::> (S 1\ K)'. The fi rst 'if ... then' clause is not much mo re difficul t. The antecedent of the clause, 'j ack is neither on the track nor in the pits', can be translated either as ' - (T v P)' o r as '- T 1\ - P'. Since the former is slightly shane r, it will be used. The consequent of the clause, ' ... his ca r has broken down or he has crashed and a caution flag is Out', is easily translated as '8 v (C 1\ Fl'. Accordingly, the whole cl ause can be translated as '- (T v P) :::> (8 v (C 1\ F))'. The entire sentence is simply the conjunction of the two clauses: (- (T v P) => (S v (e
A
F»)
A
(F => (S
Th ere are, of cou rse, other correct translations .
A
K»
~2.13: EXERCISES
24
The procedure for translating an English sentence into the symbolic language consiSts of five steps. One should: 1 Read the complete sentence carefully to grasp its meaning. 2 Determine how many simple sentences there arc. Every English seme nce should be broken up into as many simple sentences as possible in order to let the operators show the logical structure. One should replace pronouns where possible. 3 Assign letters to each si mplt= sentence, providing the reader with a clear indication of which letter represents which simple sentence. 4 Check to be sure that insignificant differences in wording have not resuhed in any redundant semence letters . 5 Write the sentence in the symbolic la nguage, being sure that this translation is a WFF and fh at if captures the meaning o f the original sentence.
One can t hen remove any unnecessary brackets. Translating English sentences into symbols is usually not very difficult if one takes things a step at a time. On occasion, a piece of English is so unclear that onc does not know which of several different translations is an expression of the intended meaning. If the context docs not make the sentence clear, the best one can do is to produce as many t ranslations as there are reasonable interpretations of the se ntence.
S2.13:
EXERC ISES
SZ.13 Pa rt I Using the given sentence lette rs, translate each of the following sentences. 8: Brian drives a BMW. M: Michael drives a Mercedes Benz. P: Paul drives a Porsche. 1 Paul does not drive a Porsche. 2 Brian drives a BMW only if Michael drives a Mercedes Benz. 3 If Michael doesn't drive a Mercedes Benz or Pau l doesn't drive a Porsche, ,hen Michael doesn 't drive a Mercedes Benz and Paul doesn ', drive a Porsche. 4 Un less Pau l drives a Po rsche, either Michael drives a Mercedes Benz or Bria n drives a BMW. 5 Brian drives a BMW if and only if either Paul doesn't drive a Po rsche o r M ichael drives a Mercedes Benz. 6 Paul d rives a Porsche; Brian, however, drives a BMW.
S2.13: EXERCISES
25
7 Michael drives a Mercedes Benz or Brian dri ves a BMW, but it is not true tha t Michael drives a Mercedes Benz and Brian d rives a BMW.
8 If Paul dri ves a Porsche, Michael drives a Mercedes Benz if Brian dri ves a BMW.
9 Provided that Michad drives a Mercedes Benz, Paul drives a Porsche only if Br ia n dri ves a BMW. 10 If it is fals e that Paul drives a Porsche and Michael drives a Mercedes Benz, then either j1aul drives a Porsche and Brian does not drive a ~MW o r Michael drives a Mercedes Benz and Brian does not drive a BMW.
p. 13 Parr 2 Translate each o f the following sentences into symbols, being sure to specify clearl y (he meanings of the sentence letters. I am not guilty.
2 My grandfather is old, but he is very active.
3 My gra ndfathe r is nor old, bm he is not very active. 4 She doesn't own her car; she leases it. 5 That exercise is neithe r enjoya ble nor useful.
6 The old are wise and cautious; the young are foo li sh and brave. 7 Neither of my pa rents has blue eyes . 8 He was an officer but no gentleman. 9 There is no heavy industry in the eastern o r northern regions.
10 T hey do not ask politdy; they either snarl or grovel. 11 Wh en you're hot, you're hot; when you're not, you're not.
12 Coppe r can be soldered but stainless steel must be welded. 13 If the rates on our loans remain the same and the rise in the ma rket does not slow down, we are bound to become mill io naires.
14 John will buy a new ca r only if he can sel l his old o ne. t5 I've nOt only made a foTtune in the market; I've made some money as a financia l jou rnali st. 16 The star doesn't sing, dance or do any comic routines.
17 If mortgage rates go down, I will buy a house if my wife still wants me to do so. 18 You will be caugh t unless you slow down and/or buy a radar detector.
19 Eithe r the theory of the supply-siders is right and that of the traditional economists is wrong, or the theory of the traditional eco nomists is right and that of the supply-siders is wrong.
26
S2. 14: DEFINING THE O PERATORS
20 Provided, but on ly provided, that twO members make an objection in writing, the chairman is obliged to reopen the question o f Smith 's membership if Smith still wants to join the cl ub.
S2.1 4: DEFINING THE OPERATORS T he five tru th -fu nctional ope rators have ~e n int roduced as being ro ughly equivalent to certain English words or phrases. There are serious problems connec ted with defining some o f the terms of a natural language such as English. Fo( CX' is tr ue while '\jI ' is false. 3 ' ' is fa lse while '1jI';s true . 4 ' " lV' is true only if '~' is true and '1j.I' is [ruc; in all other cases it is false. Although the caret suffices to express some uses of 'and', 'but' and other English expressions, it is defined by the truth table. The truth table for disjunction reflects the same four poss ibiliti es : I'bljl
4>vlV
T T T F
T T
F T F F
T F
Bearing in mind that 'v' is the rough equivalent of the inclusive sense of 'o r', there is nothing surprising in this truth table. ' y lV' is true if either '' is true or '0/' is true ; it is false only if both ' 4>' and '0/' are fals e. 'v' is defined by the truth tabl e, and this definition does in fa ct capture the sense o f man y uses of 'or' in English. So far, the truth tables for the operators have been JUSt as one woul d expect them to be from an understanding of English. Th e truth table fo r the horseshoe does not follow this pattern. As before, there are four possibilities, but the final two rows of the truth table are not what one might expect; 4>
lV
4>:J lV
T
T
T
T
F
F
F
T
T
F
F
T
The important point to remember is that the ho rseshoe is defined by this truth table. The easiest way to remember the truth table for the horseshoe is to think about how an English conditional sentence could be sho wn to be false. The SUfe way to show that a cond itional sentence in English is false is to provid e an example in which the antecedent is true and the consequent is false. If a man claims that he will eat his hat if Brown wins the election and Brown does not win, it is hard to establish whether his claim is true. If, on the other hand, Brown does win the election and the man does not eat his hat, the falsity o f his claim is clearly established.
28
S2.14: DEFINING THE OPERATORS
The horseshoe is a truth-functional operator, and the truth values o f many conditional sentences in English are not functions of the truth values of their antecedents and consequents, For example, the sentence ' If Mexico had a king, peaches would grow well in Iceland' is most naturally taken as making a causal claim which is manifestly fa lse; whether or not there is a king of M exico has nOthing whatever to do with the climate of Iceland.
However, !t ning ' M' represent ' Mexico has a king' and 'P' represent 'Peaches grow well in Icela nd', the sentence is symbolized as 'M :) P'. When this formu la is put on the following t ruth table, row 4 of the (ruth table indi(;;w:s tbat the fOrrl\ula is tfue, since as a matter of fact both the antecedent and consequent are fa lse. M P
T T T F F T
F F
M
"
P
T F
T T
It is dear that the re are many conditional sentences in English which can be judged to be true or false o n some basis o ther than the truth values of their ':Hllectdems and consequents. What this means is that the re are many sentences in English in which 'if ... then' does not function as a t ruthfunctional o perator, and this fact has raised serious problems in some areas of logic. However, for purposes of dealing with all the logic cove red in this book, these problems can simply be ignored; the horseshoe can always be used to transla te English 'jf ... then' sentences. The last o perator to be considered is the triplebar. The truth table for the triplebar follows: ~
E~
T T T F F T
T
F F
T
F F
Th, resemblance between the triplebar and the '=' sign is not entirely coincidental; '¢ =0 1jI' is tr ue when '¢' and ''11' have th e same truth values and false whe n '~. and ''P' have different truth values. This is just what one would expect from an understanding o f English phrases such as ' ... if and o nly if .. .' However, it should be kept in m ind that the symbol is defined by the truth table. Truth tables have been introduced to define the five sentence operators. In each case, all the possibilities of truth and falsity have been considered.
29
S2.tS: TRUTHTABl£S
Since eve ry complete truth table cove rs all possibili ties, there is a row on every comple te truth table which corresponds to the actual world. For purposes of doing logic, however, there is no need to determine which row that is. All the truth tables discussed so far have dealt with fo rmu las having a single operator. The next lOpic lO be considered is rhe devdopment of truth tables (or more comple x formulas.
S2.1S: TRUTH TABLES One ca n use the simple truth tables for the five sentence operators to calculate the truth values of complicated fo rmulas such as ' - (R " G)'. If borh 'R ' and 'G ' arc true, then 'R " G' is true . If 'R " G' is true, then '- (R " G)' must be false . Similarly, if 'R " G' is false , then '- (R " G)' must be true. In developing the [ruth table for ' - (R " G )', o ne has [ 0 determine the truth value of 'R " G' before one can determine the truth value of '- {R " Cr, because the '-' is the main operator of the formula. Truth tables should be developed on a column-br-column basis, with the column under the main opera lOr being done last. An example foll o ws:
G R T T T F F T F F
- (R "
F
T
T T T
F F F
G)
2
Three conventions are observed in this truth table. In assigning the different truth values for 'R' and 'G ' in the fi rst two columns, the sente nce letters are arranged in a lphabetical sequence. In principle, this sequence does not matte r, but in practice the initial columns for the sentence letters should always be arranged a l pha~tically. This convention is almost universally observed, and setting out one's sentence letters in any other way creates unnecessary work for anyone checking the truth table. 2 Still concerning the initial columns for the sentence Iwers, the right column (under'R' ) has alternate 'T's and ' F's, while the column immediately to the left of that one (under 'G') has tw O 'T's followed by two 'F's. If there were three columns, the column on the extreme left would have four 'T's followed by four ' P s. Th is pattern of arranging tile initial columns for th e sentence leners when there are three or more sentence letters is exemplified later in this section. There are other
30
S2.tS: TRUTH TABLES
arrangements o f 'T's and ' F's which wou ld ensure that all the possible combinations are included, but this is the most common way of setting Out a trut h table, and it should always be used. 3 The numbe r '1' is PUt under the column fo r the main operator of the formula. Different conventions are used [Q indicate w hich ope rato r is the main operator, but in this book the columns will be numbe red in the reverse o rder to that in which they are completed. R A C ' is a com pletdy different formu la from '- (R " Cl', and thi s diffe rence is shown in the difference between the truth table for ' - (R 1\ G )' which has JUSt been given and the truth table for '- R 1\ G' which fo llows: G R
- R
A
T T F F
T
F
F
T F T
F T F
T F
G
F
2 The uuth table for a more complicated formula follo ws: C D E
- Ie v D) :> ID - - E)
T T T T T F T F T T F F F T T F T F F F T F F F
F F F F F
F T T 4
T T T T T T F F 5
T
F F
T T T T T T
TT TF FT F F
F
TT TF FT
23
The truth ta ble was begun by setting up the initial columns in alphabetica l o rder fo r each of the distinct sentence letters which occur in the formula. In this case there were three letters, so eight rows were required on the truth table. In all cases, the number of rows on a truth table is a function of the number o f distinct sentence letters. If there is one sentence lette r, twO rows a re required; if there arc twO se ntence letters, four rows are requi red; if there a re three sentence letters, eight rows are requ ired; if there are four sentence letters, sixteen rows are required, and so on. As the number of rows increases geometrically, [ruth tables rapidly become so complex that they 3r c unma nageable. In principle, however, there is no limit to the nu m ber of rows o n a truth table.
52.15; TRUTH TABLES
31
The next step was [Q fill in the columns under the se ntence operators. T he horseshoe is the main opera lOr, so the column under it had [Q be compl eted last. The last colu mn to be completed is always colu mn 1. The first column to be completed was the column under the vel. T he truth values fo r this column were determined from the truth values of 'C' and ' D' which appear at the left side of the truth table , and the truth-table definition of 'v', There are five (;o lu!llIlS to be completed, so thi s CulUlHl1 waS labelled '5 '. One cou ld have begun by completing the column under the second '-' and determining the truth values for this co lumn by reference lO the tTUth values o f 'E' and the truth-table definitiun uf '-', There is nuthing to (;hoose between these d ifferent ways of be ~n ning the truth table, However, one could not have begun with the column under the 'E', since the truth values for th is column are determined by reference to the truth values of ' D' and the truth values of ' - E', Plainly, one must have completed the column for '- E' before onc cou ld complete the col umn for ' D E - E', Similarly, onc could not have completed the column for '- (e v Dr before one had completed the column for 'C v D', The final colum n, o f course, cou ld nm have been completed before the column for ' - (C v D)' and the column for ' D ;;; - E' were completed, One more exa mple should suffice to make t hese points clear. Since '( L ::> - - N) 1\ (M v (0 :::J L))' has fou r distinct sentence Iwe rs, [he tfu th table fOf [his fo rmula has sixteen rows; L M N 0
T T T T T T T
T T T T F F F
T T F F T T F
T F T F T F T
T F F F F F F F
F T T T T F F F
F T T F F T T F
F T F T F T F T
F F F F
(L
"
-
- N) "-
T T F F T T F
T T F F T T F
F F T T F F T
F T T T T T T T
F T T F F T T F
1M
v
(0
"
T F F T T F F T
T T F F T T F F T T T T F T F
T T T T T T T T T T T T F T F
T T T T T T T T F T F T F T F
T F T
T
T
T 3
4 56
2
L ))
32
S2. 17: T'YPESOF F ORMULAS
As in the previous example, the re is some choice in t he order in wh ic h the col umns arc com pleted. One cou ld have begun with the colum n fo r '0 ~ L', for example. Since [he truth value of '- - N ' is always the same as the truth value of ' N ', one might be tempted to skip column 6. Howeve r, a complete truth table requires that there be a complete column fo r every
occurre nce of an operaWf. Accordingly, the column for' - N' should be included; the truth tabl e would be incomplete if column 6 were missing. Producing t ruth ta bles is a co mpletely mecha nica l procedure. Once one has learned the [futh ta bles for the sentence operators and und erstands the way in which the positi on of the brackets determi nes what the operatOrs do. the whole business tends to become boring. H oweve r, some inte res ting things can be shown with tru th tabl es, and it is worth taking the t ime to complete t hem carefully.
S2.16: EXERCISES Produce a complete truth ta ble for each of the following formu las, number ing each column as was done in S2.15.
- 1M v N) 2 - - - IF . G) 3 5
(V v K )
:J
4 (5
V)
:J
K
v
5 - IP ::> - - P) 6 (5 7 S
R)
1\ 1\
(R
1\ 1\
101M
:J 1\
G)
N)
Q)
-
8 - (I F . G) 9 (R
Q
-
IG ::> H I!
A
:J
5
1\
-
(0 v K)
S2.17:
TIPES OF FORMULAS
When one produces a truth table for certain formulas, the fi nal column consists of nothing but 'T's. Such formulas are Iruth-functionally Irue. ' M v - M ' is an example o f a truth-fu nctionally true formu la:
M
M v - M
T F
T F T T I 2
33
52.17: TYPES OF FORMULAS
Another example of a truth-functionally true formula is '- (K K
- (K
A
T F
T T
F F F T 2 3
-
1\
-
K)':
K)
The mark of a truth-functionally true fo rmula is that it is true no matter what truth values are assigned to the atomic formulas o f which it is made up. No matter what sentence is represented by ' K', one can be certain of the truth of ' - (K 1\ - K)'. Whenever the final column of a truth table for a given formula consists of nothing but 'T's, that formula is truth-functionally truc. The final columns of the truth tables for some formu las consist of nothing but ' F's. Such formulas are uuth-funetionally false . 'R " - Rt is an example of a truth-functionally false formula: R
R ,,-R
T F
F F F T 1 2
'- (S v - S)' is another example of a formula which is truth-functionally false:
s
-
(S v - S)
T
F
F
F
TF TT 2 3
A truth-functionally false form ula is always false regardless of what truth values are assigned w the awmic formulas of which it is made up. No matter what sentence is represented by'S', one can be certai n that '- (S v - S)' is false. Whenever the final column of a truth table for a given formula consists of nothing but 'F's, that formula is truth-functionally false. The second example of a truth-functionally [rue formu la, '- (K 1\ - Kl', is t he negation of a truth -functionally fa lse formula, 'K 1\ - K'. Simi larly, the second example of a truth-functionally false form ula , '- (5 v - Sr, is the negation of a truth-functionally true formu la, 'S v - St, In all cases, the negation of a truth-functi onal truth is truth-functionally false and the negation of a truth-functional falsehood is truth-functionally true, Between the twO extremes of truth-functional truth and truth-functional falsehood arc formula s whose truth values arc dependent upon the truth values of the atomic formulas of which they arc made up. Such formulas are
34
S2.17: TYPES OF FORMULAS
trmh-funcrionall y indeterminate. They can be recognized by the appearance of at lcast onc 'T' and at least onc 'F' in the final columns of their truth tables. The mark of a truth-functionally indeterminate fo rmula is that its final column includes at least o ne 'T' and at least onc 'F', The fina l column of the following tfuth table has this feature. Accordingl y, '(P :::) Q) v - P' is truth-functionally indeterminate .
p Q
IP =>
T T T F F T
T
T F
F
F F
T
F F
T 3
TT TT 1 2
Q)
v - P
T he fina l column of the next truth table a lso includes at least one 'T' and at least one 'F', Thus, '( K 1\ L) :J (- L v R)" tOO, is truth-functionally indeterminate: K
L
R
(K
A
T T T
T
T F F T F F T T F T F F F T F F F
T
T T T F
F
F F F
F F 4
L) => (- L v R)
T F F F T T T T T F T F
T F
T T T
T
F T
T T
T
3
2
T
S2.18:
J5
TRUTH-FUNCfIQSAL EQUIVALENCES
51.18: TRUTH· FUNCTIONAL EQUIVALENCES Any twO formu las of sentence logic, '
MI
A
-
::> (-
K
A
::> -
GI
A
6 (( F
(L
::>
BII
-
(F
::>
MI A
(-
HII
A
K
::>
SI
(G v - SI
Sl .19 Part 2 Produce a single compl ete truth table fo r each of the following pairs of formulas. Use the truth table to determi ne whether or not the formu las in each pa ir are truth-functionally equiva lent.
(F
::>
GI
A
(F
::>
HI
F :::J (G " H )
2 (F " G) :::J H
(F
::>
3 K v
HI L
A
(G
::>
HI
K v (- K " L) R :::J (S :::J R) 4
S
::>
(R
::>
5 M :::JN - N:::J (M
SI ~
N)
6 R ::> (S v LI
(R
::>
SI v (R
::>
LI
S2.20: ARGUMENTS The term 'a rgument', as used in logic, does not refer to disputes, fights o r a nything of that SOrt. An argument is simply a collection of statements. The only arguments to be considered in this book are deductive - arguments in which the truth of some statements of the collection, the premisses, is supposed to gua rantee the truth o f another statement of the collection, the conclusion. If the truth of the premisses does guarantee the (ru th of the concl usion, the argument is valid. If the (ruth o f the premisses fail s to gua rantee the truth of the conclusion, the argument is invalid, There are no
40
52.20: ARGUME""-S
d«:grces of va lidity; any deductive argument which is not valid is inva lid and worthless. In writing arguments, the universal praClice is (Q list t he premisses, draw a line under them and write the conclusion, much in the way that sums in arithmetic are wriuen. Two examples follow: A A v B
s s =>
R
R Valid arguments are truth-pres((ving. That is to say, if all tht premisses arc true, the conclusion must be tme. To say that an argument is va lid is not to say that the premisses are in fact true or that the cond usion is in fact truc; it is only to say that there canno t be a situation in which the conclusion is fa lse while all the premisses arc true .
To determine the truth-funccional validity of an argument, onc should put all the premisses and the co ncl usion on a si ngle truth table. If there is a row on which all the premisses are true and the conclusion is (alse, the argument is truth-functionally invalid. If there is no suc h row, the argument is truth· functionally valid. It should be dear that the trurh· functional invalidity o f an a rgument can be shown by refe rence to a single rowan a truth table, whilc the truth-functional validi ty of an argumcm ca n be shown on ly by checking every row. The second example of an argument given in th is section was:
S S=> R R
T he four rows of the truth table for this argument cove r aU possibilities. S
S
S => R
R
T T T F F T
T
T
F F
F
T T F T
R
F
T
T F F
The only rowan which both premisses arc (rue is row 1. O n this row, the conclusi on is true. T hus, the argument is truth-functionally va lid.
51.20:
41
ARGUME ]\.'TS
Another example of an argument follows: - K
K ::> L - L The (ruth (able gives the truth values for both premisses and the conclusion:
K L
K::>L
- K
-L
T T
F
T
F
T
F
F
F
T
F T F F
T T
T
F
T
T
On row 3 both premisses are true while the conclusion is false. The argument, therefore, is truth-functionally inva lid. The first example of an argument given in this section will turn out to be truth·functionally valid: A
Av B The argument is checked by putting the single premiss and the conclusion on a truth table: A
A
B
A
T T
T
T
T
T F
T
F T
F F
F
F
F
v
B
T
The only rows on which the premiss is true are rows 1 and 2. On these rows the conclusion is also true, so the argument is truth-functionaHy va lid. A more complicated argument is tested next:
J K
K ::> (L (L v
v -
J) A
K
J)
42
§l.20: ARGUMENTS
As before , [he argument is checked by puuing the premisses and the conclusion on a truth table:
J
K L
T T T T
T F T F T F T F
T T F F F T F T F F F F
J
K
T T T T F
T T F F T T F F
F
F F
K ::> IL v
T F T T T T T T
- JI
TF F F T F F F TT TT TT TT 2 3
(L v
T T T T T F T F 2
J) "
K
T T F F T F F F
Row 1 is the only row a n which all the premisses are true. On this row the conclusion is also tfu e, so the argument is truth-functionally va lid. Any a rgument with a truth-functionally false premiss is truthfunc tionally valid because it is nOt possible for all the premisses to be true while the conclusion is fa lse. The fo llowing argument will be shown to be (nuh-functionally valid on a truth table. notwithstanding the fact tha t the conclusion appears [Q be completely unrelated to the premiss: A " - A
Q A Q T T F F
T F T F
A
A
- A
Q
F F F F J
F F T T
T F T F
2
Any argument with a truth-functionally true conclusion is truth-functionally valid, no matter what premisses it has. Since the conclusion cannot be false, it is clear that it cannot be the case that the premisses are true while the concl usion is false. The following argument whic h has a tr uth-functionally uue conclusion is shown to be truth-functionally valid on a tru th table:
43
S2. 21 : EXERCISES
Q A v - A A Q
Q
A v - A
T T F F
T F T F
T F T F TT TT I 2
T F T F
T he [nnh -func tional va lidity of any argu ment whatever in sente nce logic can be checked on a tr uth table. It does not matter whether th e prem isses seem to be unrc:lated to the conclusions, as is the case with t he twO arguments JUSt discussed. Simila rl y, it does not matter how complicate d the argument is. As long as there is no row of the tru th ta ble on which all t he premisses are true a nd the conclusion is fa lse, the argument is truth- functiona ll y valid.
Sl.2l: EXERCISES For each of t he following arguments, use a truth table to determi ne whether or not the argument is truth· fu nctiona ll y valid . F ::> G F
F AG
2 - (- K ::> - L) K ::> L
3 P= - Q Qv P
- P 4 M
~
(N
~
0)
M:::JN M ::JO 5 (R :::J 5)
v
- N
~
5
- R
~
(R
(5 ::J N) 1\
5)
44
§ 2.21: EXERCISES
6 (P v QJ :=l
-
R
P= - Q Q=>P 7 R R
=S ::J
(S v T)
8L =>- K
- '"
K::J ( l. v
~)
- K 9 F::>G F v - G
10 F ::::> - G F ::J C
- F 11 (8
1\
C) v (B " D )
B 12 (P => W ) v (P " - W )
p 13 K
v
L
L =J - K "
- J
J
v - L
14 (R v S) => T T ::J (R 1\ S) (R v 5)
:J
IS- KAV K V - M T::::> - V
M " - T
(R " S)
45
S2.22: TRANSLATING ARGUMENTS
§2.22: lRANSLATlNG ARGUMENTS Translating arguments from English into the symbolic language can often be difficult, but the essential principles are straightforward. A sentence lener should be assigned to every Engl ish sentence or clause which is capable of having a truth value, the meaning of each sentence lette r being stated clearly. The appropriate symbols should replace English expressions such as 'and', 'or' and 'if ... then'. The argument should then be written out, with the premisses foll owed by a line and the conclusion. The importance of using as many simple sentences as possible can be seen from consideration o f the following argument: If the coolant lea ks, the oil overheats and the o il pressure d rops. If the o il pressure drops, the bearings may be damaged. Thus, if the coolant leaks, the bearings may be damaged.
The correct way to translate this argument is as follo ws: L: 0: D: B: L
The coolant leaks. The oil overheats. The oil pressure drops. The bearings may be damaged. ~
(0
I\.
D)
D=>B L=>B The truth-functional validity of the argument when translated in this way can easily be shown on a truth table. An incorrect translation of this argument follows: L: The coolant leaks. 0: The oil overheats and the oil pressure drops. D: The oil pressure drops. B: The bearings may be damaged.
ttt
L => 0 D=>B L=>B A truth-table check on the argument translated in this way will show it to be truth-functionally invalid; using a single sentence lener for 'The oil overheats and the oil pressure drops' and another sentence lencr for 'The oil pressure drops' has hidden the logical relationship between these twO sentences.
46
52.22: TRANSLATING ARGUMENTS
A slightly different version of the argument about the coolant and bearing damage follows: If the coolant leaks, the oil overheats and the oil pressure dro ps. If the oil overhea ts and the oil pressure drops, the bearings may be damaged. Thus, if the coolant leaks, the bearings may be damaged. Translated as follows, the argument can be shown to be truth-functionally valid on a truth table: 1.: The coolant leaks.
0: The oil overheats. D: The oil pressure drops. B: The bearings may be damaged . l
:::>
to
1\
(0 1\ 0) Dj :::> B
L => B It can be seen that neither '0' nor '0' occurs except in the conjunction
'0
A
D'. and that therefore the logical role played by '0
1\
D' could be
played by a single sentence Jetter. Accordingly, the argument, for purposes of checki ng truth-funuional validity on a truth table, cou ld as well be translated as; L; The coolant leaks. 0; The oi l overheats and the oil preSsure drops. B; The bearings may be damaged.
ttt
L => 0 O=>B
L => B This translation, although it serves the immediate pu rpose well enough, is presented as an example of what should not be done. There are at least twO reasons why one should not try to translate an argument wi th the minimum complexity which preserves the necessary logical strUClUre. First, it is often unclear JUSt what the necessary logical structure is until the argument has been translated. One advantage of using a symbolic language rather than Englis h is that many aspects of the logical StruclUre of sentences and arguments are often more apparent in the symbolic language than they are in English, and this advantage can be lost when the argument is si mpli fied lx:fore it is translated. Second, there can be reasons for translating an argument othe r than allowing a truth-table check of truth-functional validity. For example, one may want to see whether the argument would still be trmh-functionally valid or uuth-functionally invalid if it were changed in
52.22: TRANSLATING ARGUMENTS
47
certain ways, and a translation which is adequate for a truth-table chec k o f truth-functional validity may not be adequate for this purpose. As a general rule, then, o ne should break up every English sentence inm as many simple sentences as possible. In dealing with a passage of argumentalive English prose, the first step is to determine how many argumentS there are. There arc often several different arguments in one short piece of English, and these arguments must be distinguished before beginning the process of translation. There are no mechanical procedures to follow; what is required is careful attention to the sense of the passage . In some cases, it ma y be unclear how many arguments are being presented, or even what the conclusions o f the arguments are. In all cases, one needs to use a little common sense and the principle of charity. This principle enjoi ns one to put the most reasonable interpre tati on on what an author says. In other words, if a passage can he interpreted in two ways, o ne interpretation being sensible and the other interpretati on being silly or false, it should be interprcred in the former way. One should be careful, however. not to distort the meaning of what is given simply because one thinks that it is ridiculous or false; the principle of charity docs not require the pretence that people never make false statements o r conS[Tuct bad a rguments. It is important to understand that statements in themselves are neither premisses nor conclusions. What makes a statement a premiss or makes it a conclusion is the part it plays in the argument. If a statement serves as a basis for believing anothe r statement, it serves as a premiss. If a statement is offered as being supported by one o r mo re other statements, it serves as a conclusion. In the fo ll owing argument, 'Bill has not studi ed logic' is the conclusion: Alice has not studied logic. Bill studied logic onl y if Alice did. Therefore, Bill has not studied logic. However, the sentence, ' Bill has not studied logic', is a premiss in the nex t argument: Bill has not studied logic. Alice studied logic only if Bill did. Therefore , Alice has not studied logic.
48
S2.22: TRANSLATING ARGUMENTS
Another difficulty in dealing with argumentative English prose is that the conclusion does not always appear at the end of a passag('. In the fo llowing three passages, the same conclusion appears in three different places. The st reets arc not wet. If it were raining, the streets would be wet, Therefore, it is nm raining.
It is not rain ing because the streets are not wet and they would be wet if it were raining. Since the streets are not wet, it is not raining, because the streets would be wet if it were raining, Having identified an argument, one can turn to the business of transladng the premisses and the conclusion. Sometimes, premisses are marked by words sUI.:h as 'bt(';3USC ' , 'since' and 'as'. Similarly, cunclusions are: sometimes marked by words such as 'thus', 'hence' and 'consequently'. Such premiss-indicators and conclusion-indicators should be dropped. They are nut parts of the: pn:rnisses Or F', exactly the same formula as the first premiss. Moreover, this inclusion distorts the sense o f the English. 'Since James is rich, she will marry him' asserts that James is rich and asserts that fr ancesca will marry him; it does not merely assert that her marrying him depends upon his wcahh. Another example should make the distinction clear. One can truly say to anyone, 'If you are the first person [Q walk on the moon, you'll go down in history.' By contrast, there is only one person [Q whom one can truly say, 'Since you are the first person to walk on the moon, yo u'll go down in his[Qry. ' The following is another examp le of correct translation: If I were guilty, I would have been in New York at the ti me of the crime. If that were true, my passport would show that I have emered the United States recently. Since this isn't so, it foll ows that I am not guilty. G: I am guilty. N: I was in New York at the time of the crime. P: My passport shows that I have entered the United States recently.
GcoN N co P
P
G The translation reflec ts the fact that 'If that were true ... ' is [Q be taken as 'If I were in New York at the time of the crime .. .' and' ... this isn't so .,,' is [Q be taken as the denial of 'My passport shows that I have entered the United States recently.' Translating arguments is much like translating single sentences except that one has to consider the ways in which the different sentences are related to each other. Even very complicated arguments can usuall y be translated without serious difficulty. Occasionally, one comes ac ross passages of what seem to be argumentative prose in which there is no recognizable logical
50 --_._---
S2.23: EXERCISES
st ruc ture. Such 'arguments' ca nnot be tra nslated, but they a re usually not wonh serious attention.
52.2 3: EXERCISES Translate the following argu ments into symbols. Provide a dear indication o f the meaning o f each sente nce luter. Separate the conclusions frOf!) the premisses as was done in S2.22.
If the cha irman were a fool, he would have fired [he manager. He is not a fool, since he did not fire the manager. 2 Either the fingerprints arc those of the crim inal or the c rimina l wore gloves. The finge rprints arc not those of the criminal. Therefore, the
criminal wore gloves. 3 T he author is either a charl atan or a foo l. If he is a cha rlata n and I take him serio llsly, I a m gullihlr. . If he is a fool, I am gullible if I rake him seriously. So, if I take him seriously, I am gullible. 4 D R) v (S " - Q))', the outer brackets afe dropped since they ace no longe r necessary. - E can be used only on a fo rmu la whose main ope rator is a '-' which is immediately followe d by another '-'. The following is incorrect:
P I , -E
Line 2 is mistaken because the main operator of the formula on line 1 is a caret, and nOi a tilde immedia tely followed by another ti lde. To obtain 'F 1\ C' from ' F 1\ - - G ', one needs to obtai n ' F' and 'G' in order to use " I. 'F' can be obtained simply by I\E. To obtain 'G', one needs to get '- - C' and use -E. O ne can then use 1\1 to obtain ' F 1\ C'. T he derivation follows: F I\--G
2 F 3
--G
4 G 5
FAG
p I, "E I , 1\£
3,-E 2, 4, 1\(
S3.5:
62
DISJU:-..:crION INTRODUCTION
Two and only two consecut ive (ildes can be removed at onc (ime. In the event that onc needs to remove fou r tildes, one needs to apply -E twice, as in the following de rivation:
1
2 3
--- G
-
P
-G
1, -E
2,-E
C;
-E is an easy rule to understand and apply. There arc twO important points [0 remember. First, two and only two tildes can be removed by a single application o f rhis rule. Second, the first of the two consecutive tild es which
arc removed must be the main operator of the formula on which -E is applied.
§3 .S, DISJUNCTION INTRODUCTION If onc knows that Peter is the captain of Ihe team, then onc is entitled to
make the weaker da im that ei ther Pete r is the captain of the team or John is the captain of the tcam. Given any sentence '¢l', one can infer '$ v IV'. The rule of disjunction introduction, 'vI', reflects the fact that a disjunctive fo rmula can never be Stronger - ca n ne ver say mo re - than either of its disj uncts,
v1 One can write the formula '(11) v 1jI)' on any give n scope line, provided ,hat either one of the following conditions is met: J The fo rmula 'q,' is already on the same scope line .
2 T he formula 'w' is already on the same scope line, T he justification fo r the new line consists o f an ap peal to the number of , he line where ' ¢' or '1jI' occu rs and the rule, 'vi', The formal statement of the rule includes Outer brackets in t he fo rmula, ahhough this requirement is ignored in what follows. This argument is t ruth-functionally valid: M (N v M ) v 0
T he derivation of the conclusion fro m the premiss consists of th ree steps: M
2 Nv,\I 3
(N v M ) v 0
P I, vi 2, v i
§ 3.5: DISJUNCTION
63
I ~'TRODUCT I O:\"
The vel which results from using v I must be the main operator of the new formula. v I ca n be applied only to an entire formula, not to a part of a formula. Although ' R :::::I (S v T )' can be derived from ' R ;:) S', it cannot be derived juSt by applying v I. (A way of deriving the former from the laner will be given before the end of the cha pter.) The following attempt (0 derive 'R ;:) (S v T V from 'R ::> S' is incorrect:
II
R ::>5 2 R::> 15
P v
TI ttt
1, vi
The next derivation, ho wever, is comple tely in o rder, because v I is applied to an entire formula:
II
P 1, v i
R ::>5 2 IR::> 51 v T
The twO following special cases should be noted. Both derivations are correct:
21~ I 2
P v A
I, v i
P
IA
I, v i
A v - A
In using v I, one need not introduce only alOmic form ulas, and the rule can be used repeatedly to crea te formulas of increasing complexity. All of thc uses of v I in th e next derivation are correct:
2
F
P
(G" H) v F
I, vI
3 IP ::> QI v IIG A HI v FI 4 li P ::> QI v II G A HI v FII v IA
= BI
2, vI 3, vi
At this point, it may not seem that v i is a ver y usefu l rule, and indeed it is true that one rarely wants lO derive a conclusion which is sim ply a disjunction having a premiss as one of its disjuncts. However, the rule will be extremely useful in the course of complex derivations. It is important in applyi ng the rule that vi introduces the main operator o f the new formula. Care must be taken that v I is applied only lO an enti re fo rmula and not to a pan of a form ula , and that the formula which results from applying v I is well formed.
64
S3.6: H ORSESHOE ELIMINATION
SJ.6: HO RS ES HOE ELIMI NATION If one knows that Sam will be elected if he runs and one knows that Sam will run, one certainly knows tha t Sam will be elected. Given any two sentences '4>' and 'Il> =:. W', o ne can infer ' \If'. The rule o f horseshoe elimination , '=:JE', is as intuitively obvious as the inference conce rning Sam's elenion. :oE
One can write the formula 'w' on any given scope line, provided that both of the followi ng conditions are met; 1 Th e fo rmula '$' is already on rhe same scope line.
2 Th e formula '($
~ \jI )'
is already on the same scope li ne.
The JUSlificatio n for the new line consists of an appeal to the number of the line where ' ~' occu rs, the numbe r of the line w here ' (¢ :J 1jI)'
occurs and the rule, ':JE', The forma l statement o f the rulc: includes outer bracken in the formula, altho ugh this requirement is igno red in what foll ows. T he followi ng derivation illustrates some uses of :JE:
r 2
G
J
F
4
G :o R
5
R
:0
(G
:0
R)
p p p 1,3, ::>E 2, 4, ::>E
On e very commo n mistake in using ::>E follows:
I
V
2
F ::> V
J
F
ttt
p p 1, 2, ::>E
The mistake is obvious in English. From the fact tha t I would be old enough to vote if I were forty and the fact that I am o ld enough to vote, it does not fo llow that I am fo n y. In order to use ::>E, one must have a for mula whose main operator is a horseshoe. T he followi ng argument is trU[h-functionall y valid: N - - (N ::> R )
R
65
53.6: HORSESHOE ELlM I~ATlON
However, the following attempt to derive the conclusion from the premisses is illegitimate: N 2
J
- - IN ::> R) R ttt
p p 1, 2, ::>E
The move at li ne 3 is wrong because the main operator of the formula o n line 2 is not a horseshoe. The following is the com~c [ derivati on of the conclusion from the premisses: 1 N
p
2 3
- - IN ::> R) N::::> R
P 2,-E
4
R
l,3 ,::>E
When :::>E is applied to complex formulas, care must be taken to ensure that the rule is applied to the main operator of the fo rmula in question. The following twO derivations are completely in o rder:
A B J C 4 A::> IB ::> IC ::> DII 5 B ::> IC ::> D) 6 C::> D 7 D 2
P P P P 1, 4, ::>E 2, 5, ::>E 3,6, ::>E
4
R K ::> (R :::> S) R ::> K IR ::> S) ::> 11M A N) • 0)
P P P P
5
K
6 7
R ::> S 1M A N) = 0
1,3,::>E 2,5, ::>E 4,6, ::>E
1
2
J
The rule of ::>E is easy to remember and to apply. Care must be taken to ensure that one has both a formula whose main operator is a horseshoe and the antc:cedem of that formula. The consequent of that fo rmula is the result of applying ::>E.
66
S3.7: TlUPLEBAR ELiMINA.TlON
53.7: 1RIPLEBAR ELIMINATION Suppose that one knows that stock market averages will rise if and only if interest rates fall. Then, if onc knows that stock market ave rages will rise, one knows that interest rates will fall. Likewise, given this background information, if one knows that inte rest rates will fall, onc knows th at stock market averages will rise. Given any twO sentences '$' and ' ¢I =: 'V', onc can infer '\V'. In just the same way, given any twO sentences 'lV' and '41 == 'V', one can infer '¢I' . The rule of triplebar elimination, 'sE', is as intuitively clear as the twO inferences concerning interest rates and stock ma rket averages.
One ca n write the formula 'III' on any given scope line, provided that both of the following conditions are met: 1 The fo rmula '11>' is already on the same scope line. 2 The formula '(ill :: W)' is already on the same scope line. The justification for the new line consists of an ap peal to the numbe r of the line where '$' occurs, the number of the line where '(ill '" W)' occurs and the rule. ':!;;E'. Similarly, one can write the formu la '$' on any given scope line, provided that both of the following conditions are met: 1 The fo rm ula
'\jI'
1 The formula '(41
is already on the same scope line. !i1
14/)' is already on the same scope line.
The juStificatio n for the new line consists of an appeal to the number of the line where '\II' occurs, the number of the line where '(41 == 'V)' occurs and the rule, '==E'. Th e formal statement of the rule incl udes outer brackets in the formula, although this requirement is ignored in what follows. The following twO derivations illustrate the rule:
K 2 K. R 3 R
P P
R 2 K·R 3 K
P
I
1,2, ==E
P 1,2,==E
All the uses of ==E in the following derivation are correct:
S3.8: I
67
EXERCISES
P P P P
F
2 G 3 R=F 4
5 6 7
R = IG" H I R
1,3, :=:E 4,5,:=:E 2,6,:=:E
G=H H
Care must be taken to ensure that =E is applied o nly to the ma in operator of a formula. Thus, the following is mistaken: I
P p
M
2 1M. NI · 0 3 N·O ttt
1,2, :=:E
It is easy to avoid mistakes in using =E if one takes can· to apply the rule only to the triple bar which is the main operator of a fo rmula.
S3.8:
EXERCISES
Each of the foll owing deri vations has severa l ju stifica tions missing. Suppl y the justifications, giving the name of the rule and the line numbers which justify the use of [he rule. Question 1
p p
A 2
3 4
5
- - G G A AG (A 1\ G) v - G Question 2
2
R ·S Q
3 IQ v PI => 1M A NI 4
5 6 7
N=>S Q v P M A N N
8 s 9 R
P P P P
S3.9;
68
CO:-lSTRUCfING DERIVATrOM
Question 3
I
P P
A
2 R 3 A
::> I R
p
= B)
4
B ::> IIR ::> Q )
5
R= B
A
IB ::> C))
P
6 B 7 8 9 10
II 12 13
I R ::> Q) R ::> Q S::> C
A
IB ::> C)
v
IF
Q C Q (Q
A 1\
C
q
1\
l)
53.9: CONSTRUCTING DERIVATIONS Fo llowing a given derivation and seeing how each step is justified is simple once onc has learned the rules. Creating a derivation from scratch is a very diffe rent thing. No onc becomes adept at producing derivations without much practice and without making many mistakes. However, unless one has some strategy in mind, it is impossible to begin to practise constructing derivations and to learn fro m onc 's mistakes; nothing is learned by staring at the page in complete bewilderment. Those who are very good at constructing derivations seem to make the right move at every poim, but to someone lea rning there seems to be no way of deciding ahead of time what the right move is. One can, of course, just apply a rule to a given formula and hope that something useful tu rns up, but this is hard ly a rational procedure. In this section a strategy will be applied and discussed in dealing with various examples. Ign oring 'categorical derivations ' (which will be discussed at the end of this chapter), the problem in a derivation is to get from the premiss o r premisses to the conclusion by a series of legitimate moves. The problem is ana logous to the problem o f getting fr om one place to another; there is no point in deciding what one's fim step is going to be until one has decided where one is trying to go and how one is trying to get there. For exa mple, in travelling from Chicago to Rome , one can fly to New York and take a direct flight fro m there. One can also fly to Paris and continue on another flight from Paris to Rome. In addition, one can take a circuitous route, going from Chicago to Tokyo and then from To kyo to Rome. No rational decision
69
§3.9: CONSTRUcrlNG D£RIVATIOr-:S
about the first step of a journey is possible withou t considerati on of the othe r steps of the journey. In constructing derivations, as in travelling, there are usually many options. Sometimes some of these options are obv iousl y absurd, and sometimes one option looks to be far more promising than any other. The following argument is truth-functionally valid, as a check o n a truth table will show. The concl usion can be derived from the premisses, using only the rules discussed (hus far in this chapter.
Q C= D Q::>--D
C The problem is to construct a derivation which begins with the premisses and ends with the conclusion. How many lines long the derivation will be is not known, so one can start the derivation by numbering the premiss lines and calling the last line 'Z' as follows: 1 Q
p
2 C= D 3 ,Q ::> - - D
P p
z ic The question is: how can one get 'C'? The only line on which 'C' appears is line 2, where it is in 'C !!O D'. If one had '0', one could use sE to get 'C'. Thus, there is a new problem of getting '0'. To keep track of what one is doing, the outline of the derivation is changed, with the addition of line Y and a justification for line Z: p
Q 2
C
ii
0
3 Q::>--D Y Z
I' D C
p p
2, Y, sE
'0', of course, is avai lable at line 2, but to get it from'C = D' would require one to get 'C', which is what one is trying to do by getting '0'. To return to the travelling analogy, getting '0' from line 2 when one is trying to get 'C' from line 2 is like trying to get to Rome in order to take a fl ight from Rome to Naples so as to take a flight from Naples to Rome. As an example of a usdess strategy, an attempt to obtain '0' from 'C ~ D' follows:
70 1
\3.9: CONSTRUCfING DERIVATIONS
Q
2 C = I) 3 , Q:J--D
w
p p p
D
x c
2, W, :;:E
Y I) Z C
2,X,EE
2,
Y ,~E
If one could obtain ' 0 ' at line W, one would have 'C' at line X. Lines Yan d Z wo uld then be unnecessa ry. Obviously, trying to get '0' by getting 'C' and usi ng line 2 is poindess. One must go back and consider another slrategy. The only other formula in which ' 0' appears is 'Q :J - - D' on line 3. If one could obtain' - - D', it would be easy to obtain '0' by -E. This looks fa r more promising:
1 Q
p
2 C s l) 3 , Q :J - - I)
P p
I)
X Y
X,-E
I)
Z C
2, Y, s E
'- - 0' can be obtai ned by =:.E from 'Q :J D' provided one has 'Q'. a nd 'Q ' is available at line 1. Accordingly, the derivation can be completed. The lines ca n now be properly numbered and the justificatio ns can refer to lines by their numbers:
2 3
Q
p
C.I)
P P
Q :J- -I)
4 5
I)
6
C
D
1,3, ::::IE
4,-E 2, 5, =E
The following argument is somewhat more complex, and the derivation will be longer. However, the complexity is no cause fo r alarm. At any point in the derivation, there is o nly one immediate problem: there are certain fo rm ulas available and another formula is needed. To re[Urn to the travel analogy, at any point in a journey there is only one immediate problem: gening to one's next destination. That this leg of the journey is only one leg among ma ny need not and should nO[ prevent o ne from concentrating on
71
S3.9: CONSTRUCfING DERIVATIONS
the immediate problem. Similarly, in co nsuucting derivations, one should concemrate on one problem at a time. Doing this will ensure that one is not complc:tely baffled merely by the fact that a Straightforward problem requires many steps for its solution. A F ::::> (R 1\ G) (A 1\ M ) ::::> F A =, --M (A A (R v 8)) ::> K K
Th e fitS( step in producing the derivation is straightforward. As in the previous example. o ne lists the premisses and notes the concl usi on:
A 2
F ::l(R I\G)
3 (A I\ M )::> F 4 A ::::>-- M 5 , (A 1\ (R v B))
::::>
K
P P P P P
Z iK 'K ' co uld be obtained from '(A 1\ (R v 8 » ::::> K' by ::::>E if one had 'A 1\ (R v 8 )'. H ence, a sensible strategy wo uld be [Q try [Q obtain 'A 1\ (R v B)'. The sketch of the derivation is developed, with a justification for line Z: A F ::> (R
2 3 (A 4
1\
p A
G)
M) ::::> F
A ::::> - - M
5 , (A
Y iA Z K
A
1\
(R v 8)) ::> K
P P P P
(R v B) 5, Y,::::>E
There is now a new problem , the problem of obtaining ' A 1\ (R v B)'. This formula appears in '(A 1\ (R v B)) :> K', but since it is on the left side of a horseshoe, one cannot use ::::>E to derive it. Since o ne cannot use an elimination rule to derive 'A 1\ (R v B)' from an available formula, one must consider an introduction rule. The main operator o f 'A 1\ (R v BJ' is '1\', so the introduction rule mUSt be 1\1. To obta in ' A 1\ (R v B)' by 1\1, one
S3.9:
72
CONSTRUCTING DERIVATIONS
n ~e ds
' A' and 'R v B'. 'A' is one of the premisses; the only problem at this point is obtai ning ' R v B'. The new sketch of the derivation follows;
1 A
P
2 F ::> (R )
GI (A " M) :;, F
4
A::J--M
5 . (A
1\
(R
X y
R v B
Z
K
A
A
p
A
v
Hl )
::J
K
(R v BI
P P P
I, X, AI 5, Y,::JE
'R v B' appears at line 5. If one could obtain' A " (R v Sl', one could use " E to obtain 'R v H'. However, 'A " (N. v H)' is the very formula one is trying (0 obtai n at line Y. Trying (0 obtain 'A 1\ (R v B), in o rder to obtain ' R v S' so (hat one can obtain' A 1\ (R v B)' is plainly a poi ntless strategy. Accordingly, one is again forced to use an introduction rule. To obtain ' R v S' by v i one needs either 'R' or 'B', There is no apparent way to get 'B', but one could use AE to get ' R' from 'R " G' which appears as part of a formula on line 2. Taking things one step at a time, the sketch of the derivatio n is changed: A
P
2 F ::> (R )
(A
4
A::>--M
5 , (A
1\
1\
(R v B))
W
R
X y
R v B
Z
K
A
1\
p
GI M) ::> F A
(R v B)
::>K
P P P
W, v i
1, X,
1\1
5, Y,::>E
S3.9;
73
CONSTRUCTING D ER IVATIOS'S
The strategy IS to get ' R' from ' R ,.. G' by AE. Accordingly, the new problem is to obtain ' R A G'; I
A
2 F => (R A G) 3 (A A M) :::J F 4
A
.\
(A A (R v B) ) => K
:::J -
-
M
P P P P p
V R A G W
R
X R
v B
y
A A (R
z
K
V
B)
V,AE W,vI 1,X, AI 5, Y, :::) E
'R A G ' could be obtained by :::)E from ' F :::) (R Accordingly, the next problem is to obtain ' F';
A
A
P
F :::){ RAG )
p
3 (A A M) => F
P P P
2 4
A :::) - - M
5
(A A (R v B)) => K
U
F
V
R A G
G)' if one had T .
W
R
X
R v B
2, U, :::)E V,,..E W,vI
y
A A (R v B)
I,X,AI
z
K
5, Y,:::)E
It should be noted that as each new problem is introduced the juslificatio n fo r a subsequent line is provided. Although the deriva tion is now look ing quite complicated. there is still only one Question mark at anyone ti me there is only one problem to be solved at anyone time. Returning to the derivation, the currene problem is obtaining 'F'. This formula appears on line 2, but since it is on the left side of a horseshoe. it cannot be obtained by :::)E. H owever, 'F' could be obtained by :::)E from '( A ,.. M) :::) F' which a ppears o n line 3. Thus. the new problem is obtaining ' A A M':
S3.9, CONSTRUCllNC DERIVATIOt\S
74
2
A F oo (R
J\
P P
G)
3 (A " M ) ::J F 4 A ::J-- M 5
(A " ( R v 8 »::J K
T
A
U
F R AG
V
AM
y
R R v 8 A A IR
Z
K
W
X
P P P
3, T,::lE2, U,:>E
V, AE W, v I V
8)
1,X,AI 5, Y,::JE
' A A \1 ' cannor hr. ohtainr:d oy an elimination rille, so one needs to lise the appropriate imroduClion rule, AI. To obtain ' A A M' by " I, one needs bOlh 'A' and ' M ', 'N is a premi ss o n line 1; the problem now is to obtain ' M '. The
revised outline o ( the derivation shows the new problem: I
P P P P P
A
2 F ::J( RAC) 3
(A " M)::J F
4
A ::::) --M
5
(A " (R
v
B)l => K
S M T A AM U
F
V
R AG R R v B
W X y
z
A
1\
1, S, " I 3, T, ::JE
2,U,:>E
V, AE W, v i
(R v B)
l,X,AI 5, Y,:>E
K
'M' appears on line 3 in the formula '(A
A
M ) ::> F', but since ' A " M' is
on the left side of the horseshoe and is thus not accessible, one cannOt use
" E to get ' M' from ' A " M' , Moreover. ' A 1\ M' is the vc:ry formula one is trying to obtain at line T. However, 'M' also appears in the formula 'A ::J - - M' on line 4. If one could obtain '- - M', it wo uld be easy to obtain ' M' by -E. The new problem, therefore. is to obtain '- - M', and the derivation is revised to show this:
7S
53.9: CONSTRUCfING DERIVATIONS
1 A
2 F ~ (R 1\ G) 3 IA A M ) => F 4
A
S (A R
~ 1\
- - M (R v 8))
~
K
P P P P P
- - M R, ~E
S M T U
A AM F
3, T, ::JE
V
RAG
2, U, ::JE
I , S, 1\1
W R X R v B
V,I\E
y
A
I , X, 1\1
Z
K
1\
W,vl
(R v B)
5, Y,::JE
T he solution to this problem is easy. - - M ' appears on line 4 in the furmu la 'A ::J - - M '. To ubtaiII '- - M', um: lIeeds 'N, and 'N is a premiss on line I. Acco rdingly, the derivation can be completed, with the
tempora ry alphabetic line na mes being replaced by numbers and [he justifications being appropriate ly amended:
1 2 3 4 S 6 7
A
F ::J (R (A A
{A ~
1\
~ 1\ ~
1\
G)
M) ::J F - - M
(R M
v B)) => K
M 8 A 1\ M 9 F 10 RAG 11 R 12 R v B 13 A 1\ (R v B) 14 K
P P P P P 1, 4 ,::JE 6,~E
1, 7,
1\ )
3, 8, ::JE 2, 9, ::JE 10, I\E
11 , vi 1, 12, 1\1
5, 13, :JE
76
S3.9:
CONSTRUcnNG DERIVATIONS
In the twO derivations which have been discussed in this section, whenever formu las we re needed, one o f the fo rmulas was immediately ava ilable. Th e result was that there was never more tha n one formula wh ich one needed to look fo r at anyone ti me. Someti mes things a re a little mo re co mplicated, but the gene ral str.uegy of dealing with onc prob lem at a time remains the only sensible a pproach. This is demonst rated in so h-jng the problem o f deriving the conclusi o n o f the foll o wing argument fro m the premisses: tw O
R R ::::> - - Q Q :0 F
R :oG IF " (j)
::J
M
M T he first step in producing the derivati o n is the sa me as ill the other
examples; o ne lists the premisses and notes the conclusion:
R 2 R :o - -Q 3 Q :o F 4 R :oG 5 (F " G ) :o M 1
P P P P P
Z iM In what fo llo ws, to save the space req uired for sho wing the development of the deriva ti o n, the mo re o bvi o us steps will not be discussed separa tely, To get ' M ' fr om li ne 5, o ne needs 'F 1\ C', This formula ca n be obtained only by using 1\1, which requ ires both 'F' a nd 'G', Thus, there are twO pro blems in the fo llo wing o utline of t he derivation:
1
p
5 . (F
p p p
R 2 R :o --Q 3 Q :o F 4 R :oG 1\
G)
~
M
p
F X G Y F AG
W, X, 1\1
Z
5, Y,::JE
W
M
77
§3.9: CONSTRUcrlNG DERIVATIONS
There are two problems: obtaining ' F' and obtaining 'G'. Which problem is tackled first does not matter. As an arbitrary choice, one can approach the problems in alpha betical order. Hence, the immediate problem is obtaining 'F'. Th is formula can be obtained fro m 'Q :::> F' by :::>E:
R 2 R :> - - Q 3 Q:>F 4 R :>G 5 (F" G) :>M
,
p p p
P I'
V Q
w F
3, V,::::>E
G Y F AG
W,X, AI
Z M
5, Y, ::::>E
X
The obvious candidate for obtaining 'Q' is 'R ::::> - - Q' on line 2. One should obtain ' - - Q' by ::::>E and use -E to obtai n 'Q'. A glance at the formula o n line 2 shows that the next formula needed is 'R', and this is
available at line 1. The sketch of the derivation is amended:
R 2 3
R ::::>--Q
X
G FA G
Q:>F 4 R:>G 5 IF A G) :> M 6 --Q 7 Q 8 F Y
Z M
P P P P P 1, 2, ::::>E 6, -E 3,7, ::::>E
8, X, AI 5, Y,::::>E
The first problem, that of obtaining 'F', has been solved. There remains the problem of obtaining 'G'. This is easily solved. 'G ' appears on line 4 in 'R ::::> G', and 'R' is available on line 1. The derivation can now he completed. Dealing with the two problems separately allowed one to concentrate on a single manageable problem at each step of the derivation. The completed derivation follows :
S3.9:
78
CO,,"STRUCTING DERIVATIONS
R 2 R=> - - Q 3 Q=>F 4 R =>G 5 (F A G) => :VI 6 - - Q 7 Q 1
P
p P p
P 1, 2,
~E
6,-E
8 9 10
F
3,7, :::)E
G F, G
1,4, =>E
8,9, AI
II
M
5, to, ::::lE
The various steps in working out the de riva tions in this section have been deliberately shown in excessive detail with a degree o f neatness hardly to be expected in the course o f actually solving a problem. In practice, there would be a 10[ of erasing or scratch ing out, and probably use o f some son of 'shorthand', As a practical maUer, it is important to keep rough work neat enough to read. In (he course of a long derivation, onc can easil y bc:co me confused, even forgetting what fo rmula onc is trying to obtain at a particu lar point. In dealing with complex derivations, [here are ohen several equally good ways of gening to the conclusion , and without some written record it is easy to lose track of the overall strategy. In S3.10 there a re exercises that require the construction of derivations. In deali ng with these exercises, the following advice should be kept in mind :
One should not try to prod uce a final copy straigh t away. There shou ld be a rough COpy of the derivation on a separate sheet. This need not be neat, but it must be legible, since in all probability it will be needed fo r reference. 2 One should deal with one problem at a time. 3 One should nOt try to complete a derivation by bending the rules a bit to obtai n what is needed. A problem half solved by legiti mate use of the rul es is better than a problem 'solved ' by cheating.
79
53.10, EXERCISES
S3.10: EXERCISES For each of the following arguments, derive the conclusion from the premiss(es).
A
B A=>--(B=>--C)
C
2 B BE - - A
(A " B)
:::'I
R
R
J B BE - - A
(A v S)
::::l
R
1\
A
R 4 AA B (8 v
q
5 A A => G BER B => S G E - - I ((F v G) A (H v I II => R A A S
6 P P :::'I---S A E (R v S) (A v B) • (P A Q ) -$::J F F => R (P A Q ) A R
80
SJ.Il: ASSUMPTIONS
-------
SJ.ll: ASSUMPTIONS In arguing for a conclusion, it is often useful to assume something in order to sec what follows from the assumption. This is a legitimate move in derivations. h is, of courSe, necessary to keep in mind that anything one assumes is only an assumption, and in a derivation this is done by keeping the assumption to the right of a new scope linc. Every scope line except the primary scope line which is d rawn against the line numbers is a secondary scope line. Every secondary scope line of a derivation in sentence logic begi ns with an assumption. By the end of a derivation, one wants to be back against the primary scope line; that is, what o nc has established should nor depend upon the [ruth of assumptions which have been introd uced provisionally, Later in this chapter there will be rules for justifying fo rmulas on the primary scope line by appea l to information which is on secondary scope lines. For [he moment, howe ver, the important point to remember is that the secondary scope line serves to keep information away from other scope lines; it is a reminder that the info rmation on the secondary scope line is dependent on the assumption with which that secondary scope line began. In the following derivation, 'S ::::> R' is derived from ' R', In addition to the rule of assumption, the derivation uses twO rules - Rand ::::>1 - which have not yet been discussed. For the momeO!, appeals to these rules can be ignored; what matters at present is the way the new scope line is begu n.
1 Q
P
2 3
A
4
2-3, ::::>1
1, R
It is possible to make one assumption and then go on to make another assumption. The re are no limits to the numbe r of assumptio ns which can be made in a deri vation. In the following deri vation, four assumptions are made, each assumption beginning a new scope line:
S3.1 1: ASSU MPTIONS
81
3
KA B
P P A
4
B B::> C
3.AE 2, R
C D
4.5. :::)E 1, R
CAD
6,7, AI
1 D 2
5
6 7 8
9 10 11 12 13 14
B ::> C
(K A B) ::J (C A D) F
r~D
G:::> (H:::> D)
15 16
F ::> (G ::> (H ::> D))
17
11K A B) ::> (C A 0 )) A IF ::> IG ::> IH ::> D)))
3-8 , ::JI
A A A 1, R 12-13, ::>1
11-14, ::>1
10-15, :::>1 9, 16, AI
Again, appeals to Rand :::)1 can be ignored for the moment. These rules will be discussed later in this chapter. The derivation JUSt considered has fou r subderi vations. Lines 3- 8 and 10-15 are subderivations o f the main derivation. Lines 11-1 4 are a subderivation of the main deri vation as well as a subderivation o f the subderivation which consists of lines 10-15. Simi larly, lines 12- 13 are a subderivation of the main derivation, a subderivation of the subderivation which consists of lines 10-15, and a subderivation of the subderivation which consists of lines 11-1 4. Secondary scope lines show the extent of subderivations just as primary scope lines show the extent of complete derivations. Assumptions, once made, need not be ma intained fo r ever. When an assumption is no longer need ed, fo r any reason whatever, t he scope line which began with that assumption is ended. The rules for end ing scope lines will be discussed shortly. These rules will allo w one to ma ke use of assumptions and the formu las which appear in the subde riva tions. When a scope line is ended, any new formula and any new scope line should be placed as far left as possible. It should be noted t hat :::::lE, AE and AI work in exactly the same wayan secondary scope lines as they do on primary scope lines. The other fa miliar rules - v i, =E and -E - ca n also be used on secondary scope lines in exactly the same way in which they are used on primary scope lines.
82
53.12: R EITERATION
Th e for mal statement of the rule of assumption follows:
A Ar any point in a derivatio n, o ne can introd uce any well-fo rmed
fo rmu la as an assumption, provided [hat the foll owing condition
IS
mct;
The fo rmula is written at the beginning of a new scope linc . The formula which is assumed should be followed by a short horizontal line to disti nguish it fro m other formu las which may appea r
on the new scope line. Th e jus ti fication for the new line is simply the rule, ' A',
T he re are tw O crucial points wh ich req uire care in using rhe rule o f ass umptio n. First. what is assumed must be a well-fo rmed fo rm ula, but of cou rse unnecessary outer brackets can be dropped. Second, eve r y assumption must be the first fo rmula on a new scope lin e.
p .ll: REITERATION If one knows something, that piece of knowledge is usable no mane r what else onc kno ws and no ma tter what assumptions one makes. Fo r example,
if onc knows that John is married, then , if one assumes that John is an accountant, onc still kno ws that Jo hn is ma rried. This is o bvious. a nd the rule of reiteration is equally obvious.
R At any point on a scope line. onc can copy a ny for mula which already appears in the derivation onto the scope line, provided that the following condition is met: T he scope line fro m which the fo rmula is copi ed has no t been ended. Th e justi fi cation fo r the new line consists of an appeal to the number o f the line from w hich the fo rm ula is reite rated and the rule, ' R '. In dealing with predicate logic. some restrictions will be placed on the rule o f reiteration. Ho weve r, for purposes of doing sentence logic, the rule of reiterati o n as JUS t stated is complete ly adequate . Some examples of proper uses of the rule of reiteration a ppeared in S3. 11. T he re are more examples in thC' following derivation which usC's some rules which have not yet been discussed. The details o f these rules can
83
§3.12: REJTERATION
be ign ored for the moment; the derivation is presented to provide examples of reiteration. 1
o
2
O::> C
3 4
B ~ (M
v B ~
(F ::> C))
P P P A
0
5
M
A
6
0
7
O::>C
8
c
4,R 2, R 6, 7,
9 10
fZ
11
F ::> C
12
M~ ( F ~C)
~
13 14 15
16
~E
A
8, R 9-10,
~I
5-11 , :::)1 A
::> (M ::> (F ::> C)) M ::> (F ::> C)
M ::> (F :> C)
3, R 13,14,:::)E 1,4-12,13-15, vE
In the following derivation, some common mistakes in reiterati on are shown:
M"N 3 4
F
5
F::> K
P P A 1, R 2,R
6
F
3, R
7
K
3, 5,
ttt
7,R
2
8
F:>K
M
K
9 10 11 12
F
ttt
ffttt ttt
F A K
~E
A 3, R 10,R 8,1 1,,,1
The reiteration at line 6 is in order. However, since ' F' is already on the scope line, no purpose is served by its repetitio n. The reiteration at line 4 is incorrect because only part of a formu la is reiterated . All three of the reiterations at lines 8,10 and 11 are wrong because in each case the formula
S3.13:
84
HORSESHOE INTRODUCfION
is reiterated from a scope line which has ended. This is by far the most common mistake made in reiteration.
§ 3.13: HORSESHOE INTRODUCfION The way the horseshoe is defined on the truth table gua rantees that any fo rmula 'o,f) :::J IV' is true if '\If' is true, no maHer what the truth value of . ~. may be. Given any sentence ''V', one is entitled to infer ' ::::> IJI'. Inferences of this son sometimes have odd consequences. From the fact that Shakespeare is (he author of Hamlet, it fo llo ws (har, if today is Tuesday, rhen Shakespeare is the a uthor of Hamlet- a claim that wou ld strike most people as very strange. However, the inference is licensed by [h e truth-table definition of the horseshoe, and such inferences are reflected in the rule o f horseshoe introduction, '::>1'. :>1
One can end any subderivation and write the formula '(¢l ::> 'V)' on the scope line immediately to the left of the ended subderivation , provided (hat bo th of the foll owing conditions are met:
1 The subderivation begins with the assumption
'¢l',
2 The formula 'IV' appears in the subderi vation. The justification for the new line consists of an appeal to the range of numbers in the ended subderivation of 'IV' from '¢I' and the rule, '::> 1', The formal statement of the ru le includes outer brackets in the formula, although this requirement is ignored in what follo ws. Some examples of correct uses of :::>1 follow: A 2
B
3 4
C
5
6 7
8 9 10
~
D :> A D :> B I):> C
P P P A I, R 2, R 3, R 4-5, :::>1
4-6, :::> 1 4-7, :::>1
The formulas involved need not be atomic fo rmulas. ::>1 can be used several times in the same deriva tion to obtain increa singl y complex formulas.
85
)3.13: H ORSESHOE INTRODUCTION
Nothing in the ru l~ requires that the two fo rmula s joined by the horseshoe be distinct, so formulas such as 'A :> A' can be introduced by use of:>1. The following derivation ~x ~ mplifies thes~ points: 1
((S :> L) :> M ) :> R
2 3
P A
5
A
L
4
A
M
~:>
5
6
A
L
7
(S :> L) ;:, M
8 9
((5
15
L) :> M ) :> R
R M :> R L :> (M :> R)
10
11 12 13 14
:>
S :> (L :> (M :> R)) P (P
:>
A
P P)
1, R 7,8, :>E 4-9, :>1 3-1U, :>1
2- 11 ,:>1
~ :>
4,R 5--6, :>1
A
(5
:>
(L
:>
(M :> RII)
13-13, :>1 12, 14, 1\1
Li ne 13 and line 14 of this derivation may s~em a li ttle strange. The scope which b~gan with th~ assump tion at lin~ 13 was end ~d at line 13. This provides a o ne-lin ~ subderivation, which is appealed to at line 14. It would b~ correct, but unnecessa ry, to reit era t~ ' P', providing a two-l ine subderivation. The derivation would then be sligh tl y longer: l i n~
86
\3. 13: H ORSESHOE INTRODUcrJON
((S :> L) => :\1 )
::J
R
S
2
J
L
4
M
~:JL
5
6
M L) ::> M
7
(5
8
(( 5 ::> L )
9
R
10
II
12 13 14 15 16
:J
::J
M ) ::> R
M :> R L ::> (M ::> R) S ::> (L ::> (M ::J R))
~ P :> P (P :> PI
A
(S :> (I. :> (M :> Rill
P A A A A 4, R 5-6, ::II I, R 7,8, :::>E
4-9, ::>1 3-10, ::>1 2-1 1,::>1
A 13, R 13-14, ::>1 12, 15, ,,)
In using ::>1, it is important that on ly the scope line on which the ass umption
is introduced be cnded and that the new form ula be written against the scope line immediately to the Idt of the ended scope line. If mo ves suc h as the
mistaken one in the following attempt at a derivation were all owed, one could 'derive' anything whatever fro m any premi ss: I
2 J 4
P
A
r~
5 A:>B ttt 6 B
A A 2, R 3-4 , ::>1 1,5, ::> E
'S' was properl y introduced as an assumption at line 2 with a new scope li ne. Tha t scope line serves to show that anything derived after line 2 may depend on'S', a fo rmula which has been only assumed. In end ing the two scope lines simultaneously at line S, the fact that 'B' was only an assumption ha .~ neen completely ignored .
87
~3.13: HORSESHOE INTRODUCTION
It is important to remember that :JI introduces a horseshoe as the main ope rator of a new formula. With this rule, as with others, it is necessary 10 pay attention to brackets, to avoid mistakes such as the following:
2
3 4
IB :> C) :> R
P
~:> C IB :> C)
A
:> R
R
5 B :> IC :> R) ttt The correct way
10
3 4
2, 3, :JE 2-4, ::JI
derive '8 :> IC :> R), from 'IB :> C) :> R' follows:
IB :> C) :> R 2
I, R
B
P A A
C
~
5 6 7
B:>C (8 ~ C) ::J R
8
R
A
3, R 4-5,:J1 I, R
6,7, :JE
9
C:>R
3-8, :::>1
10
B :> IC :> R)
2- 9, :JI
It should be noted that 'B' occurs at line 4 as an assumption and is not reiterated from line 2. The fact that a formula appears earlier in a deriva tion does not mean that it cannot be assumed at the begin ning of a new scope line. H one pays attention to brackets and is careful to end onl y one scope line at a time, it is easy to avoid mistakes in using :JI.
S3.1 4:
88
TRIPLEBAR INTRODUCfION
53.14: lR lPLEBA R INTRODUCTION ' ¢I ;: \II '
is trut h-fu nctiona lly equivalent to '($ :J IV) A (\fI ::) ¢I)', Accordingly, if one can derive ''V. from '¢>' and derive ' $' from ''V', one should be able to dn ive '$ :: W'. It is not surpri sing, therefore, that the rule of triplchar jmroducrion, '=1', is very similar to twO applications of
horseshoe introduction. .1 One ca n write the fo rmula '( ;;; IV)' or the formula '(IV "" CIl )' on any
scope li ne, provided that both of the following conditions arc met: On a scope li ne immediately to the right of the given scope li ne, t here is a subdcrivation of 'lV' from ' ¢>'.
2 U n a scope line immediately to the right of the given scope line,
there is a subderivation of '$' from
'\jI' ,
Th e justification for the new line consists of an appea l to the range of numbers of the subderi vation of 'w' from '<J>', the range of numbers of the subderivation of 'Ill' from 'w' and the rule, ':;;:1'. The formal statement of the rule includes outer brackets in the fo rmula , although this requi rement is ignored in what foll ows. The follo wing derivation shows some legiti mate uses of :;;:1; 1
1 3 4
5 6 7
8 9
A ::> B B ::> A
~
::> B
~::> A:;;:B
to B . A
P p
A
A I, R 3, 4, ~E A l,R 6, 7, :lE 3-5,6-8, , I 3- 5,6- 8, . 1
89
S3 .14: TRIPLEBAR INTRODUCTION
Th e fo llowing deri vation provides anmher example o f the correct use of =1:
1 M 2 (- R v M) :J « F A G) A - S) J F " (- S " - R)
4 5 6 7 8 9 10 I1 12 13
-R
- R
v
M
4, v i
(- R v M ) " (( F A G ) A - S)
2, R
(F A G ) A - S
5, 6, :JE
F AG
7, AE
- S
7, AE
F F A - 5
8, AE
filR M
14 - R:::> M 15 F" - S 16 F 17 F " (- S :> - R) 18 -5" - R 19 - 5 -R 20 21 - R . (F A - S) 22
P P P A
(- R
= (F A - 5)) A ( - R " M )
9,10,AI
A I, R 12-13, :::>1 A
15, AE
J, R 16,17, :::>E 15,AE
18, 19, :::>E 4-11 , 15-20, =1 14, 21 , ,,1
It should be noted that the use of EI at line 21 is completely in order, notwithstanding the fact that the two subderivations which are appealed to are separated by the derivation of '- R :J M' which extends from line 12 to li ne 14.
90
SJ.14: TRIPlEBAR INTRODUCTION
Some common mistakes involving ~I are made in the following attempt to derive ' A :;: S' from 'A ::J B'. As in this example, mistakes are often made not in applying the rule o f triplebar introduction but in illegitimate moves before the ...1 step. I 2 3 4 5
~
II 12
I, R
::> B
2, 3, ::lE A
I~
7
10
A
B
6
8 9
P
A ::> B
B::>A
A ttl A ttt
ttt
A A 6,R 7-8, :::>1
2, R 6,R
5,9, 'JE
13
A A. B
14
A=B
2-4,5-10, =1
15
A·B
2--4,5-11, =1
16
A=B
2-4,5-12, . \
ttt
2-4 ,7-8 , =1
Lines 14, 15, and 16 are not in themselves wrong, but each of them relies on
an occu rrence of 'A' which was obtained illegitimatdy earlier in the derivation. The mistake at line 9 is a mistake in ::>I. The scope line which begins at line 6 and the assumption 'A' are simply ignored. The reiteration at line 10 is ilkg.itimatc because ' A' was rtimau!d from a scope line which has been ended. The reiteration at line 11 is illegitimate for the same reason. Line 13 is incorrect because the subderivation of 'A' from 'B' is not immed iately to the right of the scope line on which ' A == B' appears. To use triplebar introduction, one has to derive o ne formula from another and then derive the latter from the former. If both of these formulas are leg.itimately derived on scope lines immediately to the right of the scope line o n which one introduces the formula whose main ope rator is a triple bar. it is hard to go wrong in using =1.
~3.15: NEGATIO:-;O INTR oDucno~
91
p .1S: NEGATION INTRODUCTION If one kno ws that Peter cannot see, one knows that the assumption that Peter saw the accident is false. The assumption leads to a flat contradicti on, such as that Peter is both blind and not blind. Similarly, given the premisses '4>' and '4> :::J - W', the assumption of 'W'leads to a contradiction: 'lV' and '- W'. The rule of negation introduction, '-I', allows one to derive '- 41 ' when the assumption of '$' leads to a contradiction. -I
One can write the formula '- 41' on any scope line, provided that both of the following conditions are meC: On a scope line immediately to the right o f the given scope line, there is a subderivation which begins with the assumption '41' .
2 On the scope line which begins with the assumption appear twO fo rmulas whic h form a contradiction:
'¢I' there
'w' and '-
IV'.
The justification for the new line consists of an appeal to the number o f th e li ne where the assumption '¢' is int roduced, the number of the line where 'W' occurs, the number of th e line wh ere' - W' occurs and t he rule, '-I'. The following derivation illustrates some correct uses of -I: - (A v HI
2 3 4
5 6 7 8 9 10
~
p A
A v H
2, v i
- (A v HI
I, R
-A
~
2,3,4, - I A
A v H
6, vi
- (A v HI
I, R
-H
-A,,-B
6,7,8, -I 5, 9, 1\1
It shou ld be noted that the pair of formu las o n lines 3 and 4 and on lines 7 and 8 are a genuine co ntradil.:tory pair, even though no brackets are written around' A v 8'. Strictly speaking, the contradictory pair consists of '(A v H)' and '- (A v Hr.
S3.15;
92
NEGA nON If'..'TRODUcrION
The assumption itself can be one of the contradictory pair of formulas, so the following derivation is also correct:
P A
B
fiB
2 3 4
I, R
2, 2, 3, ~ I
- - B
In the following derivation, 1
F :::) G
~::>
4
5 6 7
- f
8
- G ~ - F
C ::> - F' is derived from 'F
:J
C':
P A A
-G
2 3
'~
G
-G
I, R 3,4, ::;IE
2, R
3, 5, 6, -I 2-7, ::>1
The formula derived by - I must be the formula which results from adding a tilde to the assumption. If the assumption is a negated formula, it is incorrect simply to remove the tilde; a tilde must be added. It is, of course, legitimate to remove the twO tildes by -E in a subsequent step. The following is an incorrect use of -I: 1
p
2
-N P r;-NN - P
3 4
5
6 7
::> -
::> - P
p
Nttt
P P
A 2,R 3,4, ::>E
I, R 3,5,6, - I
S3.15:
The prope r way to derive 'N' from ' P' and '- N 1
P
2
-N i;N -N - P
3 4 5
6 7
8
93
NEGATION I NTRODUcnON
=> - p => - P
p
--N N
::::> -
P' follo ws:
P P A 2, R 3, 4, ::::>E 1, R
3, 5, 6, -I
7,-E
In using - I, one must pay careful anemion to brackets. T he fo llowing derivation illustrates this point:
-A 2 3 4 5
P
~' B - A - A , B t tt
A
2,I\E 1, R 2,3 , 4, -I
The Tuk of - I allows one to obtain the negation of an assumption, not the negation of part of an assumption. Line 5 is mistaken because the entire assumptio n is not negated. '- A 1\ B' ca nnot be legit i malC~ ly derived fr om '- N, although ' - (A 1\ B)' can. This is done in the following derivation: 1
2 3 4 5
-A
~'
B
- A - (A , BI
P A 2,I\E 1, R
2,3, 4, - I
It should be noted that in this derivation the tilde negates the entire formula which is assumed o n line 2. In using -I, one mUSt take care to ensure that one really has a contradictory pair. Most mistakes in using this rule involve mistakes about the contradictory pair o f fo rmulas. To illustrate this sort of mistake, three mistaken attempts to derive '- C' are exemplified in what follows. In each case, what is appealed to as a contradictory pair is not a genuine contradictory pair.
S3.t5:
94
NEGATION INTROD UCTION
1
P
P
2
Q
P
3 P " - Q
P
4
A
C
~
5 6 7
A
2, R 5-6,
p" Q
~p
8 9 10 12 13
- P " Q P "- Q P P v R
14
P
15 16 17
ttt -c ttt -c ttt
1I
- C
v
-
R
~I
A
2,R 8-9, ::>1
3, R I, R
12, v i 12, v i 4,7, 10,-1 4,7, 11 ,-1 4, 13, 14,-1
The rule of negation introduction is very easy to usc and remember, but care must be taken to ensure that onc has a genuine contradictory pair under the assumption.
S3.16:
95
DISJUNCTION ELIMINATION
p.16: DISJUNCTION ELIMINATION If one knows that taxes will go up if Jones wins the election, that taxes will go up if Smith wins the election, and that either Jones or Smith will win the dection, then one certainly knows that taxes will go up. Given any disjunctive formu la '4> v lV', if one ca n derive some formula 'n' fro m '11>' and one can derive '0' from '\II' th(:n one is entitled to infer '0'. The rule of disjunction elimination, 'vE', reflects the propriety of this son of inference.
vE One can write the formu la '0' o n any scope lin(:, provided that all of the following cond itions are met: There is a disjunctive formula '(4) v l.jI)' on the given scope line. Z On a scope line immediately to the right of the giv(:n scope line there is a subderivation of [he formula 'n' from '11>'.
3 On a scope line immediately to the right of the giv(:n scop(: line there is a subder ivation of that same formula '0' from '\jI'. The justification fo r the new line consists of an appeal to the number of the line on which the disjunctiv(: formu la '(41 v \jI)' appears, the [wo ranges of numbers o f the cwo subderivations and the rul e, 'vE'. The formal statem(:nt of [h(: rule includes outer brackets in the form ula, although this r(:quir(:mmt is ignor(:d in what follows. The follow ing derivation shows the correct use of vE: 1
A v B
P P
2 A ::>C l B ::> C 4
5 6 7
8 9 10 C
~::> ~::>
p A C
2.R 4,5, :lE A
C
l. R 7, 8, ::>E 1, 4-{l , 7-9. vE
v E is often used to obtain one of tbe disjunCls from a disjunctive form ula . In the following derivation, 'F' is derived from 'F v G' and '- G':
96
§3.16: DISJUNCTION ELIMINATION
Fv G 2 -G
P P
I
L!..
3 4
A
~F
5 6 7 8 9 10
A
G
A
4, R 2,R
-G - - F F
5,6, 7, -1
8,-E
F
I, 3-3, 4-9, vE
As wi th the other roles, v E can ~ used with co mplex fo rm ulas as wdl as atomic formulas. The sequence in which the disjunctive formu la and the tw O subderivations appear does nO[ matter. Sometimes (here are disjunction eliminations within the slIbderiva tions. These points are exemplified in the following derivation:
I
- M
2
- K
P P
3 4 5
- L
P
=> R
(K v L) v (M
fi'
6
A
A A
~K
9,10,11,-1
L
A
- L
-M -M
- M ~ R
R
A
8, R 2,R
- M
~
R
P 5, ,,£
II 12 13 14 15 16 17
R)
A
R
Kv L K
7 8 9 10
18 19 20 21
A
A
13, R 3, R 14,15, 16, - 1
7,8-12, l3-1 7,vE I, R 18, 19, =:IE
4, 5-6, 7-20, v E
97
§ 3. 17: EXERCISES
The most common mistake in using vE is tr yi ng to use it on a for mula whose main operator is not a vel. It is only th rough fai lu rc to look carefully at thc brackets that someone wo uld think of applyi ng vE to a fo rmula suc h as 'P :::::l (R v $)'. Th e o ther mistakes in using vE are usuall y mistakes in the subderivations. T he di sc ussion of mistakes in using =I in §3.14 illustrates thc SOrt of mistake which is common. Ca re must be taken to ensure that vE is applied to a disju ncti ve formula, tha t the s ubderivations are legitimate and that t he su bderivati ons are o n sco pe lines immediately to th e right of the di sjunctive formula. If this is done, use o f vE is completely straightforward.
53.1 7: EXERCISES Suppl y the missing justi fica tions in the following deri vations. Question 1
G ::> IF ::> IH ::> I)) 2 F J G 4 G ::> IF ::> IH ::> III 5 F ::> IH ::> II F 6 7 H ::> I 8 G ::> IH ::> II 9 F ::> IG ::> IH ::> I))
p
Q uestion 2
I 2 J 4 5
6 7
8 9
p
K K ::> R
~ :> R ~ K. R IK ::> RI ::> IK • RI
S3.17:
98 Qu~st ion
J
1 D ::> (R ::> M ) 2 - N :J- M 3 4
P P
D ::> R D
5
-N
6 7
- N ::> - M - M D
8 9
D ::> R
10 11 12
R
R :J M
?
13 14
M
?
15 16 17
?
D ::> (R ::> M )
- - N
?
N D ::> N (lJ ::> R) ::> (D ::> N)
Qutstion 4
1 2 3 4 5 6 7
P
5. T
~' T 5 ::> T - 5
8 9
~' T
10
- 5
11 12
13
- T - 5 ~ - T (5::> T ) A ( - 5 ::> - T )
EXERCISES
99
S3.18: MORE DERIVATION CONSTRUCTION
Question 5
1 (F ::l - G) ::l H 2 - F v - G 3 4
p
- F F
~F
5 6 7
-G
8 9
F::l-G
G
rF~GG
10 11
12 13 14 15
p
F =>
-G
? ?
?
H
5].18 : MORE DERIVATION CONSTRUCTION In the discussion o f constructing derivations in S3.9, the general strategy was to try to use an elimination rule first. If that did not work, o ne turned to an introduction rule . This strategy will have to be changed somewhat, since there are now more rules to thi nk about. Th e first additional rule to be considered is reiteration. If a formu la is needed and it can be reiterated onto the scope line on which one is wo rking, the obvious thing to do is simply to reiterate it . As a gene ral guide, th en, when one wants a formula '¢I', one should: 1 See if one can reiterate '¢I'. 2 See if one can use an elimination rule to get '(%>'. Usua lly, but by no means always, immediately using v E when a disjunctive formula is available results in a shorter derivation. 3 See if one can use an introduction rule to get '¢I'. The introduction rule to use is determined by the main operator of '¢I'. If '¢I' has no main operator - if it is an atomic formula - there is, of course, no way in which one can get '¢I' by an introduction rule. The point of approaching problems in this order will become clearer as some derivations are constructed in this section. Beginners at the business of producing derivations frequently make the mistake o f immediately assuming what is needed. As a strategy fo r deriving
100
S3.18: MORE DERIVATION CONSTRUCfION
' R ::J (S v T)' fro m ' R :::::l S', introducing 'R does nothing to solve the problem: 1
R :::::l S
2 ~ => IS v TI 3 IR => IS v Til => IR => IS
:J
(S v T )' as an assumption
P A v
Til
2-2, :ll
' R :J (S v T)' muSt be derived against the primary scope line, but this formula is on a secondary scope line. '( R ::J (S v T)) ::J (R :::> (S v T )l' was legitimately derived on the primary scope line by =>1, but that is not what is needed. The above attempt at a deriv,Hion fails because the assumption wa s introduced for no good reason. In introducing an assumption and starting a new scope line, o ne should have a clear idea of why one is doing it and a clear idea of what rule is [0 be used to end the scope line. To de rive 'R ::::> (5 v T)' from 'R ::J S', one need only think a bit about possi ble strategies. The first thing to do is to lay out a plan of the derivation, showing what is availab le and what is wanted: I R :::> S
Z
iR
p
:::> (S v T )
There are no secondary scope lines, so the Question of reiteration does not arise. 'R :::> (S v T )' does not appl"ar anywherl" in what is availabll", and it Sl"ems dl"ar that one cannm obtain it by an elimination rule. The main opl"rator of 'R :::> (S v T ), is a horsl"shol", so the introduction rule to be uSl"d is :::JI. One should assume th e antl"cedent of the formula and try to obtain the consl"quent:
R => S 2 :1
Y'I Z
~
is v T R :::> (S v T )
P A
2-Y, :::>1
Th e formula which is needl"d at line Y cannot be obtained by reitl"ration or an elimination rule. The main ope rator of'S v T' is a vel, so the appropriate introduction ruk is v I. To obtain'S 1/ T' by v I, OOl" needs either'S' or 'T'. There seems to be no way of obtaining 'T', but'S' appea rs in 'R :::> S' on linl" L It seems plain that one should try to obtain'S', and then go on to obtain '5 v T' from'S' by v I. Th e outiinl" o f the derivation is modified to show this:
101
§3.18: M ORE D ERIVATION CONSTRUCTION
lI R ;:)S
P
~
A
2 :
~ ' i~ Z
v
T
X,vl
2-Y, ::JI
R Ol (5 v T I
To obtain '5' from ' R ::J 5' on ~ n~~ds 'R', and this fo rm ula is available at line 2. Once ' R ;:) S' is r~iterat~d Onto the secondary scope line, it is clear where 'S' is to come from. It can sim ply he obta ined by ::::>E. Th e derivation can thus b~ completed:
P A
ROl5
2
~~
3
ROl 5
4
1, R 2, 3, ;:)E 4, v i 2-5, ::JI
v T 5 6 R Ol (5 v TI
It should not be supposed that it is never a sensible strategy to assume what one needs. This is done in d~riving the conclusion of the following argument: (R Ol 51 Ol R
5 R Th~
derivation
b~gins
;:) 5) ;:) R
with
th~
premisses and conclusion:
p p
The obvious strategy is to try to obtain 'R' by using;:)E on line 1. To do this, om: needs ' R ::::> 5':
I
1 (R ;:) S) ::::> R
p
2
5
P
Y
i~
Z
Ol 5 1, Y,OlE
The only way to o btain ' R ;:) S' is by::::>1. One must assume ' R' and try to get'S'. Since '5' is availabl~ by re i t~ration from line 2, the derivation ca n be completed without difficulty:
~3.18:
102
MORE DERIVATION CONSTRUCTION
1
(R :> S) :> R
p
2
5
P A 2, R
3 4
~
5
R => 5 R
6
3-4 , ::::)1
1, 5, ~E
The pain[ about che assumption of 'R' at line 3 is that it was assumed in order to obtain 'R :;I S' by:>!. The fact that 'R' is the conclusion of tht argument which onc was trying 10 obtain as the last line of the derivation is JUSt a coincidence. Another problem is deri ving the conclusion from the premiss of the following argumen t: (M , 0) • (5 v N)
S :> (0 v R)
The derivation is begun by writing the premiss at linc 1 and the conclusion at line Z: 1
! (M
Z
i s:>
1\
0)
ii5
(5 v N )
p
(O v R)
'5 :::l (0 v R), does not appear anywhere, and it is clear that it cannot be obtained by an elimination rule . The ho rseshoe is the main operalOr, so the appropriate introduction rule is ::::>1. The outline of the derivation is developed accordingly: 1 1 (M " 0)
5
(S v N)
~
2 :
P A
Y'I Z
io v R 5 => (0 v R)
2-Y, ::II
There seems to be no way to obtain what is needed at line Y by an elimination rule. To obtain '0 v R' by v i, one needs either '0' or 'R', There is no easy way to obtain ' R', so the alternative is to obtain '0', The skeleton of the derivation reflects this:
103
S3.18: MORE D ERIVATION CONSTRUcnO:-..l
1
I (M
0)
1\
(S v N )
~
2:
p
A
~ . i~ v Z
!Ii
R
5 => (0 v R)
X,vl 2-Y, :)1
The only way to obtain '0 ' seems to be: by I\E from ' M 1\ 0' which appears on li ne 1 as pan o f '(M 1\ 0 ) ;;; (S v N)t. If o[t(: had'S v N', a ile co uld o btain'M 1\ 0 ' by ""E. Accordingly, '( M 1\ 0 ) =: (S v N)' is rei terated and the derivation is developed: (M A 0 ) • (5 v N)
z
~
3
P A
A
0) .. (S v N )
I, R
V
5 v N
W
MI\O
3, V, EE
X
0 OvR
W,I\E X, v l
Y
Z
S :=I (0 v R)
2- Y, ::::ll
'S v N' cannot be obtained by an elimination rule, but it is easy to obtain 'S v N' by using v!. To do this, one needs either'S' or 'N'. and '5' is already on the present scope line at line 2. All that rema ins is to com plete this step and number the lines and justi ficatio ns of the derivation. (M
0) = (S v N )
P A
3 4
5 (M 1\ 0 ) = (S v N) 5 v N
5
M AO
6 7
0
Z
8
1\
OvR S :=I (0 v R)
I, R 2, v i 3, 4, =E 5,I\E 6, v I 2-7, :=II
A more: d ifficult problem is that of obtain ing the concl usion from the premisses of the following argument: (F => G ) => (G A I) F => H (H => G ) => G
S3.18:
104
MORE DERIVATION CONSTRUCfION
As befor~, the sktleton of the derivation is constructed from the premisses and the conclusion:
I IIF " GI " IG
A
II
p p
2F ::> H
Z i IH " GI " G Since there are no secondary scope lines, reiteration is not a possible strategy. Th e:: formula '(H ::> G) ::J G' does not appear as part o f one of the premisses, and thus it ca nnot be obtained by any elimination (uk Turning to introduction rules, since the main operator of '( H ::l G) ::J C' is the second horseshoe. one should try to use ::>1. One shou ld assume what is to the left of the main operator, and try to obta in what is to the right. The o utline of the derivation is altered (0 show this:
I IF " GI " IG 2 f ::> H
I-':!. "
3 Y Z
i
A
II
P P A
G
iC (H
~
3-Y, ::>1
G) ::> G
'G ' cannot be reiterated. Turning to elimination rules, 'G' appears in '(F ::> G) ::l (G " I)' and in the assumption 'H => G', If one tried to get 'G' from 'H :::> G', one would need ' H ':
IF " GI 2 3
X y
:J
IG
II
p P
F:::> H
I-':!. "
A
G
i~
Z IH " GI " G
A
3, X,::::>E 3-Y, =>1
The only reasonable strategy here is to obtain 'H' from 'F ::::> H', which can be reiterated from line 2, The sketch of the derivation is suitably amended:
105
S3.18: M ORE DERIVATION CONSTRUCTION
(F ::> G) ::> (G 2 F ::> H
P A
f¥
3 4
W X
Y Z
p
I)
A
::> G F ::> H
2,R
F H
4, W ,~.E
3, X, ~E 3-Y, ~I
G (H
~
G)
~
G
Th e only places in whic h 'F' appears are on the left side of a horseshoe; in both places it is completely inaccessible. Plai nl y, some other strategy m ust be tried. When the pro blem of geuing 'G' at line Y was fi rst posed, there were twO options: trying to get it from ' H ~ G' and trying to get it from '(F :;, G) ::. (G 1\ I)'. The first o ption has led nowhere, so one must try the second opnon. The step from 'G 1\ I' to 'G ' is obvious, as is the step from ' F :;, G' and '(F :;, G ) ::> (G 1\ Il' to 'G 1\ 1', so the proble m is obtaining 'F ::::> G'. The sketch of the derivation is appropriately amended:
I
(F ::> G) ::> (G
A
p
I)
2 F ::> H 3 ~::> G 4 W
X y
Z
. (F ::::> G) ::::> (G
F ::>G G A I G (H::::> G)::::> G
P A 1\
I)
I, R
4, W,::::>E
X, AE 3-Y,::::>1
It is impossible to reiterate ' F ::::> G' and it cannot be obtained by an elimination rule. Accordingly, one must try the appropriate introduction rule, ::>1. One should assume 'F' and try to obta in 'G'. The revised sketch of the derivation shows this:
106
\3.18, MORE DERIVATION CONSTRUCfIO:"l
--- -- - -
r
1
(F
2
F => H
3
::J
G)
(G
:J
1\
P
J)
P
A
(F => => G G) => (G A 1)
4
:
5
V W
1, R
A
~
iG
F
5-V, ::>1
G
::J
X y
GA l
4, W, =:>E
G
X,,,E
Z
(H => G) => G
3-Y,::JI
What onc needs at line V is exactly the same as what onc needed earlier at line Y. It might be supposed that. since onc could not get 'C' from ' H :J G' at line Y, one cannot get 'G' from 'H :::> G' now. This would be a mistake, but the mistake is worth exploring.lf onc cannot get 'G' from ' H :J G',onc must get it from 'C A I' - and this is juSt what is being attempted now. Accordingly, the steps involved in getting 'G' from 'G 1\ I' will be simila r to the steps one has already taken. The sketch of the derivati on is revised [Q show this strategy:
2
(F => G) => (G A 1)
p
F => H
P
3
H =>G
A
4
(F => G) => (G A 1)
1, R
r
5 6
A
=> G)
7
S T U V
:
~
'
Ie
:J
(G
A
1)
1, R A
F => G
7-5, ::Ji
G A I
6, T,::>E
U,AE
G
W
F => G
5-V, :.>1
X
G A I
4, W, ::>E
y
G
X,,,E
Z
(H
:::J
G)
::>
G
3-Y, ::JI
With the outline of the de rivation on paper, it is clear why the present strategy is a mistake. One needs 'G' at line S, juSt as one needed 'G' at line V. At line S there is no mo re information available [han there was when one
107
S3.18: MORE DERIVATION CONSTRUCTION
first tried to get 'G' at line Vi if there is a strategy which se rves to obta in 'G' at tine S, exactly the same strategy would have served to obtain 'G' at tine V. It is time to back up a step:
IF " GI " IG A 11 2 F" H 3 4
5
r"
IF " G GI " IG A 11
:
~
p P
A 1, R A
iG
V
W
5-V, ::JI
F" G GA l G
X
Y
4, W,::JE X,AE
3-Y, ::JI (H " GI " G Although what one needs at line V is exacdy the same as what o ne first needed at line Y, and although one could nOt get 'G' from 'H ::> G' at line Y, another attem pt to gel 'G' from 'H :::> C' is now a sensible move. The reason fo r this is that one has more information available at line V than one had previously. Whenever one innoduces a new assumption, ir is always possible that a strategy which did not previously work will now work with the new infor mation available from the new assumption. To get 'G' from ' H ::J C', one needs ' H', and the sketch of the derivation is amended accordingly;
Z
IF " GI " IG A II 2 F" H 3 4 5 6
U V
W X
Y Z
r"
P
~::J G
A I, R A 3,R
I~
6,U,::JE
(F ::J G)
'
p
::J
G
5-V, ::JI
F" G G AI G (H ::J C)
::J
(C A I)
4, W ,::JE X,AE G
3-Y, ::JI
The pro blem of obtaining 'H' is easily solved. A glance at what is available shows that 'H' is available from 'F ::J H' on line 2. To obtain 'H' from 'F ::J H' one needs 'F', and 'F' is the formula which was introduced as an
108
S3.18: MORE DERIVATION CONSTRUcnON
assumption at lin~ 5. All that is needed is to reiterate 'F :) H' Onto th(' present scope line and then use :::IE. The derivati on is completed in this way:
1 IF ::> GI ::> (G 2 F ::> H H ::> G 3 4
S
6 7
8 9
10 1I 12
13
A
(F ::::> G ) ::. (G
F H ::> G F ::> H H G F ::> G G A I G IH ::> G) ::> G
P P A
II
A
I)
I, R A
3, R l,R 5 , 7, =>E
6, 8, ::J E 5-9, :>1
4, 10, :::,E 11, AE
3-12, :>1
The genual strategy o f trying an elimination rule bdore trying an introduction rule usually results in a shorter derivation. This is not always SOl however, and the re is often a good reason [Q stan a derivation with an attempt to use an introduction rule . There are tWO perfectly sensible strategies in trying to obtain the conclusion from the premisses of the foll owing argument: M:::>G
N ::> G M v N S ::> G It does not take much logical insight to see that the disjunctive premiss is essential. Consequently, somewhere in the course of the derivation, o ne will have to use vE on the third premiss. A perfectly sensible approach is to start by assuming one of the disjuncts and obtaining the conclusion. One can the n assume [he other disjunct and obtain the conclusion. T his approach is taken in the following derivati on:
S3.18: MORE DERIVATIO:\, CONSTRUCfION
109
:\1:=lG
p
2 N::>G 3 M v N
p
1
4
~
w
i5
X
~
P A
:=l G
is:=lG Y Z S ::> G
A
3, 4-W, X-Y, vE
The twO problems of obtaining '5 ::> G' at li ne Wand at line Yare very similar. In neither case can '5 :=l G' be reiterated or obtained by use of an elimination rule. It foll ows that, in both cases, an introduction rule must be used. The sketch of the derivati on shows the strategy: 1 M:=lG
2 3 4
5 T
U V
W X y
Z
N::>G MvN
f~
i
iG S ::> G
f~
i
iG 5 ::> G S ::> G
P P P A
A
5-T, ::>1
A A
W-X , ::>1
3, 4-U, V-Y, v E
As might be expected, the twO new subderivations are also very similar. In both cases one needs to get 'G' under the assumption '5 '. In neither case can one reiterate 'G', but in both cases a strategy with an elimination rule is evident. In the first new subderivation, one can reiterate 'M' and 'M :=l G'. In the second new subderivation, one can reiterate 'N' and 'N ::> G'. The derivation is completed according to this plan:
110
I
SJ.18: MORE DERIVATION CONSTRUCfION
p p
M ::::>G
N::>G 3 M v N 2
M
4
6
7
8 9 10 II 12 14 15 16
A
~
5
13
P A
::>G
I, R 4,R 6,7, =:IE
S ::> G N
5-8, :)1 A A
G G::> G S ::> G S ::> G
2, R lO,R
12, 13,:::;IE 11- 14, ::JI 3, 4-9,10--15, vE
Since the main operator of'S::::> C' is a horseshoe, another approach to the problem would be to start by trying to usc horseshoe introduction. The der ivation would then be developed as foll ows: I
2 3
M=>G N::>G .\1 v N
4
Y Z
i
~
P P P A
ic
S => G
4-Y. :::JI
The problcm of obta ining 'G' at line Y is similar to the problem of obtaining 'G' in the previous derivation. One cannot obtain what one needs either fr om 'M ::::> C' or fro m 'N => G'. Thus, one needs to usc vE on 'M v N' once this has been reiterated:
III
53.18: MORE DERIVATION CONSTRUCTION
M=>G 2 N=>G
p
3 M vN
P A
I
4
5
~
p
vN
3, R
:
~
A
V
I
iG
W
I
~
6
iG
X
y
A
iG
5, 6- V, W-X, 'IE 4-Y, ::II
Z S::I G
Obtaining 'G' in the twO subderivations is a simple matte r. 'G' can be derived under the assumption 'M' by reiterating 'M ::I C' and 'G' can be derived under the assumption 'N' by reiterating 'N ~ G'. The tWO reiteratio ns and twO applications of :JE are enough to complete the derivation:
M=>G 2 N=>G 3 M vN
P P P
4
S
A
5
MvN
3, R
6 7 8
~ => G
9
~ => G
10 11
12 G 13 S => G
A
" R 6, 7,
~E
A
2,R 9, 10, ::IE 5,6-8,9-11 , v E 4-12, ::11
An imponam thing to remember in constructing derivations is that one should never do anything without some strategy in mind. In particula r, one should never imroduce an assumption without having a clear idea of how the new scope line which is introduced with the assumption is going to be ended.
S3. 19:
112
EXERCISES
53. 19: EXERCISES for each of the followi ng arguments, derive the conclusion from the
premiss(cs). - (A v HI - A
2 - AA - H - (A A HI
3 F D
~
(E
4 A ~ (8 - C
:J
F)
A
C)
-A
5
J =>
(K => LI
J => J =>
K L
6 K ::> L (M
::J
K)
(M
::::>
::l
7 (S => T I ::> T
T.U (S::::> U)
:::>
U
8 A ::::> B
H ::> C
C=> A
9 A
v
D
(- H A - C) • D H ::> - (C ::> AI
- H 10 K => (L ::> MI
(K " L)
::l
M
L)
S3.20:
113
INOIRECf PROOF
I1N::JO
O.p N·S 5 :> P
12
U v K) ::J L
J :>
L
13 (e::J 0 ) v (e ::J E)
C
D vE 14 ( - F v G) :> (1 :> H ) F:> G
F
:>
(1 :> H)
15 A. B A
v
B
A
A
B
S3.20: INDIRECf PROOF The co nclusion of the following argument can be obtai ned from the premisses; F ::J-G
G - F
The outl ine of the derivation looks like th is:
~ I ~ ::J Z
i-
-G
P P
F
'- F' cannot be o btained by reiteration or by an elimination rule, so one must try - I, the appropriate introduction rule:
114
S 3.20: INDIRECT PROOF
p
1 F :::)- G 2 G 3
P
~
;
X I y
Z
A
I-
i-
4>
F
3, X, Y,-I
The problem now is to find a pair of contradictory formulas. Any pair whatever would serve. ' H ' and '- H' are a contradictory pair, but since neither appears anywhe re in the derivation, it would be ridiculous to look for such a pair. A more promising approach would be to look for a fo rmula which is readily available and its negation. The formulas which afe readily available are ' F ::::> - C', 'G' and ' F', 'F' and '- F' wo uld be a contradictory pair, but since '- F' is what one is looking for, th is option is the least promising. 'G ' and '- C ' are a contradictory pair. 'G' can be reiterated from line 2. ' - C' can be obtained fr om 'F :> - C· provided one has 'F', and 'F' is on the present scope line. The most sensible strategy, then, is to complete the derivation us ing 'G' and '- G' as the contradictory pair:
2 3 4
5 6 7
F :>-G G
~:>
P P A
2,R
-G
-G
- F
I, R 3,5, =:IE
3, 4,6, -I
The problem of obtaining the conclusion of the fo llowing a rgument is a little more complicated :
-IF:>G)
-G As before , the oudine of the derivation is started: 1 I - IF :> G )
p
Z ;- G The only way to obtain' - G' is by -I, so one should ass ume 'G' and look for a contradiction :
11 5
S3.20: INDIREcr PROOF
21- IF "
G)
~
X I Y Z
I'
I' A
;
i-
¢
2, X, Y,-I
- G
'G' and '- c.' arc a contradictory pai r, hilt the problem is the same as that enco untered in (he previous example; one member o f this pair ;s what one is now (Tying to obtam. A bener bet would be [0 try to get some other contradictory pair. one member of which is readily available _ ' F :J G' and '- (F :J G )' 3fe such a pair. One shou ld reiterate' - (F :J G )' and try to get ' F :J G'. The sketch of the derivation shows this strategy: I
-IF"G )
p
2 3
~IF
A
Y
Z
i
" G)
i F::JC -G
1, R
2,3, Y, - I
'F ::> C' cannot be reiterated and it cannot be obtained by an elimination rule, so one must use ::>1. One assumes 'F' and tries to o btain 'G '. Th is is easi ly don e, since 'G' ca n be reite rated from line 2. The complete derivation follows: 1
2 3 4
5 6 7
- IF " G) G - IF " G)
fZ F"G -G
P A I,
R
A 2, R 4-5, :JI 2,3,6, - I
Pro ficienc y in finding contradictory pairs increases with practice. This skill is extremely useful in constructing derivations whenever one needs a formula whose main operator is a tilde. T he fact that every formu la is truthfunctionally equivalent to the negation of its negation, that '¢' is tfuthfunctionally equivalent to ' - - ¢ ', suggests that negation introduction might be useful in some less obvious ways. One o f the most powerful strategies in constructing derivations is indirect proof. If onc nceds a formula '¢I', and all the standard ways o f obtaining it fail, it is always a possible strategy to try to obtain ' - - ¢ ' and usc -E to
116
\3.20: INDIRECf PROOF
get '¢l'. Since the main operator of '- - 4>' is a tilde, the indirect proof strategy first uses - I and then uses -E. To obtain an unnegated formula '¢' by an indirect proof, one should assume '- ct>' and try to obtain a contradictory pair of formulas. Any contradiction allows one to obtain '- - ct>' by -I. It is then a simple matter to use -E in order to obtain '1%>' . If
the formula one needs is a negated formula '- 41', one should of course el iminatc d1C -E step by assuming '¢I' rather than' - - ¢I' . In such a case, the negated for mula is obtained by -I directly and indirect proof is not used. Deriving the conclusion of the following argument is a good example of the use of indirect proof: - (M v N) == 0
M " - p
o "
p
The obvious suategy is to obtain '0 => P' by :::>1, and the derivation is started accordingly: 1
- (~ v N)
2 M 3 Y
z
1\
ii!
0
P
-
~
P P
A
ip
io
=> P
3-Y, :::>1
'P' canno t be reiterated and it cannot be obtained by an elimination rule. Since 'P' is an atomic formula, no introduction rule can work. The only possible strategy is indi rect proof. One should assume '- P' and look for a contradictory pair: 1
- 1M
2 M 3 4
V
1\
v -
p P
P
r~p
A A
i
I
i-
W
X y
NI _ 0
- - p p
Z O"p
4>
4, V, W, - I X.-E 3-Y, :::>1
There are no obvious candidates for the contradictory pair of formu las to replace '4>' and '- 4>'. One could start by loo king for an atomic form ula and
S3.20: INDIRECT PROOF
117
its negation. It is pointless to look for 'P' and '- P' since o btain ing ' P' is the whole point of looking for a contradictOry pai r of formulas. Neither 'N' no r ' - N' appears as a compkte formula anywhere in the derivation , and there seems to be no easy way to obtain eith er. '0' and ' M' are easily obta ined, but the only way to obtain either '- 0 ' o r '- M' is by an indirect proof. Obtaining either fo rmula in this way would require fi nding a contradictory pair of formulas, and this is exactly the problem bei ng add ressed. Accordingly, one must consider more complex formulas. One could reite rate the formula on line 1 and try to obtain '- (- (M v N ) '" 0)'. The only way to obtain this for mula is by -I, and the prob lem of finding a contradictory pair of formulas seems just as baffling as it did when considering atomic formulas. One could reiterate the fo rmula on li ne 2 and try to obtain '- (M 1\ - P)', but once aga in the problem of finding a contradictory pair of formulas seems insurmou ntable. Using part of '- (M v N ) '" 0 ' seems to be th e only strategy remaining. If one reiterated '- (M v N ) '" 0' and '0', one could obtain '- (M v N )'. One would then need 'M v N' to have a con tradicto ry pai r. This loo ks far more promising. 'M v N' is a dis junctive formula which could be obtained by vi if o ne had either 'M' or 'N'. It is worth amending the outline of the derivation to pursue th is strategy: 1
2 3
- 1M v N) M A - P 0
4 5 6 7
P P A
; (M ~P
3, R
, - 1M v N )
5, 6,,,,E
v N ) ", 0
A I, R
iM v N
W X Y
Z
=0
o
- - P P => P
4,7, W, -I
X, - E
3-Y, ::ll
To obtain 'M v N' by vI at lin e W, one needs either ' M' or 'N'. The re is no obvious way to obtain 'N', but 'M' can be obtained from ' M 1\ - P' by AE when 'M A - P' is reiterated from line 2. The derivation can now be completed:
118
S3.20: INDIRECT PROOF
1 - 1M v N I 2 M , - p
E
0
0
3 4 5
- P
P p
A A
- (M v N) . O
1, R
h
0
3, R
7
- (M v N)
5,6.=£
M " - P
2, R 8,I\E
8
M M vN - - P P 0 => P
9 10 11 12 13
9, v i
4,7,10, -I 11, -E 3-12, ~ I
Indirect proof is very I.lsd ul wht=n one needs a disjunctive formula but cannot obtain either of the disjuncts. Fo r example , since '- (- A 1\ - 8 J'is truth-functionally eq uivalent to 'A v B', it should be possible [0 derive the latter from the fo rmer:
1
! - (-
z iA
A , - 81
p
v 8
It is impossible to derive: either ' A' or '8' from (- A 1\ - BJ', so v i cannot be used to obtain ' A v S' . Accordingly, one must use indirect proof. One assumes' - fA v 8)' and looks for a contradiction :
1-
1 2 :
1- A , - 81 I-=-I A v BI
p A
W i
i4>
X Y
i-
- - (A v 81
2, W, X,-I
Z
A v B
Y,-E
In considering what contradictory pair of formulas one could use instead of '4>' and '- 4>', one must consider whal formulas one cail get Ollto the Cutrt llt scope line. In the presem example, no atomic formulas can be obtained, and the only formulas available are the formulas on lines 1 and 2. The choices seem to be either the formula on line 1 and its negation or the formula o n line 2 and its negation.
119
\3.20: 1l'.'DIRECT PROOF
One possible strategy would be to look for the negation of the formula on line 2. Then one would want either ' A v S' or '- - (A v B)'. There is no way of obtaining either formula except by indirect proof, and th is would involve assuming '- (A v 5)' and looking for a con trad iction. The sketch of [he derivation is appropriately amended: B)
P
2 3
f-=- (A v B) : I-=- (A v
A
V
I
1
W X
- (- A
I-
A
-
B)
A
i
i-
¢l
- (A v B)
Y - - (A Z A v B
3, V, W,-I 2,2, X,-I
v B)
Y,-E
A glance at this outline is sufficient to show that the present strategy is hopeless. If one could obtain a contradiction on the scope line which begins at line 3, one could obtain exactly the same contrad iction on the scope line which begins at line 2. Clearly, some o ther contradictory pair must be considered. The o nly other pair which looks at all promising is the fo rmula on line 1 and '- A " - B'. Accordingly, the mistaken step is deleted from the sketch of the derivation and the new strategy is outlined:
1
- (- A
A
B)
-
P
2 3
f-:- (A v B)
A
, - (-A " - B)
1, R
X
i -A
A
-
Y - - (A v B) Z A v B
B
2,3, X,-I
Y,-E
The obvious way to obtain '- A " - S' is by " I from ' - A' and ,_ Bt. Neither of these fo rmul as ca n be reiterated or obtai ned by an elimination rule, so in both cases one should try the appropriate introduction rule. One should assume 'A' and try to find a contradiction, and assume '8' and uy to find a contradiction. The contradiction under the assumption 'B' need not be the same as the contradiction under the assumption 'A', but of course it does not matter if it is. In the expanded sketch of the derivation, one contradictory pair is shown as 'cpt and '- 41' and the other contradictory pair is shown as ''V' and ' - 'V', leaving open the possi bility t hat 'eI>' and ''V' may turn out to be identical formulas.
S3.20;
120
I
2 3 4
Q R 5
I-
I NDlRECf
P
- (- A A - 8) (A v 8) - (- A A - 8)
: 0-
A
I. R A
! !$ i-
¢
-A
4.Q. R.-I
T
l!!-
A
V
!IV
?
V
i -IV
W
- 8
T, U, V,-I
X
-AA-B
S, W, ....1
2,3, X,-I
Y - - (A v 8) Z Th~
PROOF
Y.-E
A v 8
usefulness of v I is shown again at this point. One needs a contradictory
pair under the assumption' A' and a contradictory pair under the assumption 'B'. Since '- (A v 8)' can be reiterated, it is easy (0 obtain the required contradictions. One can get 'A v S' from 'A' by vi and one ca n get 'A v B' from'S' by vI. The derivation can now be completed:
I 2 3 4
- (A v 8) - (- A
A -
8)
I. R A
A v 8
4, vi
- (A v 8)
2. R 4,5 ,6, -I
-A
~
8 9 10 II 12
1\
~
5 6 7
P
- (- A A - 8)
A
A v8
8, vi
- (A v 8 )
2.R 8,9, 10,-1 7,11,AI
- 8
-AA -8
13
- - (A v 8 )
2,3,12. - I
14
A v 8
13, -E
53.21, EXERCISES
121
It should be clea r that if one contrad ictory pa ir of formulas can be obtained on a given scope line, any con tradictory pair can be obtained si m ply by using indirect proof to obtain each member of the pair. However, using indirect proof twice to obtain an a rb itra r ily chosen contradictory pair is a poor strategy. When onc needs a contrad ictory pair of form ulas, onc shou ld spend some time in exa min ing w hat is easily available before atte mpting [Q o btain a particular pair. In producing derivations, it is not a good idea (Q try indi rect proof too 500n. In general, the fo llo wing procedure will result in the shortest derivation. At each stage o f a deri vation, when onc needs a fo rmula '$', one shou ld go through these steps in order;
t See if o ne ca n reiterate '$'. 2 See if one can usc an elimination rule to obta in '4>' . In most cases, if one has a disjunctive formula available, one should first fry v E.
3 Sec if o ne can use an introduction rule to obtain
'¢' .
4 See if one can obtain '4>' by an indirect proof.
S3.21:
EXERCISES
For each of the following arguments, derive the concl us ion from (he prcmiss(es).
F G
:J -
(G v H )
- F 2 M
-- -:M -:-"----=-O 3 M :> (N :> S) M ::::> N
-S::::>- M 4 F ::::> (G ::::> H ) - H
- IF
A
G)
5 P " IQ A - R ) P " (S :> R) - 5
S3.2 1;
122 6 F :::::l (G
A
H)
(- G
-
H)
v
F
:::::l -
7 F v G
Fv - G F
8 A ::> IB v C)
-c A
B
::>
9 (A v B)
:::::l
C
(- C
D)
1\
v
A
o 10 - li P v QI -S::>- Q
A
IR
V
- Q 11 -A::J- H
A => G - IG v F)
- Bv F 12A (B ::> C) ::> 0 (C
Al
::::>
:::J -
D
B
13 - P
Q
V
IS ::> PI ::> IS ::> QI
14 IA
V
BI ::> IC
C:::>-E D :::::l - A
A
A
- IC
E V
01
15 M ::::> R S::>K - S => M - R :::J K
V
01
S))
EXERCISES
123
§3.22: CATEGORICAL D ERrvATIONS
16 - IA
v
B)
A·B 17K
- Q v - S (- K v S) 1\ Q
-Q 18 IR v S) :> F
- IF A - G) - R::::o M
-G M t9 -H I\G
F
:>
IIG
E
A
F
iii
(S v T )
20 R
A
E)
:>
IH
v
I))
- S T=R
S3.22: CATEGORICAL DERIVATIONS The most common son of derivation starts with the premiss or premisses of an argument and ends with the conclusion. However, it is possible to derive certa in formulas given no premisses whatever. A derivation o f this sort is a categorical derivation. A fo rmula which is truth-functionally true does not depend on the truth of any other formula . In other words, given any information whatever, or no information at all, a truth-functiona lly true formula is bound to be true. Thus, it is possible to construct derivations having such fo rmulas as their final ste ps without any premisses at the beginni ng. Since o ne cannot appeal to a premiss for information in a categorical derivation, one cannot immediately use an eli mination rule because the re is no operator to eliminate. Every ca tegor ical derivation, therefore, must be based on some introduction ru le. A few examples will make this clear. One of the most obvious examples of a truth-functionall y true formula is 'A ::::0 A'. The derivation is very simple:
II
2
~
A:> A
A 1-1, ::::01
S3.22:
124
C ATEGORICAL DERIVATIONS
The derivation of 'A ;;; A' is equally simpk: t 2
I ~
A
A. A
1-1 , I -I , =1
Another example of a truth-functionally tru e formula is ' A
:J
(8
::l
A )'.
One ca n use :::11 to obta in it: t
2 3
4 5
r~
A A
t, R 2-3, ::>1 1-4, ::II
A ::> 18 ::> AI
Th e method of indi rect proof is very useful in constructing categorical deriva tions. It is used here to derive 'A v - A':
A A
- IA v - AI 2
3 4
5
~
Av - A - IA v - AI
2. vI t, R 2,3, 4,-1
-A
7
Av -A - - (A v - AI
5, v I 1, 1,6,-1
8
A v - A
7,-E
6
T he categorical derivation o f '- (F functiona ll y false;
t 2
3 4
~A
-
F
- F - IF A - FI
1\
- Fl' shows that ' F
A
A AE 1, AE 1,2,3,-1 1,
- F' is t rut h-
125
§3.22: CATEGORICAL D ERIVATIONS
Categorical derivations are often used to demonstrate truth·functional equivalences. In each of the foll owing three derivations, an intuitively clea r truth· func tional equivalence is shown: A v B
2 3 4 5
6 7
8
~vA
ff
v A
B v A B v A
f-i
v B 9 10 B 11 AvB 12 13 (A v B) iI!i (B v A )
~v
1
2
r:A B
3 BAA
5
~A A
7 8
9
2, vi A 4, vi 1,2- 3, 4-5, vE A A 8, vi A 10, vi 7,8-9,10-11 , vE 1"";,7-12, . 1 A
4
6
A A
AA B (A A B) E! (8 A A )
1, /I.E 1, AE 2,3, AI A 5,AE 5,AE 6,7, AI 1-4,5-8 , ::1
S3. 23:
126 1 2 3 4 5 6 7
A => B - B
~
A A A I, R => B
- A -B => -A
9
-B:J-A
2,R 3,5, 6, - I 2-7, :JI
A A A 9, R 11 , 12 , :JE
A
10
15 16 17 18
3, 4, :JE
-B
8
11 12 13 14
~B
- B => -A - A A - - B B A => B (A :::> B) B ::l - AI ;Ii
10, R
11 ,13, 14, - 1 IS, -E 10-16, ::II
1-8,9-17, =1
(-
S3.23:
EXERCISES
Provide categorical derivations of the following formulas. M . (M
A
MI
2 A v (- A v BJ
3 L . (L
v
LI
4 (R => SI => (R => (R
A
SII
5 ((F => GI => RI => (G => RI 6 « R :J S) 1\ - S) ::::> - R 7 - F => (F => G) 8 (A v 81 => (( A => 8 1 => BI 9 (M => NI => (- M
v
NI
10 - ((F A GI A ( - F v - Gil 11 (- M v NI => (M => NI 12 G v (G
13 (F
::::>
v
H)
G) v (G
14 (P => (Q 15 (( P
:::>
A
EXERCISES
:::>
F)
RII => ((- Q
v _
RI => _ PI
QI v RI • (P v (Q v RII
S3.24:
127
SUMMARY OF TOPICS
S3 .24: SUMMARY OF TOPICS With a few possible exceptions to make the summary cleare r, the topics appear in the order in which they appear in the text. Derivation (Sec p. 55): A nu mbered se quence of formu las, each occu rrence of a formula bei ng justified. Scope line (See p. 55): A vertical line which shows the extent ('scope') of a derivation or subderivation. Primary scope line (Sec p. 56): T he scope line wh ich is furthest a derivation. See Secondary scope line.
to
the left in
Introduction rule (See p. 56): For eve ry operalOr, there is an introducti on rule which se rves to create a formu la which has th:H o perator as its main operator. Eliminarion rule (See p. 56): For every ope rato r, there is an el imination rule which uses the main operator o f a given formula to create a differem formula. Conjunction introduction (See p. 57): A rule, " I, which serves to create a formula which has a ca rel as its main operator. Every usc of conj unction introduction req uires an appeal to exactly twO lines. The gem:ral pattern of " I fo llows:
m '" n
ljI
m, n, " I If 'e!>' and ''V' arc not distinct, if o ne is creating a for mula such as 'e!> " e!>', the twO li nes which arc appealed to would be idemical. The appeal would loo k like 'm, m, "I'.
S3.2 4:
128
SUMMAR Y OF TOPICS
Conjunction elimination (See p. 59): A rule. I\E, which ser ves to obtain onc of the conjuners from a formula which has a caret as its main operato r. Every use of conj unction elimination requires an appeal to e xactly onc li ne. The genera l patte rn of I\E follows:
m 1 n
A II'
i
m, I\E
0'
m n
1 A II'
i ll'
m, /\[
Negation elimination (See p. 61 ): A rule. -E, which allows the removal of tW O consecutive tildes when the first o f these tildes is the main ope rator. Every use of negation elimination requires an appeal to exacdy onc line. The genera l pattern of -E fo llows: m l - - ¢l
m,-E Disjunction introduction (See p. 62): A rule, vI, which ser ves to create a formula which has a vel as its main o perator. Every use of disjunction in troducti on requires an appeal to exactly one line. The general panern of v I follows:
m l m, v I m
ml n l lV v ¢,
m, vI
S3.24: SUMMARY OF TOPICS
129
Horseshoe elimination (See p. 64 ): A rule. =:lE, which serves to obtain the consequent of a formula which has a horseshoe as its mai n operator. Every usc of horseshoe elimination requires an appea l to exactly twO lines. The gene ral pancrn of::::>E foll ows: m 1$::::> IV n j$ o
i lV
ITI,
n, =.,E
Triplebar eliminatiOn (See p. 66): A rule, :E, whidl ServeS 10 ublain Olle uf the formu las flanking the '::' in a formula which has a triple bar as its main o perator. Every use of triple bar elimination requires an appeal to exactly two lines. The general pattern of ::E fo llows:
m, n,=E 0'
m, n,EE Secondary scope line (See p. 80): A scope line which serves to keep informati on which depends on an assumption away from the scope lines to the left of the seconda ry scope line. Assumption (See p. 80): A formula which is assumed in order to produce another formula. Every assumption begins a new secondary scope line. Subderh·ation (See p. 81): A derivation on a secondary scope line. Reiteration (See p. 82): A rule which allows one to copy a form ula from a scope line which has not ended ontO a secondary scope line.
S3.24:
130
SUMMARY OF TOPICS
Horseshoe introduction (Sec p . 84 ): A ruk, ::lI t which serves to create a formula which has a horseshoe as its main operator. Every use o f horseshoe introduction requires an appeal to exactly onc range of lines. The gene ral pattern of ::Ji foll ows: A
ml
n
i m-n, =:.1
If ' ¢I' and '\II ' are not distinct, if onc is creating a formula such as '41 :) ' , t he range ca n consist of a single line. The appeal would look like 'm-m, :;}I '. Triplebar introduction (Sec p. 88 ): A rule, . 1, which serves [0 create a for mula which has a triplebar as its main o perator. Every use of tri ple bar introd uction requires an appeal to exactly two ranges of lines. The general pancrn o f =1 fo llows: m
(.!
n
i ljl
0
~
P
i
q
A
A
. 1jI
m-n, o-p, =1
If '¢I' and -'II' are not distinct, if one is creating a formu la such as '¢l ;;: ¢l', [h ~ twO rang~s lOg~[h ~ r can consist of a singl~ lin~ . Th ~ app~al would look
lik~
'm-m, m-m, !!!!I'.
131
S3.24 : SUMMARY OF TOPICS
Negation introduction (See p. 91): A rule, - I, which serves to create a formula which has a tilde as its main o perator. Every usc of negation introduction requires an appeal to exactly th ree lines, an assumption and a contradictory pair of fo rmulas on the scope line which begins with the assumption. The general pattern of - I foll o ws:
m!
~
n!
! IV
0 1
1 -\1'
P i - ' and ' \If' do not have to be distinct. Categorical deri vation (See p. 123 ): A categorical derivation is a derivation which does not begin with premisses. Ca tegorical d(:rivations in sentmce logic can b(: used to derive truth-functional truths. Every categorical derivation in sentence logic begins with an assumption. A simple categorical derivation fo llows: A
1, vi o
i ell :J
(ell v 'V)
I-n, :JI
CHAPTER FOUR
PREDICATE LOGIC
S4.1: INDIVIDUALS AND PREDICATES There are some severe limitations to sentence logic. This can be seen from consideration of the following argument: Since all Frenchmen arc:: Europeans a nd Pierre is a Frenchman, Pierre is a European.
Th e argument is obviously valid in some sense; if the premisses are tfue, the conclusion is bound to be tfue. However, when one represents each sentence by a sentence lener, all appearance of validity vanishes: F: All Frenchme n are Europeans. P: Pierre is a Frenchman. E: Pierre is a European. F
p E
The argument as represented is trurh -functionally inva lid. The difficulty is that the original argument in Engl ish can be seen ( 0 he valid because of the internal structure of the sentences rather than because of (ruth-functional
133
134
§4.1: INDIVIDUALS AND PREDICATES
connections between sentences. Predicate logic is a way of deal ing with such internal structure. Ordinary English grammar texts make a distinction between the subject of a sentence and the predicate. In the sentence 'The book is expensive', 'the book' is the subject and 'is expensive' is the predicate. This simple example is sufficient to imroduce twO logica l (erms: 'th e book' refers to an individual and 'is expensive' is a predicate. An individual ca n be a person, a place, an idea or collection - virtually anything about which onc can say something. Although mass nQuns such as 'silver' and 'coal' Ojj '. Wh en o ne uses the five sentence operators in predicate logic, it is important to bea r in mind that these operators must be used only with sente nces. 'John does not despise Mary' is properly translated as follows: j: John
m: Mary 0_ 1 _ 2: - I despises - 2 - Di m Both ' 0 - jm ' ttt and ' OJ - m' ttt are mista ken because the tilde does nOt negate a sentence . Us ing the same symbolization, ' Mary desp ises Joh n only if Jo hn despises Mary' can be translated as ' Omj ::::> Oim'. However, ' Omj ::::> jm' ttt is not a well-formed fo rmu la .
S4.3: CHOICES IN lRANSLATION In predicate logic, to a much greater extent than in sentence logic, there are choices to be made in translation. Th ere arc many ways in which the sentence 'John sent the parcel to Kate' can be translated, including the follo wing seven: S: John sent the parcel to Kate. S 2 j: John S _: _ sent the parcel to Kate Sj 3 k: Kate S _: _ was sent the parcel by John Sk 4 p: the parcel S _ : Jo hn sent _ to Kate Sp 5 j. John k: Kate 5_ 1 _
Sjk
2: _ \
scnt the parcel
[Q- 2
S4.4:
138
-
EXER CISES
- -- - - - ----6 j: John k: Kate p: the parcel
5_ 1 _
2_
3: _
\
sent
-2
[Q _ )
Sjpk
7 j: John k: Kate p: the parcel
S-I- z-f _l wassemto_ l bY _J Spkj
As a genera l principle, translations should be only as complex as is required in the context of argumentation and deriva tion. Without some knowl edge of argumems in predicate logic (which wi ll be discussed later in this chapter) and some knowledge o f derivations in pred icate logic (which will be discussed in Chapter 5), onc often has no adequate basis for deciding how comple x a trans[a lion is nceded. However, (here is no need fo r excessive caution. For example, if one needs to translate 'Catherine loves Michael and Catherine is loved by Peter', onc should nOl use two predicates, one for 'loves' and onc for 'is loved by'. Sim ple considerations of cconom y suggest that one should use a single predicate, either 'loves' or 'is IO\'ed by '. Moreover, the English sentence asserts that the same relation which holds between Catherine and Michael also holds between Peter a nd Catherine, and the translation should reflect this.
S4.4, EXERCISES Translate each o f the following sentences into symbols using the given predicates and constants. c: Carl p: Peter f: Frank A_ I _ z: D_ I _ 2_
-
I 3:
annoyed_ z discussed _zwit h _
-I
3
1 Peter discussed Carl with Frank. 2 Peter was discussed by Frank with Carl. 3 Carl discussed frank with Peter. 4 Carl annoyed Frank jf he discussed Frank with Peter. 5 Carl discussed himself with Frank. 6 Carl discussed Peter with Frank but did not d iscuss Frank w ith Peter. 7 Carl discussed himself with Frank but not with Pe ter.
5 4.5:
1]9
Q UA:-JT1I'If.H.S
8 If Frank discussed himself with Peter, then Peter discussed Frank with Carl. 9 If Frank discussed Peter with Car! and Carl discussed fr :lI1k with Peter, then Frank di sc ussed Carl with Peter. 10 Carl discussed himself w ith Frank and annoyed him. 11 Pete r disc ussed himself with neithe r fr ank nor Carl. 12 Peter did not d iscuss frank with Carl unless Peter discussed himsel f with Fran k. 13 Peter discussed himself with neither Fran k nor Carl and annoyed both of them. 14 Assuming (hat Peter discussed Frank with C:HI, then neither Fran k nor Carl was annoyed by Peter. 15 Fra nk discussed himself with Carl only if either Pner or Carl discussed himself with Frank.
54.5: QUANTIFIERS In S4.1 it was poi nted o ut that using a sentence letter to translate ' All Frenchmen arc Europeans' made it impossible to StaTe a n ohviously vali d argument in terms of sentence logic in a wa y which ma intains any appearance of validity. It is possible in principl e to usc sentence logic and still avoid the problem. There are a fi nite number of Frenchmen, an d in princip le it is possible to state that all French men a rc Europeans by using a fini te number o f se ntences . I.etting ' FI ' represent 'French man nu mber o ne is a Euro pean', ' F2' represent ' French man numher two is a Euro pean' and so on, a conjunction of several million sentences wi ll sen ·e . Pla inly, no human being could enumerate all Frenchmen in this way and no human being cou ld write out suc h a conjunction, but in principle it is possible to do wit hout the word 'all' in this case. In much the same way. as long as there are a finire numbe r o f drunken fools, 'Some drunken fool is shouting in the street' can be thought o f as a very long disjunction: '({ D I v D 2) v Dj ) v ... Dn'. In principle, it is possihle to express 'some' in terms of a disjunction, but of cou rse in pr:Kt ice this is o ften imposs ible. It sho uld be remembe red that th e vel expresses the inclusive sense of 'or' and that the long disjunc tion IC:lvr.s open the possibility that mo re than one drunken fool is shou ting in the meet. When one is dea ling with a small number of th ings, It is often easy to express ' all' and 'some' in sentence logic. If I have o nly three sisters, it is casy to use a co njunction to say that all my sisters a re married. Similarly, if I have just fou r cousins it is easy to use a dis junc tion to say that at least one o f m y cousins plays golf. Problems a rise when la rge numbers of things arc involved
140
S 4.6: THE EXISTESTIAL Q UASTIFIER
o r when o ne lacks kno w ledge . r or exa mple, even [hough I may not kno w very much about my local po lice force, I may well want to say tha t a policeman is on rhe scene. Simil arly, I ma y want to say thai none of the men o n (he field are tall, even though I ma y no t know ho w many men there 3rc on the field. As a practical matter, quantifiers, words like ' all', 'some' and 'no', arc indispensable. Aside fro m [he practical need fo r quantifiers, there arc theo retica l reaso ns wh y quantifiers arc needed. Many scientific claims a re concerned with em ities which cannot be enumerated. Much of mathematics involves infinities. One docs nO{ need to be a very so phisticated malhcmatician ro understa nd sc ntcnt.:cs ahou l I II rationa l num bers. a ll mu ltiples o f 3 a nd all odd numbers. Quantifiefs, therefore, a rc indispensable both in natura l languages suc h as English and in the artificial languages which are used in sy mho lic logic.
§ 4.6: THE EX ISTENTIAL QUANTIFIER Using 'b' fOf ' this book' and ' R _' for ' _ is red', one ca n represem the clai m that this book is red by ' Rb'. The clai m that something is red can be represented by '(3x)Rx', which can be read as 'There is at least o ne thing
wh ich is R', or in ra ther stilted English as 'There is an x such that Rx.' The ' x' is not a co nstant because it docs not name a pa rticular indi vid ual ; it is a variable. '(3x)' is an existential quantifier. '(3y)' and '(3;0:)' a re a lso existential qua ntifiers because 'y' and 'z' are also va riables. As with sentence letters and constants, the stock of variables can he increased by use o f subsc ri pts. T h us, ' X l' and '7.3 ' are also variables. Using 'b' for 't hi s book', ' R _ ' fo r ' _ is red' and ' H _ ' fo r ' _ is heavy', o ne ca n represent the claim that this book is red and heavy by ' Rb " H h'. T he claim that something is red and heavy can be represented by '( 3x )(Rx " H x)', which can be rcad as 'There is an x wh ich has properties Rand H ' o r as 'There is a n x such that Rx and Hx.' ' Rb' a nd ' H b' arc sentences and can be joined to fo r m mo re complex sentences such as 'Rb " Hb'. In the formula '(3xH Rx " H x)', 'Rx' and ' Hx' a re joined by a sentence o peraro r, even tho ugh strictly speaking they a rc not sente nces, no mo re than ' R _' and' _ is red' are sentences . Suc h expressions a fe nm sentences because they can be neithe r true no r false . Ho we ve r, it has beco me common to call such formulas 'open sentences'. An open sentence is a fo rm ula with this property: if all the variables in i! were replaced by constants, the resul! would be a 'closed seme nce '. A dosed sentence which is the result of such a replacement is a genuine semence ca pable o f possessi ng a (ruth value. The legi timate use o f sentence o peralOrs has been extended to a llow these o peralOrs to be used with o pe n sentences.
§4.6: THE EXISn:.~TIAL QUA!\''11FIER
141
If '(3x)Rx' represents 'Something is red', '- (3x)Rx' represents 'It is false that something is red', while '(3x)- Rx' represents 'There is something which is not red .' The difference between '- (3x)Rx' and '(3x)- Rx' ca n be seen by considering the real world. '- (3x )Rx' denies that there is something which is red, and that is manifestly false. By COntrast, '(3x)- Rx' asserts that there is something which is not red, and that is manifestly true. The cla im that my dog is as large as Harry's dog can be represented by 'L mh'.1f my dog is as la rge as Harry's dog, something is as la rge as Harry's dog, which can be written as '(3x)Lxh', Similarly, if my dog is as la rge as Harry's dog, my dog is as large as something, which can be written as '(3x)Lmx '. Finally, if my dog is as large as Harry's dog, something is as large as something. Because this claim involves twO 'somethings', twO existential quantifiers and two dist inct var ia bles must be used: '(3x)(3y)Lxy'. If one wrote '(3x)( 3x)Lxx' ttt, there would be no way of telling which qua ntifi er goes with which variable. The following twO claims have cumpletely difftrtm mea nings, Tin: first is a simpl e conjunction, whil e the second asserts that one and the same thing is both red and warm. 1 Something is red and something is warm. 2 Something is red and warm. The former sentence can be expressed as '(3x)Rx 1\ (3x)Wx' o r '(3y)Ry J\ (3y)Wy ', while the latter sentence can be expressed as '(3x)(Rx 1\ Wx ), or ' (3z)( Rz 1\ Wz)'. Since 1 is a conjunction of two quantified formulas, each formula being capable of standing alone, it can also be expressed as '{3x )Rx 1\ (3y)Wy', The claim that something is red and something is warm neither asse rts nor denies that the red thing is identical to the warm thing, and this is reflected in every translation of 1 which has been given. Assertions of identity and non-identity will be discussed in S4.12. To see why there is nothing to choose between '( 3x )Rx " (3x)Wx' and '(3x) Rx J\ (3 y)Wy', it is necessa ry to learn about the 'scope' of quantifiers. The scope of a quantifier determi nes what is governed by Ihe quantifier. If one remembers how brackets extend what is negated by [he tilde, it is a simple matter to determine the scope of a quantifier, since the twO situations arc analogous. If a quantifier is followed by 3n unbracke ted open sentence - something which would be a WFF if the variables were replaced by constants - the scope of the quantifier is that unbracketed open sentence. 2 If a quallt ificr is folluwed by a left bracket, [he scope o f (he quantifier is that left bracket, its corresponding right bracket and what is between those twO brac kets.
142
§4.7: T ilE UNIVERSAL Q UA\Tl fIER
3 If a quantifier is followed by a tilde, the scope of the quantifier is the ti lde and whatever the tilde negates. There is no need to make an cHon to memorize these four points. Working through the examples in the next few sections will lead to an understanding of the scope conventions.
§4.7: THE UN IVERSAL QUANTIFIER One way to assert the manifestly false claim that everyth ing is liquid is ro say {hat it is false that something is not liquid. One way to assert the manifestly false claim that eve rything is non-liquid ;s by saying Ihat it is false that
something is liquid. Similarly, one can deny that everything is liquid by saying that something is no n-liquid, and one can deny that everything is non-liquid by saying that somcrhing;s liquid , T hus, in Engl ish one could get by without terms such as 'everything', and use negations and terms like 'something', T hIS wou ld be cumbersome in English a nd it w ou ld be cumbersome in rhe symhnlic language. The rlifficul ry is avoided by having a universal quantifi er, Using 'R ~ , to represent ' ~ is red ', one can represent 'Everything is red' by ' ('' xHOx :::J - Rx )'. Although in English a single word distinguishes 'Some dogs are brown' from 'All dogs are brown', th e symbolic representation requ ires a change in the operator as well as a change in the quantifier. 'Some dogs are brown' asserts tha t there is at least one thing which is both a dog and brown and is prope rl y translated as '( 3xH Dx" Bx)'. It should be noted that '( 3x HD x :::J Bx ), ttt is not a correct translation of this sentence, since it asserts that there is something such that if it is a dog then it is brown. It follows from the truth-table definition of the horseshoe tha t '(3xHDx :::J Bx )' will be true if there is something in the universe which is not a dog. It should also be noted that ' (Vx )( Dx " Bx)' ttt is not a correct translation of 'All dogs are brown', since it asserts that eve ryt hing is both a dog and brown. Unlike the existential quamifier, the universal quantifier docs not assert the existem:e of anything. '(\>'x)( Dx :J Bx )' l1either asserts nor denies the existe nce of either dogs or brown things. It is natura l to wonde r why anyone would have anything serious to say abom non-existem things . In fact, chere are often occasions on which one makes truc claims which IUrn out to be about non-existent emi ties. For example, there cou ld be an award fo r anyone who develops a workable fusi on reactor. One could then make the true claim that anyone with no kno wledge o f science or tec hnology who develops a workable fusion reactor is eligi ble for an award, knowi ng full well that it is extremely un likel y that there is any such person. Altho ugh the uni versally quantified form ula '(\>'x)( Px :::J Qx )' does not assert or deny the existe nce of anything which is P or anything which is Q, one can use the existential quantifier to do this. Thus, the two fo rmulas '(3x)Sx' and '(\>'x)(Sx :::J Dx)' can be used to translate a sentence such as 'There are sharks and all sha rks are dangerous.' Sim ilarl y, the two formulas ' - (3x )Gx' and '(\>' x)(Gx ~ Fx )' can be used to translate a sentence such as 'The re are no ghosts, but all ghosts are frightening.' Now that both quantifiers have been introduced, it is possible to expl ai n what an atomic formula in predicate logic is. An atomic formula in predicate logic is any WFF which contain no operators, neither sentence operators nor quantifiers. Th us, ' Rc', ' Lms' and 'Gksd' are all atomic formulas.
144
)4 .8: USI:-":C BOTH Q UAl'\TIFlERS
S4.8:
USING BOTH QUANTIFIERS
Many English sentences use bOlh the existential and the universal Quanti fie r. An example is ' Everything is larger than something.' This sentence means somethi ng Quite di fferent from 'Something is larger than everything', a nd one must be ca refu l to maintain this difference in translations. 'Everything is larger (han something' asserts that England is la rger than something, that the moon is larger than something and so on. The claim that the moon is larger than something is the claim [hat there is someth ing such that the moon is larger than it. This is easily translated as '(3 x)Lmx', The claim that everything is la rger tha n something, however, is nm a clai m just abo U( the moon . Accordingly, one should usc a universal qua ntifier a nd replace 'm' with a variable which does not al ready appea r in '{3x)Lmx ': '(Vy)(3x)l yx'. h should be noted that the English sentence begins with a universal quantifier, as does its symbolic representation. 'Somet hing is larger than everyt hing' asserts that there is something which is larger than England, larger than the moon a nd so on. 'The moon is larger than everything' is easily translated '(' Lss'. The cla im that anything wh ich is la rger than everyth ing is large r than itself is not a cla im JUSt about Spain. Accordingly, one needs to add a universa l qualltifier. Bedon: doing this, however, Olle: ne:cds to look carefully at the formu la w hich is to be un iversally qua ntifi ed. ' ('9' x )Ls x ::> Lss' is not Strictly speak ing a Wff; before one can change the claim abou t Spain to a claim about everything, one needs to add brackets: '(( V'x )Lsx ::> Lss)'. Once this is do ne, the translation is straightforward; one simply prefaces the formula with a un iversal quantifier using a va riable
146
S4.8: USI:\C BOTII Q UANTIFIERS
which does not already a ppear in '{( V'x)Lsx ~ Lssf, A correct translation. therefore, of 'Anything which is la rge r than everyth ing is larger than itself' is '( \fy)«' Lyy )"
' An ything which is larger than anyth ing is larger than itsel f' also makes a claim ahout cvcrythinp.. It asscns th at if Spa in is larger than POrlugal then it is larger than itself. that if my car is larger t ha n m y briefcase then it is large r than itSelf anJ so On. One 1..:311 tra nslau: 'If Spa in is larger than Ponugal [hen it is larger than itself' as ' Ls p :> Lss', As t he next step in translation, o ne can mo ve to the claim that Spa in is larger than itself if it is larger than anything, ra ther than JUSt larger than Portugal. There a re (wo ways in which this claim can be translated: '(V' xHl.sx =:. Lss )' o r '(3x )Lsx =:. Lss'. 1t should be noted {hat {he ma in opera{Or of the forme r translation is a universal Quantifier, whilt: the lIIail1 opnatur of du.: latter is a horscsh m:. As wi th tht: tra nslation in the previous paragraph, one needs to add a unive rsal quantifie r to make the tr;:mslations rcfb:t the fan that the claim ma kes an assenion about eve ryt hing rather tha n JUSt Spain. In addition, JUSt as with the translati on in the previous paragra ph, one needs to be sure that the re are no bracke ts missi ng. '( V' xHLsx =:. 1.ss)' needs no attention, but brackets must be added to '(3 x)l.sx ~ Lss' before rhe formula ca n be universally qua ntified. Accord ingl)" the semencc, ' Anyt hing which is larger tha n any thing is larger than itself' ca n be correctly translate d as either '(V'y)(V' xIl Lyx J Lyy)' or '(Vy)( (3 xlLyx :l t yy)'. The way in which brac kets ca n alter meaning can be seen fr om consideratio n of {he sc mence 'If John becomes imercsted in anything, Mary will be happy.' One wa y to para phrase this sentence is ' For anyth ing o ne chooses, if John becomes interested in that thing, Ma ry will be happy.' Anot her way to paraphrase th e semenee is 'If there is something in which John becomes interested, then M ary will be happy.' The fi rst paraphrase is a uni versa ll y quamified semencc, while the second is a conditional sentence. T he scmcncc can bc sym bolized in eithcr way, but thc brackcts m ust show whcther the mai n o perator is the exis tentia l q uantifier or the ho rseshoe. The tWO translations of ' If John becomes interested in anything, Mary will be happy' follow: Translation L (V'x)( ljx
:l
Hm )
Translation 2: (3x)ljx :> H m To sec rhe impo rtance of brac ke ting, o ne can consider what wo uld be asserted if one changed the brat.:kets. Translation I would become: Translation L
{V x)ljx :> Hm
Translation l ' says that if John becomes inte rested in everything, then Mary will be happy, and th is is not the meaning of the original sentence . Changing the bra ckets in T ranslation 2 also results in a radically changed meaning:
147
\4.9, EXERCISES
Translation 2': (3x)( Ijx :) Hm ) Translati on 2' says that there is something such that, if John becomes
interested in it, Mary wil l be happy_ As long as there is something in the world in which John does not become interested, this claim will be triviall y
[fue. To see that this is so, suppose that Joh n does not become interested in the history of shoelaces, whic h can be represented by the constant '5', Th en, since ' Ijs' is fa lse, it follows (rom the definition of the horseshoe that
'lis::> Hm' is nue. Accordingly, if John does not bc,ome interested in the history of shoelaces, '(3x)(l jx ::J Hm )' will be t ri vially true.
S4.9, EXERCISES Using the given nmatian, translate each of the following into symbols. e: the Empire State Building t: the Tai Mahal 0_ I _ f -1 is older than -2 1 Something is older than something.
2 Everything is older than something. 3 Something is o lder than everything. 4 Nothing is older than everything. 5 Nothing is older than itself. 6 Nothing is older than the Taj Mahal. 7 Something is not older than itself. 8 Something is older than the Empire State Bui lding. 9 Anything which is olde r than the Tai Mahal is older than something. 10 Something is o lder than the Empire Sta te Building but not older than the Taj Mahar. 11 Everything which is older than the Taj Mahal is o ld er than the Empire State Building. 12 Something is neither older than the Empire State Building nor older than the Taj Mahal. 13 Nothing which is not older than the Empire State Building is olde r than the Tai Mahal. 14 Anything which is older than everything is older than itself. 15 Anything which is older than itself is older than everything.
S4 .10:
148
PEOPI.E Ai\D THI:-.:GS
§ 4.1 0: PEOPLE AND THINGS There is an additional poim which requires carc when onc translates many English semences. The English terms 'everybody' and 'somebody ' afC used to refer only to people, unlike 'everyt hing' and 'some thing' which can refer to anything whate ver. 'Everything' can be transl ated as '(Vx)' and 'something' can be translated as '(3x)'. 'Everybody despises somebody' can be paraphrased as ' For any perso n one c hooses, there is a person whom that person despises.' One (.:an expand the paraphrase lO 'For anything one chooses, if that thing is a person, there is a person whom that thing despises.' T he paraphrase can be expanded yet again to 'Fur allyrhillg o ne chooses, if that th ing is ;J person, there is somet hing which is a person and whic h the first thing despises.' The final para phrase, of course, is ha rdl y an example of elegant English, bm it docs display th~ logic of till' s~rnrn(X in a way whidl suggests a proper translation. 'For anything one c hooses, ii that thing is a person, there is something which is a person and which the first thing despises ' is a universally quantified sentence. It assens that if Bill is a person, then the re is something which is a person and which Bill despises and that if Na nq is a person, then lhere is something ..... hich is a person and wh ich Nancy despises. Since Bill is a person, then acco rding to this sentence there is a person whom Bill despises. However, the original sentence begins wit h 'For an.ythin.g one chooses .... Thus, it :1150 makes the apparentl y bizarre assertion that if thc Nile is :l pe rson, then there is something which is :l person :lnd which the Nile despises. Since the Nile is not :l person, one is not forced to concl ude that the re is a person whom the Nile despises. Using ' P _' for' _ isa person' and '0 _1_2' for ' _I despises - l ' , o ne can begin the translation. II should be noted that this beginning is neither English nor the symboli!.: language; il is a concoction created purely for illustrative purposes. ::J (the re is something which is a person a nd which x despises))
('1x)( Px
'There is something which is a person and which Bill despises' can be translated as '(3y )( Py " Dby)'. This formula would be true if Bill despises hi mself or if Bill despises someone else . Substituting 'x' for 'b', one can complete the translation: (Vx){ Px
::J
(3yHPy " Ox y))
It should be noted that a variable other than 'x' was used with the existential quantifier. T he use of different va riables does not preclude the possi bility that the person despising is the same person who is despised . If different variables had not been used, the resul t would not have been a WFF.
149
§4. 11 : EXERCISES
The fo llowing formula is not a WFF because there is no way of telling which variable is lin ked to which quantifier. ("x)(Px " (3x)(Px
A
Oxx) )
ttt
'Somebody despises everybody' is an existentially quamified sentence. It asserts tha t there is somethi ng which is a person and which despises everything which is a person. Expanding the paraphrase, the sentence asserts that there is something which is a person and, for anythi ng one chooses, if that thing is a person, the fi rst thing despises the second. Using the same symbols as before, one can begi n the translation. Again it should be noted that th is beginning is neither English nor the symbolic language. (3x ){ Px 1\ (for anything one chooses, if that thing is a person then x despises [hat thing)) As before, one mus t use a different va riable with the next quantifie r. T he person who despises everybody is, of course, bound to despise himself. However, if the world contains more than one person, there arc other people whom he despises. 'Bill despises everybody' can be translated as '(Vy)( Py => Db y)'. Again substituting 'x' for 'b', one can complete the translation: (3x)( Px
1\
(Vy)(Py => Dxy ))
It should be noted that brackets we re added in the new fo rmula, since '(3x)Px 1\ (Vy)( Py => Dxy)' ttt is not a WFF because the 'x' in 'Oxy' is nO[ within the scope of a quantifier.
S4.11: EXERCISES Usi ng the given notation, translate each of the follow ing into symbols . c: the d ock b: the book j: Jim k: Kate P _: _ is a pe rson 5_ 1 _ 2 _ 3 : - I stole
-1
from
1 Jim stole the book from Kate. 2 Kate stole the dock from Jim.
3 If Jim stole the dock from Kate, she stoic the book from him. 4 Jim stole something from Kate.
S Jim stole something from someone. 6 Someone stole some thing from Jim. 7 Everyone stole something from Jim.
S4.12:
150
_
I O[~'TITY
._ 8 Jim did not steal anything from Kate. 9 Jim did nOt steal anything from anyone.
10 Anyone who stole the clock from Jim stole the cloc k from someone.
S4.12: IDENTITY Identity is a relation which can be expressed by a two-place predicate. English ex pressions suc h as 'is identical to', 'is the Ycry same thing as' and often simply 'is' are used in doing such things as slating that the criminal who robbed a bank o n Thursday is the same criminal who robbed a bank on Monday. It is useful to be able to translate such sentences . In addition, refe rence to identity is necessary in order to translate such ph rases as 'only John', 'no one but j ohn', 'only onc' and 'at least tWO', Express ions of identi ty in the symbolic language have a unique notation . Instead of writing '1- 1 - 2' for ' _ I is identical to -2', the usual notation uses the fam iliar '=' sign. Thus, 'a is identical to b' is expressed as 'a = b'. Nevertheless, identity form ulas are negated in just the same way as Other formulas; the ''Ic-' symbol is not used. Thus, ' a is not identical to b' is expressed as '- a = b'. A clarification of the use of constants can now be made. At the beginning o f this chapter, constants were introduced as standing for indiv iduals. J USt as a single ind ividual can have twO names - 'Istanbul' and 'Constantinople', for example, refer to the same place - so different constants can represent the same indi vidua l. When one comes across tWO or more disti nct constants, one should not assume that they refer to different individua ls; the fact that more than one constant is used says nothing about whether or not the discussion involves one o r more than one individual. To address that question, one needs to make an identity claim. The simplest use of the iden tity predicate is in translating sentences which assert that twO names refer to the sa me individual. Thus, using ' m' for 'M ark Twain' and's' for 'Samuel Clemens', one can express the fact that these twO names refer to the same pe rson by the simple formu la 'm = s'. Expressions of ide ntity are commonly used in translating sentences which claim that only one thing has some property. For example, to say that o nly Karen knows the solution is equivalent to saying that Karen knows the solution and no one but Karen knows the solution. Using 'k' fo r 'Karen' and '5 _ ' fOf ' _ knows the solution'. all three of the fo llowing formulas afe correct translations of the sentence:
151
S 4 . 12: IDE:\,TITY
Sk A (V'X)(SX => x = k) This formula sa ys that Karen knows the solutio n and that a nyth ing which knows the solution is identical to Karen. 2 Sk A - (3x)(S x A - x=k ) This fo rmula says that Ka ren knows the solution and that thefe is nothing w hich knows the solUlion wh ich is not identical to Kare n. 3 (V'x)(S x ;: x = k) T his formula says that, for anythin~ one picks, that thing knows the solution if a nd only if that thing is identical 10 Ka ren. As mentioned, id enti ty expressions make possihl e Ihe transbt ion of such phrases as 'only one· and ' at least t wo'. An example is the statement that th ere is o nly one popr.. Using 'P _' as ' _ is a pope', the statement that there is a po pe can be translated as: (3x )Px Ho weve r, this says that the re is al least one pope; it leaves open the possibility t hat there is more than one pope. What is needed is Ihe additional statement that there is at most one pope. Saying this amou nts (0 saying t hat, for any x and any y, if x is a pope and y is a pope, the n x :lnd y arc identic:l!. This can be expressed as: (\l x )(\ly )(( Px
A
Py) => x = y)
Thc sta tement that there is only one pope can be expressed by simply conjoining the twO fo rmu las: (3x) Px
A
(V' xHV'y){ (Px
A
Py) :::)
X
= y)
Another way of stati ng that there is only one pope is the fo llow ing: (3x:)( Px
A
(V' y)( Py :::) x = y))
Th is formula says th at there is something which is a pope and that anything which is a pope is identica l to that someth ing. T his reading may be easie r to foll ow if o ne replaces ' x = y' by'y = x' a nd rewrites the formul a as: (3x: )( Px
A
(V'y)(Py => y = x))
An even sho n er way o f making the same claim is: (3x)(V'y)(Py ;: y = xl This formu la says that there is something such that for anything one chooses, the thing that one c hooses is a pope if and only if it is identica l to that somethi ng. Som~ of these transla tions a re fa r from inmitively obvio us. Wh ether formulas a re quantificationally cq uivalcm or not qua nrificationally equ ivalent will become clearer after deal ing with ' inte rpreta tions' at the end of thi s cha p [~ r and derivations in Chapter 5. T his will mak~ the translations seem less confusing. It will remain true, however, [hal predicate logic simply
152
S4.12: IDE1'.'TITY
is more difficult than srlllcncc logic; there freque ntly is no way to pa raphrase an ordin ary English sentence into unSliltcd English which
mirrors the logica l struclUre of a formula in predicate logic. A st:.ltement of non-identity is used in translati ng 'James owns morc than
one piano.' Using 'j' for 'James', 'P _ ' fo r ' _ is a pia no' and '0 _ for' _ I owns _ 2' , onc can write: (3x)(3y)( - x = y
1\
« Px /\ Py)
1\
(O Jx
1\
I_
2'
Ojy)))
This formula tells one that x a nd y arc d istinct, that they James owns both x and y.
3rt
pialtus and tha t
It can be secn that the symbolic representation of some ordinary English sen tences can become quite long and rather confusing, particularly w hen a specific number of things afC im'olvcd. Th e claim that at least one thing has {he property f is simply the claim that somethi ng is r~ '(3x)fx', To say (hat at Ic Lxm ) " Lmi ) => (3x )( Px " Lxm)
A translation wh ich is quantificati onally equivalent is: (('tfxHPx => (Lmx => Lxm))
1\
Lm j) => (3x )( Px
1\
lxm)
The use of 'everyone', 'someone ' and names which are normally used only for people shows that the domain is simpl y the set o f people. Accordingly, a far simpler translation is poss ibl e: Domain: {XIX is a person} (( 'tfx )( Lmx :::> Lxm )
A
Lm j)
~
(3x)Lxm
Wh enever one can reSfrict the domain ro o ne son of entit)" th ere is the possibility t hat translation can be simplified. It shoul d be noted , however,
)4. 1 3: TRANSLATION PROBLEMS A;.;D DO~AI"' S
157
that specifying a domain does not always el iminate the need to specify the kind of thing being disc ussed. 'Some French wines are appalling and no Frenchma n drinks them' is a case in point. Wit hout specifying a doma in and using some obvious symbols, this can be translated as: (3x)(((Wx " Fx ) " Ax ) " (V'y)( ( ~y " ry ) =:. - Dyx ))
Since everything under discllssion is French, it is possi hle to speci fy a domai n and dispense with the predica te ' is French' : L>omain: {XIX is frenchl (3x )((Wx 1\ Ax ) 1\ ( V'y)(My =:. - Dyx )) Although this translation is simple r than the previo us one, it is still necessary to distinguish between men and wines. Restricti ng a domain restricts the predicates wh ich can be used in the translation. 'Eve ry Englishman loves all dogs' can be translated using 'E _' to represent 'is English' and '1. _' to represent 'loves all dogs': Domain; (XIX is a man) (V'x)( Ex =:. I.x ) Th is sentence can also be translated without festricting the doma in. Using 'E _' [Q represent 'is an Englishman', 'D _ ' to represent 'is a dog' and 'L - 1 - 2' to represent' - 1 loves 2' , the sentence C3n be rranslated as:
l"x)(Ex ::> l"yllD, ::> Lx y)) It wou ld be a mistake, however, to use '(V'x)( Ex =:. (V' y)( Dy =:. Lxy »)' while restricting the domain to men for [he simpl e reason tha t dogs are not elements of the set of men. It would also be a mistake to rest rict the domain to English things, because the claim that every Englishman loves all dogs is different fro m the claim that every Englishman loves all English dogs. Many beginners, when faced with a translat ion problem, seem to decide first what quantifie rs are needed and then go on to fill in the predicates. Thus, one sees a sentence such as 'Some Frencb wines afe appalling and no Frenchman drinks them' translated as: Domain: {XIX is French) (3x)(V'y)( (Wx /\ Ax ) /\ ( ~ y =:. - Dyx » In fact, this tra nslation is quantificationalty equivalent to the translation which was given earlier in this section. However, the beginner wou ld do well to introduce the quantifiers as they are introduced in the English . Nothing is gained by forcing the qua ntifiers to the front of the formula, and it is very easy to make mistakes in doing this.
158
S 4. 13: TRANSI.ATION PROBLEMS AN D D OMAIM
Th is poim is shown even m OfC clearly in the tra nslation o f 'Someone loves Carl and someone loves Jane, bur no one loves them both.' This is simply a conjunction of three claims. Using the SCt of people as the do main and some obvious symbols, the sentence call be translated without any
diffil;ulry at all as: Do main: {XIX is a pe rson} ((3x)Lxc
A
(3x)L xj) " ('v'x)- (Lxc " Lxj )
T he followi ng translation is quantificationally equivalent to the previous one, hut it has [inlc to recommend it: Domain: {X IX is a pe rson} (3x)(3y)(' Rx ) (\>'x)( Px ::::> - B x) " (Vx )( Rx ::::> Hx )
(V x )(C x
:::J
(Fx ::::> (- Bx " Hx )))
In the following a rgumem, it is possible to make the translation simpler by restricting the domain to the set o f people:
Since Paul's fa ther has (he combination, he must be the bank manager. We know this because only o ne person has the combination.
Domai n: {XIX is a pe rson} h: the bank manager f: Paul's fa ther has the combination C Cf ( 3x )(Vy)(Cy
= x = y)
f =b Translation of the next argument can also he simplified by restricting the domain [Q t he set of people: Every golfer is sillier than someone, but no golfer is sill ier than everyone. Bob and Mike are golfers. Mike is not sill ie r than Bo b. T herefore, Mike is sillier than someone other [han Bob. Domain: lXIX is a person I
b, Bob m: Mike
G_: _ is a golfer 5_ 1 _ 2: _ I is sillier than (' (x :: a 1'1 y:: b))' is a universally quantified form ula , to show that it is true on the given interpretation, one mUSt deal with everr item in the doma in. Si nce thcre arc two items in the domain, '{Vx)(Vy )( Rxy a :J (x :: a 1'1 y:: b))' can be Undt'TSIOOd as Ihe conjunction of two formu las: (V'y)( Raya :::) (a ", a
1'1
y '" b))
1'1
y:: b))
,nd 2 (Vy )( Rbya
::>
(b:: a
Bmh of these formu las, (00, are universally quantified and can oe expanded as conjunctions. 1 beco mes the conjunctio n of: 1.1 Raaa:::) (a : a
1'1
a:::b )
A
b:: b )
,nd 1.2 Raha => (a:: a
When expanded, 2 becomes the conj unction of: 2.1 Rbaa
(b::a
1'1
a::b)
2.2 Roba :::) (b = a
1'1
b = b)
:J
,nd Since 'Raaa' is false and 'a::: a 1'1 a ::b' is true, 1.1 is true. Since ' Raba' is true and 'a ::: a 1'1 b::: b' is true, 1.2 is true. Since ' Rbaa' is false and 'b '" a 1'1 a = b' is true, 2.1 is true. Fi nally, since 'Rhba' is fa lse and 'b ::: a A b::: b' is true, 2.2 is truc. Thus, 1.1, 1.2, 2. 1 and 2.2 arc all true. Because any conjunction of four true formu las IS true, '(Vx )(V y)( Rxya :::) ( x = a 1'1 y '" b))' is tr ue on th e given interpretation. Since in an interpretation, universally quantified formulas oecome conj unctions and exi stentially quantified fo rmulas occome conjun...:tions, one ma y wonder wh y truth tables aTe nOl used when dealing wit h interpretations. T here arc twO reasons. The practical reason is Ihat truth tables used for this purpose arc messy and hard to co nstrue!. The theoretiCal reaso n is (hat trUlh tables give bizarre results when two or more items in t he doma in arc identical. For example, wh en 'a::: b· and 'Fa' afe bmh true, a complete truth table will have at least one rowan which ' Fa' ;lIld 'Fb' have diffe rent (rut h values. Alt hough the quantificational equ ivalence of two formulas cannot be shown as simply as t ruth· fu nctional equivaicnce , it is quite easy to usc an interpretation to show that two formulas arc not quantificationally eq uivalent. One need only provide an interpretati on on which the formul as
)4 . t 7; SI~PLF. r ~TERPRETATlONS
170
have different truth values. An interpretation is used to de monslrate that '(3x)Gx " (3x lH x' is not quamificarionally equi valent to '(3x)l Hx " Gx)' in the following example:
Domain: la, bl Extension of G: (a)
Extension of H: {bl On this interpreta tion, '(3x) Hx 1\ (3x)Gx' is true whi le ' (3x)( Hx 1\ Gx )' is falsc. As another example, an interpretation is used to demonstrate that '(3x)R x :J Ma' is not quanti fi cationally eq uiva lent to '(3x)( Rx ::::> Ma J':
Domain: la, bl
Extension of M: {bl Extension of R: {a) On this interpretation, '(3x)Rx ::J Ma' is false while '(3x)( Rx :J MaJ' is true, as can be seen by system::ltic:J.lIy substituting constants for variables. As mentioned, Quan tificational va lidity cannOt be shown in the simple way that tr uth-functional validity can be shown . Howeve r, it is relatively easy to use interpreta tions to show quamificati onal invalidity. If one can prod uce an interpretation on which all the premisses are [ru e and the conclusion is fa lse, one can show that the argument is quantifica tionall y invalid. T he quantificational invalidi ty o f the fo llowing argument can be shown by means of a simple interpretation: (V'xHFx :J Gx) (3xl - Fx (3x)- Gx
To show tha t this argument is quanti fi cationally invalid, an interpreta tion with a domain consisti ng of two items suffices: Domain: la , bl Extension of F: lal Extensi on of G: la, bl On this inte rpretation, '(V'x)(Fx :J Gx)' and ' (3xl- Fx' a re true, while '(3x)- Gx' is fals e. The argument, therefore, is quantificationaHy invalid. To show that the foll owing argument is quantificationally invalid, o ne again needs two ite ms in the domain: (3x )( Ax 1\ Bx) (3x HBx " Cx) (3x)(Ax
1\
C x)
The trick is to set up the extensions o f 'N, ' S' and 'C' so that something is both A and S, something is both Sand C, and nothing is both A and C.
S4.18: EXERCISES
171
Setting up the extens ions in this way allows all the premisses to be true while the conclusion is false: Domain: Ii, kl Extension of A: Iii Extension of B: {i, k} Extension of C: Ikl In trying to develop an interpretation, o ne shou ld firs t consider whether a domain with one object in it would be adequate. If not, the domai n should be expanded only as necessary. With increasingly com plex for mulas requiring larger domains, t he use of interpretations to show qua ntificati onal invalidity becomes impracticable. The method of using inte rpretati ons discussed in this section is cum bersome, and it frequently requires much trial and error to arrive at the necessary domai ns and extensions. There a re more sop histicated procedures. In fa ct, there are ways to use interpretations to do such th ings as show quantificational validity so long as one is dealing on ly with one-place predicates. Howeve r, these also arc difficult to apply in practice, and a discussion of how interpretations can be used in such ways is beyond the scope of this book. Learning something abou t inte rpretations can be useful for seeing how the qua ntifiers work . However, as a technique to be methodically applied to p roblems, the use of imcrpretations has serious limitations.
54.18: EXERCISES S4.18 Part 1 Determine t he truth value of each of the following fo rmulas o n the given interpretatlon. Domain: {a, b, c, d} Extension of F: (bl Extension of G: I(b, a), (c, a)} Extension of H: {(b, c, a)J Extension of K: { I Extension of =: {{c, d)1
1 (3x )- Fx
2 (3x)Fx 3 (Vx )( 3y)Gxy 4 (V'xl- Kx
S (3x )( Fx v Kx )
6 (3x Il3yIl3z)H xY'
172
) 4 .18: EXERCISES
7 (3x)-x =b 8 (' (VYIGy ( V'xHFx ::> (' (3y)Gy) 7 (3x)( Mx v Ra j (3x)(Mx v Rb)
8
((3x )Fx y (3x )Gx ) ::::> ('v'y)Ry
(3x) Fx v ((3x)Cx ::::> (' l \1y)Rxy ) 13x)(Qx A l\1y) Rxy )
10 l\1x)( 3y )(3z)Rxyz (3x )( 3y)(V'z)Rxyz
S4.18 Parr 4 For each of the following argu ments, provide an interpretation to show that (he argument is quantification ally invalid. - a=b - b= c
--- a=c
~
2 (V'x)(Fx
Gx)
-h - Ga
3 l \1y)( 3x )K xy 13x )( \1y)Kx y
4 l\1x )( Fx :> Gx ) (V' x)- Fx (V' x)- Gx 5 ( V'x )Fx :> (3y)Gy Fa
~
(3y )Gy
6 (3x)( Fx
~
Gx)
( V'x)( Fx
~
Gx)
7 ( V'x)( Fx
~
(x = a v x = b))
Fa v Fb ~ Mx) (3x)(Fx " - Kx )
8 ( V'x )( K x
(3x)( Fx " - M x)
9 (3x)Rxa (V'x)R xa
~
(3x)Rax
{3x)Rax
10 (V'x)( Hx
:::> Fx ) (3x )( Fx /\ - Gx )
(V'x )( H x
~
- Gx )
174
54.19: SUMMARY OF TOP[CS
54.1 9: SUMMARYOFlDPICS Wit h a few possible exceptions to mah the summary cleare r, d1C topics appear in the o rder in which they a ppear in the text.
Individual (See p. 134): An individ ual can be a person, a place, an idea or collection, Mass nouns such as ' water' and 'sugar' should be treated as individuals. Predicate (See p. 134): A predicate ca n deal w ith age, si ze, shape or com plexity - anything one ca n say a bo ut an individua l. It is rep rese med by an upper-case letter distinguished from a sentence letter by bei ng immedia tely followed by a lo wer-case letter.
Constant (See p. 134): A constant is a lo wer-case iener whic h names a pa rticular individual and follows a predicate lette r. All the lower-case letters from a to t can serve as constants. In the formulas, 'Gb' and ' Mdhk', 'b', ' d', 'h' and 'k' a re constants.
One-place predicate (See p. 135): A predicate which is represented by a predicate letter followed by a single lower-case letter. 'Fa' and ' Rc' arc examples of one-place predicates.
Multi-plaCe predicate (See p. 136): A pred icate which is represented by a predicate letter followed by more than one lowe r-case lener. ' Fac', ' Hejk', ' M ek k' are exampl es o f multi-place predicates. Quantifiers (See p. 140 ): Words such as 'each', 'all', 'some ' and ' no' are quantifiers. Variable (See p. 140): A single lower-case letter which does not name a particul ar individ ual as a constant does. 'x', 'y' and 'z' (with subscripts if needed) are the only va ri ables used in this book. In a WFF, every variable is within the scope of a qua ntifier. Existential quantifier (See p. 140): Expressions like '( 3x)' and '(3y)' are used to represent quantifiers such as 'some', Open sentence (Sec p. 140): An ope n sentence is nOt a WFF. If all the variables in it were replaced by constants, the resu lt wo uld be a WFF. ' H x' and ' R xy' and examples of open sentences. Closed sentence (See p. 140): A closed sentence is a WFF. 'Hat and 'Rdg' are examples of d osed sentences.
S4.19: SUMMAR Y OF TO]']CS
175
Scope of a quantifier (Sec p. 141 ): The scope of a qua ntifier de lermines wha ( is governed by (he quantifier. Brac kCls ex (end (he scope o f:l quantifier in mu(b the same way as brackets exte nd what is negated br the tilde. Universal quantifier (See p. 142): Expressions li ke '(V x)' and 'IVy )' arc used to represen! quantifiers such as 'all' and 'ever)". Q uantificationally equivalent (See p. 142): '- (3x )Fx' and '(\t' x)- Fx' can easil y be seen to be quantificationally equivalent. In Chapter 5 there arc rules which allow the derivation of o ne membe r o f a pair of quantificatio nall}' eq uivalent formulas fnml the other. Atomic formula (Sec p. 143) : An atomic fo rmula in predic;ll(' logil' is any WFF which contain no operators (sentence o perators or quantifiers). '\.1a ', 'Qba ' and 'Pdck ' arc all atomic fo rmulas. Identity (Sec p. 150): Ex pressions o f identity usc the familiar '=' sign. T hus, 'a is identical to b' is ex pressed as '3 '" b', Neverth eless, identity formulas arc negated in iust the same wa y as othe r fo rmulas; the ';t' symbol is nO! used. Thus, 'a is no! identiClllO b' is e xpn:s~ed as '- a = b'. Expressions of identity arc necesS3ry to tran~late such phrases as ' no one but Paul', 'only Paul', 'only one' and 'at least lWO '. Definite descriptions (Sec p, 153 ): Expressions such as 'the sma rtest th ief o f the gang' and 'the youngeSI student at the college' arc typical definite descriptions. Sets (Sec p. 156 ): Sets arc collections of cle ments. Elements (Sec p. 156 ): El ements are the mem bers which cons titute sets, List notation (Sec p. 156 ): Listing the clements of a set between set brac kets as in 'la, b, The elements in this notation aTCseparated by comm:ls.
cr.
Property notation (Sec p, 156): Specifying some property of the clements, as in '{ XIX is green or X is bluel' , wh ich can be read as 'the set of all Xs which arc green o r blue'. Domain (Sec p. 155): To specify a domain is to restrict the d iscussion by limiting it to include on ly certa in things. T his is done by specifyi ng :l SCt which constitutes the doma in. Q uantificationall y valid (Sec p. 16 1): An argument in predicate logic is quantificati onally va lid if thc conclusion can he derived fr om the premi sses using the rules of Chapter 5.
176
)4. 19: SUMMARY OF TOPICS
Quantificationally true (See p. 165): A formula in predicate logic is quantificationally true if it can be derived in a categorical deriva ti on using the rules of Chapter 5. Quancificationally false (See p. 165): A formula in predicate logic is quantific3ti onall y false if its negation can be derived in a categorica l derivation using the rules of Chapter 5. Interpretation (See p. 166): An imerpretation uses a domain to show that a formu la is quantificationally indeterminau: or that an argument is quanrific3riona lly invalid. Empty set (See p. 166): The set having no elements. Domain of the interpretacion (See p. 166 ): The doma in of the interpre tation is a non-empty sct which is 3ppealed to in determining the truth or falsity of formula s in the interpretation. Every interpre tation has a single domain. Extensions of the predicates (Sec p. 166): A list of what predicates are nue of what items in an interpretation. Ordered pairs (Sec p. 167): In an interpretation , orde r is sho wn in the extensions of predicates by the pointed brackets: '(' and ')'. Thus, if ' Rab' is true and 'Rba' is false, the extension o f ' R' is '{(a, b} j', If'Rab ' and ' Rba' arc both true, the extension of 'R' is '{(a, b), (b, a)l'. Ordered triples (See p. 167): Ordered triples are just like ordered pairs but arc used with three-place predicates. 'I(a, b, c), (b, b, a)J' is an example of an extensio n including o rdered triples. Quanrificationally indeterminate (See p. 165): A formula in predica te logic is Quantificationally indeterminate if it is possible to provide both an inte rpretation on whic h it is tfue and a n interpre tation on w hic h it is false. Quanrificational invalidity (See p. 166): The quantificarional invalidity o f an a rgument can be shown by prov iding an interpretation on which the premisses aTe true and the conclusion is false.
CHAPTER FIVE
DERIVATIONS IN PREDICATE LOGIC
S5. 1: PREDICATE LOGIC DERIVATIONS De ri vations in pred icate logic can be used to derive the concl usions of quantificationally va li d arguments and to derive quantificationally [rue formulas categorically. Since the negation of a Quantificationally true fo rmula is quantifica tiona lly false, deri vations in predicate logic ca n also be used to show q ua ntificational falsehood. Using the derivation rules from Chapter 3 togethe r with th e rules w hich will be prese nted in this chapte r, one ca n derive the conclusions of all quantificationally valid a rgumentS and only quantificationally va lid arguments from their premisses. In addition, the rules will allow the categorical deriva tion of all quantificati onally true formulas and onl y quantificationall y true for mulas. That th e rules a re logica lly respectable in this way can be demonstrated, but such a demonstration is a matter of metat heory. Derivat ions in predicate logic ma ke use of six rules in addition to the derivation rules o f sentence logic. The new rules are introd uction and elimination rules for the existential quantifi er, the un iversa l quantifier and identity. Interpretations ca n often be developed by tri al and eTTor to show that an argu ment is quantificationally invaliet. Interprerations ca n also be used to show that a formula is quanti ficationally indeterminate, and thus neithe r
177
178
SS.2: EXISTENTIAL QUANTIFIER INTRODUCTION
quantificationally true nor quantificationally fa lse. However, the method of interpretations docs not provide a mechan ica l proced ure to determine such things in the way that a check 0 11 a tfuth table does. O ne may not know that an argument is quantificationally invalid or that a formula is quamificationally inde terminate simply because onc has not devel oped the necessary interpretation. Accordingly, there is an important diffe rence between derivations in sentence logic and derivations in predica te logic. In sentence logic, if one is tryi ng without success [0 derive a formu la, it is always possi ble (0 refer to a truth table to make sure that the formula is in fact derivable. In predicate logic, it is possible in principle to work for a lifetime o n a derivation with no way of dete rmining with certainty whether the formula one is trying to derive is in fact derivable. In practice, worki ng seriously at a de rivation usua lly leads either to success or to an undemanding of why the formula one has been trying to derive cannot be derived. All the derivation exercises in this book have been checked to ensure that the required derivations arc of a reasonable length.
S5.2: EXISTENTIAL QUANTIFIER INTRODUCTION If one knows that grass is green, one knows that something is green. Given any conStant 't' and any predicate 'eI>' in a sentence 'cJ>t', o ne can infe r '(3x)¢x'. Simila rly, if one knows that somethi ng is be hind the house, o ne knows that something is be hind some thing. Given any constant 't' and any predicate '¢t in a sentence ' (3x)lbxl', one can infer '(3y)(3x )lb xy'. T he rule of existential quantifier introduction, '31', reflects the propriety of such inferences. Before stating the ru le (or introducing the existential quantifier, it is necessary to look ahead a link bit. Some of the rules which wi ll be discussed later in this chapte r will use dummy names, much as the name 'John Doe ' is o ften used. These dummy names are called 'parameters' and their use will be ex plained in the appropriate place. For the moment, it is sufficie nt (0 recognize that o ne will have to deal with twO kinds of names, constants and parameters. In what follows, the word ' term' will be used to refer to both constants and parameters.
SS.2:
179
EXISTENTIAL QUAl'TIFIER 1r-.'TRODUCfIO",
Although using existential quantifier introduction is really quite easy, the formal statement of the rule is somewhat forbidding.
31 One can write an existentially quantified formula '¢I ' on any give n scope line, provided that all of the following condi tions are met: A for mula which contains one or more occurrences of the term ';: ' appears on [hat scope line. 2 The variable 'X' does nOt appea r in this original formula. 3 The existentially quantifie d formula '¢I' is formed by replacing one or more occurrences of the term 't' in the original formu la by the variable 'X' and writing '(3X)' in front of the result of the replace ment. Th e justification for the new line consists of an appeal to the num ber of the line where the original formula containing 't' appears and the rule, '3 1'. Th is may seem difficu lt to follow, but a few examples will make matters dear. All uses of 31 in the fo llowing derivation are correct:
1 Rab 2 Fe 3 (3z)Fz :J Qi 4 (3z)F7. 5 Qi 6 (3x)Qx 7 (3y)Ryb 8 (3x)(3y) Ryx 9 (3x)(3y)Ryx to. (3x)Qx 10 (3x)((3z)Fz ::> Qx )
P P p 2,31 3, 4, :J E 5,31 1,31 7,31 6,8, AI 3,31
It should be noted that the brackets were added to '(3z )Fz :J Qj ' when 31 was used at line 10. Strictly speaking, 31 was used on '((3z)Fz :J Qj)'. If this had not been done, the appea l to 31 at line 10 would have been illegitimate, since the 'x' in ' Qx' would not have been within the scope o f a quantifier. In using 31, one can replace as many occu rrences of the sa me term as one li kes. All of the following uses of 31 are correc t:
\5.2,
180
EXISTEl'.'TIAL Q UANTIFIER INTRODUCfION
1 Raaa
P
2
(3x)Rxaa
1, 31
3
(3x)Raxa
1, 31
4
(3x)Raax
1, 31
5 6
(3x)Rxxa
1, 31
(3x)Rxxx
1,31
7
(3y)(3x)Rxya
8
(3z)(3y)(3x)Rxyz
2,31 7,31
T here are three importam poims to remember in connection with 31. First, the new variable must not already occur in the formu la which is to be Quantified. Second, on e cannot replace different terms with the same variable. Third, the existential Quantifie r which is introduced must Quantify the entire formula. This means that any outer brackets whic h have not been included must he added before the existential quantifier is introduced. Some common mistakes in using 31 are exemplified below: 1
- Ma
2
(3x)Fx
3 4
Jm n
HOI
::J
P P P P P
Gb
::J
Hb
5 (3x)Kx " lm 6 - (3x)Mx ttt
1, 31
ttt
7
(3x)jxx
8
(3x)(3x)Kx " Lx )
9
(3x )Hx => Hb
4,31
(3x)Fx =>
2,31
10
ttt
ttt (3 y)Gy ttt
3,31 5,31
Line 6 is completely mistaken. It is possible to obtain '( 3x)- Mx' from '- Ma' by 31 but not '- {3x)Mx'. The error in line 7 is that twO distinct terms, ' m' and 'n', are replaced by a single variable. This error wou ld be obvious in English: letting 'Jmn' represent 'Michael played a joke on Norman', it certainly does not follow that Michael or anyone else played a joke on himse lf. Line 8 is mistaken because the variable which is used in the new quantified for mula has already been used in the fo rmula which is to be Quantified . The result is nOt a WFF. Line 9 is mistaken because the complete for mula is not Quantified. h is possi ble to obtain '(3x)( Hx ::::J Hb )' from 'Ha ::::J Hb' but not '(3x)Hx ::::J Hb'. This error, toO, would be obvious in Eng!ish. From [he fact that Bill is ha ppy if Ann is happy, it does not follow that Bill is happy if somebody is happy. Line 10 is mistaken because only
S5.3: U;';IVERSAL Q UANTIFIER
EU~INATIO~
181
pan of a formula is quantified . '(3x)Fx :::J (3y)Gy' is obtained legitimatel y from ' (3x)h :::J Gb' in the fo llow ing derivation: 1
(3x)Fx :::J Gb
P
2
~X)FX (3x) Fx
A
)
:::J Gb
1, R
4
Gb
2,.1, :::J E
5
(3y)Gy
4,31
6
2-.1, :::J I
(3x )Fx :::J (3y)Gy
The rule o f existential quantifier introduction is easy to remember and usc. In introducin g the existential quantifier, one mU! Pcb
:J
Pzy)
P J, VE 2, liE 3, liE
The conStants or parameters which replace the diffe re nt va riables do not have [Q be different. T he next derivation is JUSt like the previous one, except that the constant 'k' is used in each application of VE: (\fxHVy)(V'zH Rx y
2 3 4
:J
(lIy)(lIzIIRky::> P,y) (ltzIl Rkk ::> pzk) Rkk:::> Pkk
Pzy )
P 1, li E 2, 'liE
3, \IE
\I E and 31 are used [Q derive '(3x )Gx' from '(\fxHFx fo llowing derivation:
::J
Gx)' and ' Fa ' in the
3 4
Ga
P P 1, liE 2,3, :::>E
5
(3x )Gx
4,3 )
1
(V'x HFx
::J
Gx)
2 fa Fa :::> Ga
T he universa l quantifier which is eliminated must be the main o perator o f the formula. In addition, every occurrence o f the variable in question must be replaced in order to avoid being left with a variable which is not within the scope of a q ua ntifie r. Another important poi nt is (hat all (he occurrences of the variable being replaced mu st be replaced by the same term. Some exa mples o f common mistakes in using VE are shown bel ow: - (\fxJFx
2
Fa
3 4
Fa :;:l (\fx )R x (lIx )( lIy)px y (\f x)Fx :;:l Rk (\fx)Bxx
5 6 7
Bjk
t tt
- Fa t tt 9 (lIx)Pxa ttt 10 Fa ::> Rk ttt 11 Fa :;:l Ra ttt
8
P P
P P P
P 6 , li E 1, liE 4, V'E 5, liE 3, liE
S5.4: EXISTENTIAL QUANTIFIER ELIM INATION
183
The same variables in a universally quantified formu la are replaced by different constants in the formula on line 7, and this is nm a legitimate use of "IE . This error is intuitively clear in English if one letS '(Vx)8xx' represent 'Everything is as big as itself.' Plainly this does not a llow one to in fer that Cambridge is as big as Oxford. Line 8 is incorrect because the main ope rator of [he formula at line 1 is a tilde. Line 9 is incorrect beca use the main operator of '(Vx )(Vy)Pxy' is '( "'1 x )' and not ' IVy)'. Tbe main Operator of the formula at line 5 is the horseshoe, and thus VE is misappl ied at line 10. Th ere is exactly the same problem with the appeal to "IE at line 11. It is possible [Q derive ' Fa ~ Ra' from 'Fa ~ (Vx)Rx' legitimately, and this is done in the fo llowing derivation: 1
2 3 4
Fa
:::J
(Vx)Rx
F:
~
P A
:>
IIIx)Rx
(Vx)Rx
Ra 5 Fa :J Ra 6
I, R 2, 3, ::JE 4, 'lfE
2-5, ::Jl
The r ule of universal quantifier elimination is very easy to remember. In mOSt caSeS, mistakes arc the result of fa ilure to look carefully for the main operator of the formu la under consideration.
S5.4:
EXISTENTIAL QUANTIFIER ELIMINATION
The rule of existential quantifier elimination, '3£', is more complex than the other rules discussed so fa r, but the underlying idea behind the rule can be explained easily. If one can establish a conclusion from the fact that something has the property with no appeal to m her kn owledge about that thing, o ne is entitl ed to infer that conclusion from ' (3x)x '. There is a very common way to make sure that it is only the fact that the thing unde r consideration has the property which allows one to infer the conclusion. That is [Q assign a dummy name like 'j ohn Doe' [Q that thing. Since 'john Doe' is a dummy name which does not stand for an actual thing O f person, anything which can be shown to follow from the fact that j ohn Doe is is established. The lower-case lette rs 'u', 'v', 'w', supplemented if need be by subscr ipted letters like 'w 2 ', are parameters and serve [he same SOrt of purpose as do dummy names. The best way to see how parameters work is to examine a simple derivation which uses 3E:
184
1
S5.4; EXISTEI\.'TIAL QUANTIFIER ELIMINATION
(3x )Fx
2 ("x )lFx 3 4 5
6 7
:> Gk)
"[
P P
A 2, R
(" x)lFx :> Gk ) Fu => Gk
4, 'VE
Gk
3,5, :::JE
Gk
1, 3--{), 3E
Th~
assumption at line 3 is the to ugh equivalent of sayi ng 'Let us give the dumm y name "u" to the thing which is E' The scope line which begins with this assumption is flagged with the parameter 'u', Th!: flagged scope line serves as a param eter barrier, in this case a 'u' -barrier. This indicates that n o information about ll, no formu la contain ing 'u', is allowed to he moved across the scope line. Although '('9' x)( Fx :::> Gk )' was reite rated across the 'u'- barri er, this move was legitimate' beca use the dummy name 'u ' does not occur anywhere in '('t' is replaced by 'X ', where variable which does not occur in '¢tt'. 4 The same variable 'X' follows the 'V' in '( VX)41X '.
'x' is any
Th e justification for the new line is the number of the line on which '1' appea rs and [he rule, "vI'.
189
S5.5: UNIVERSAL QUANTIFIER If'frnODUCfION
'VI can be used seve ral times in the same derivation, as the fo llowing example shows:
1
2 3 4
(lI xl( lIyl( IIz)Rxyz u v w (II xl( lIyl( IIz)Rxyz
P 1, R
(lI yl(lIz)Rwyz
2, 'dE
('dz)Rwvz
3, liE
Rw vu 5 (Vz)Rzvu 6 (lIyl(lIz)Rzyu 7 8 ('Vx)('Vy)( Vz)Rzyx
4, VE 5, VI 6, 'VI 7, 'V I
Only one parameter at a time can be replaced by a vari able . The pa ra mete r which is replaced by rhe variable o f the universa l qua nti fier must be th e parameter of the barrier which is ended. Some mistakes in using 'VI a re shown in what foll ows: ('dx )('dy)Kxy
2 3
4 5 6 7
v ('dx)('dy)Kx y
(lIy )Kuy
Kuv ('dx)Kx x ttt (V x)Kxv ttt (lIyl(lI x)Kx y ttt
P 1, R
2, liE J, If E 4, 'V I 4, V I 6, 'V I
Line 5 is mistaken because both of the parameters ' u' and 'v' are replaced by a single va ria ble. Line 6 is wrong beca use the parameter which is replaced is not the parameter of the barrier. Line 7 is wrong for e xactly th e same reason. In introduci ng the un iversal quantifier, one must remember that the scope of the Quantifier is the entire form ula. It is important, therefore, to be sure that the formulas which a re going to be universa lly Quantified are pro perl y bracketed. If one keeps in mind the reason for having pa rameters and ba rriers - that parameters must function as dummy names and that parameter barriers prevent info rmation about parameters from being reiterated - the use o f 'V I is completely straightforward.
190
SS.6: IDENTITY INTRODUCTION
SS.6: IDENTITY INTRODUCTION The predicate used to express identity differs importantly from other predicates in that there are introduction and eliminati on rules connected with ie. The rule of identi ty introduction, '=1', is extremely simple. It is far morc usefu l than it might appear at first:
=1 At any point in a derivation, one can write 't = 't', where 't' is any constant or parameter. The justification for the new line is simply the rule, '=1'. As 3 n example o f the correct use of :I, the conclusion o f the fo ll owing argument is derived fr om the premiss: ('d x )( Rx ::::> - a = a)
- Ra I
("x)(Rx
::::> - a= a)
2 Ra ::::> -a=a J 4
5 6 7
P 1, 'v'E
Ra:::>-a=a
A 2, R
- a=3
3,4, ::::>E
a=a
=1 3,5,6, -I
~a - R.
The rul e of identity introduction is so simple that mistakes in using it are extremely rar~.
191
SS .7: IDEl'.TITY EUM[:-JAllON
SS.7: IDENTITY ELI MINATION The rule of identity elim ination, '=E', is easy to l!:arn: =E
One can write (he formula '(J>t2' on any given scope line, provided chat both of the following conditions a r!: m!:t: On that scope line there is a formula '$t ]' 2 On that scope line there is a formula 't] = t2' or 't2 = t]' 3 '$t2 ' is the result of replacing all or some of the occurrences of 't] ' by 't2' in '4>"t]'. The justification for the new li ne consists of an appeal to the number of the line where 't 1 = t2' or '""[2 =""[,' occu rs, the number of the line where '$t 1' occurs and the rule, '=E'. All of the following uses of =E are correct: 1 m=j 2 Rj 3 Kmj 4 Rm 5 Kmm 6 Kjj 7 Kjm
P P P
1,2, =E 1,3, =E 1,3, =E 1,6, =E
The rule of identity elimination is intuitivdy obvious, and mistakes in using it ar!: usually the r!:suit of simple inattention.
SS .8:
192
S5.8:
EXERCISES
Supply the m issing justifications in the foll owing derivations.
Question 1 I
P
- ( 'Vx)Fx
- (3x)- fx
2
u
J
~FU (3x)- Fx
4 5 6 7 8 9
- (3x ) - Fx
- - Fu
Fu ("xIFx - (Vx )f x
10
- - ( 3x)- Fx
II
(3x l- Fx Question 2
(3x )lxc 2 (3xlLxj
P p p
I
J 4 5 6 7 8 9 10 II
12
(,V'x)- (l xc U
1\
Lxi)
Lue
?
(3x) Lxj
v Lv j W
(V'x )- (Lxc " Lxj )
- (Lwc Lue
A
LwiJ
Lv j
Lue " Lvi (Lue " Lvi)
IJ
('', one should go through tht following steps in order:
1 See if one can reLterau: ''. 2 See if one can use an elimination rute to obtain '¢l'. In general, wheneve r one has an existentially quantified for mula available, the best strategy is [Q usc 3E immediately. 3 See if one can use an introduction rule to obtain '¢l'. 4 See if one can obta in ' ¢l' by an indirect prooL The best way to learn the art of producing derivations in predicate logic is to work through some examples. Deriving the conclusion of the following argument is a good beginning: ('Vx)( Fx ::> Gx) (Vx)( Gx ::> Hx ) ('Vx)(Fx ::> Hx ) as with derivations in sentence logic, the first step is to set up a skeleton of the d erivation;
JUSt
I
I ('Vx)( Fx ::> Gx) 2 , ('Vx)(Gx ::> Hx) Z
p
p
i (Vx)(Fx ::> Hx)
There are no secondary scope lines, and thus reiteration is not possible. The only elimination rule one can use is 'VE. One could obtain something like ' Fa::> Ga' from line 1 and something like 'Ga ::> Ha' from line 2, but neither of these formulas would be o f any help in deriving the conclusion . One must turn to an introduction rule, in this case VI. If one had 'Fu ::J H u' against a ·u'·barrier, one could obtain '(\fxllFx ::> I-I.x )' by 'VI. The sketch of the derivation is expanded to show this:
195
S5.9: DERIVATION STRATEGIES
2
3 Y
Z
('Vx)(Fx :::I Gx)
p
('Vx){GX:::l Hx)
p
UI 'I
i Fu
:::I Hu ('Vx){Fx:::l Hx)
Y,III
Once again, one cannot use an elimination rule to obta in what is needed, so one must look to an introduction rulc. The appropriate introduction rule is =>1, and so one sho uld assume ' Fu' and try to obta in 'Hu'. The oudine of the derivation shows this:
P P
('Vx)(Fx => Gx) 2
3 X
Y
('Vx){Gx :::I Hx)
.
~
IFu
A
i Hu OJ
Hu
3- X, :::II Y, III
Z ('Vx)(Fx => Hx )
A glance a t what is available shows that ' Hu ' can be obtained only from the
second premiss. If one used 'VE on the fo rmula on line 2. one could have 'Gu :::) Hu', but a formula with 'u' in it cannot be reiterated across the 'u'-barrier. Accordingly, ' ('Vx)(Gx => Hx )' should be reiterated bdore VE is used. What is needed now is 'Gu', and the sketch of the derivation is amended appropriately: 1
('Vx)( Fx
:::J
p
Gx)
p
2 ('Vx)(Gx => Hx) 3 4
5 W X
Y
Z
u
~
(lIx)(Gx
OJ
GU:::l H u
Hx)
A 2, R 4, 'VE
GU
i
Hu Fu :::I Hu
(Vx)(Fx => Hx)
5, W, =>E
3- X, =>1 Y, VI
To obtai n 'Gu', one must reiterate the first premiss and use ' Gx)
1, R
Hx )
Fu " Gu Gu Hu Fu " Hu
8 9 10 11
::J
(V'x )(Fx
::J
6, 'E S, 8, ::>E
3-9, ::>1 10, 'VI
Hx)
In dealing with derivations in sentence logic, it was suggested that if one has a disjunctive formula available, it is a good strategy to apply vE [Q that formula immediately. Since an existentially quantified formula is like a disjunction - it claims that a predicate is nue of one thing or another in the domain - it should be no surprise that in general one should immediately use 3£ when one has an existentia lly quantified formula . This is done in deriving the conclusion of the following argument: (\fx ){ Kx ::> Ks) (3x)Kx
Ks As before, the premisses and conclusion dt=rivatio n:
I ('Vx )( Kx
afC
sct out in the sketch of tht=
2 ,I3xJKx
p p
Z i Ks
,
1
::J
Ks)
Since ont= has an t=xistt=ntially quantified formula, one should try to obtain ' Ks' by 3E. The skt=leton of the dt=rivation is suitabl y amt=ndt=d: ('Vx)(Kx ::J KsJ
p
2
(3x )Kx
P
J
u~
A
Y
i Ks
Z
i Ks
2, 3-Y, 3E
It should be noted that moving 'Ks' from line Y to lint= Z is It=gitimate, because the paramt=ter 'u' does not occur in 'Ks'.
197
S5.9: DERIVATION STRATEGIES
What is needed now is a way o f getting 'Ks' from ' Ku '. ' Ku ::> Ks' can be obtained from '('9'x)(Kx ::> Ks )' by '9'E. Bearing in mind the ' u'·barrier, '(V'x)( Kx ::> Ks)' must be reiterated from line 1 before '9'E is used. It is then a simple matter to complete the derivation by obtaining 'Ks' at line Y by :::lEo Once this is done, all that remains is to number the lines and justifications:
P P
(\fx)( Kx ::> Ks) (3x )Kx
1
2
~u
3 4
A
(\fx)( Kx
5 6 7
Ku
:::l
Ks
K, K,
::>
Ks )
1, R
4, '9'E 3, 5, ::>E 2, 3-6, 3E
In S5.8, '( 3x )- Fx' was derived from '- ('9'x)Fx ' with the missing jl.lstificalions Idt to be supplied as an exercise. It is intuitively obvious that these two formu las are quantificationally equivalent; saying that something is not F comes to the same thing as saying that it is false that everything is F. Accordingly, it should be possible to derive '- ( ...... x)Fx' from '(3x)- Fx'. Since one has an existentia lly quantifi ed formu la, using 3E would be a
sensible strategy. On the other hand, since what one needs is a
nega[(~d
formula, it would be reasonable to assume '('9'x) Fx' and look for a contradiction. Both strategies work, and both will be developed. If one startS out by using 3E to obtain ' - (\fx)Fx', one should assume '- Fu' against a 'u'·barrier and try to obtain '- (\fx)Fx' on that scope line. Since there is no 'u' in '- (\fx)Fx', the scope line can be ended once '- (\fx)Fx' is obtained and this formu la can be mo ved one scope line to the left. This strategy is sketched in what follows: 1
(3x )- Fx
P
2 :
ul-=- Fu
A
1
Y Z
I'
j - ('9'x)Fx - ( \fx)Fx
1, 2-Y, 3E
What is needed at line Y cannot be reiterated or obtained by an elimination rule. The appropriate introduction rule is -I, since the main operator of what is needed is a tilde. One shol.lld, there fore, assume '(\fx)Fx' and look for a contradiction:
S5.9:
198
(3x l- Fx
Uf-=- Fu
2
3
:
~xlFx
W
!
! If>
1
i-
X
DERIVATION STRATEGIES
P A A
- (V x )Fx Y Z - (\lxIFx
3, W, X,-I I, 2-Y, 3E
An obvio us ca ndidate for a contradictio n is the pai r 'Fu' and '- Fu'. '- Fu' can be reitera ted from line 2 and 'fu' can be obtained by 'o'E from line 3. The derivation is completed using this contrad ictory pair:
(3xl- Fx
1 2
U
_
Fu
r
XIFX Fu - Fu
3 4 5
6
- (Vx )Fx
7
- (\lxIFx
P A A 3, \IE 2,R 3, 4, 5, - I 1,2--6, 3E
The other possible Strategy in deriving '- ('v'x) Fx ' fro m '( 3xl- Fx' is [0 assume ' (\fxlFx' immediately and look for a contradiction. The o utline of [he derivation follo ws: 1
I ( 3x ) -
Fx
2 :
~ x lF x
X t
i
Y'I Z
i-
P A
¢
- ('v'x)Fx
2,X,Y,-I
'(\fx )Fx ' by itself holds no promise of a contradiction. Accordingly, one must reiterate ' (3x) - Fx ' and use 3E to try to get a contradictory pair. Once '( 3x)- Fx' has been reiu: rated, and '- Fu' assumed against a 'u' -barrier, an easy conuadiction is apparent. One has '- Fu' and one can obtain 'Fu' simply by reiterating '(Vx )Fx' from line 2 and using \:;f E. The skeleton of the deri vation is amended to show this:
S5.9: DERIVATION STRATEGIES (3x )- Fx (\>' x)Fx
2
3
(3x)- Fx
4
U~FU
(\>'x )Fx Fu
5 6
Z
P A
1, R A
2, R
5, \>'E
i
X y
199
I
i - - (\>'x)Fx
2,X , Y, - I
'- Fu ' and 'Fu' are certainly a contradictory pair, but they are on the wrong scope line. The contradiction has to appear on the scope line which begins at line 2, under the assumption of '(\>'x)Fx'. Since both formulas contain 'u', neither can be moved onto that scope line by a legitimate appeal to 3E. The problem o f obtaining a contradiction on that scope line remains. It may appear to be insoluble . Under the assumption of '(\>'x )Fx' at line 2, a contradictory pair could not be obtained without reiterating '(3x) - Fx' and assuming '- Fu' against a 'u'-barrier. As soon as this is done, one is on the wrong scope line. Although it may seem that either one cannot get a contradiction at all or one can get It only in the wrong place, there is no need to abandon the derivation; the contradictory pair, 'Fu' and '- Fu', can be used. Neither '- Fu' nor 'Fu' can be moved to the scope line which begins at line 2 b«ause they both contain the parameter of the barrier. Any contradictory pair which drn=s not contain 'u', however, could be moved. Fo r example, if one o btained 'Fa' and '- Fa' on the scope line which begins at line 4, o ne could move both of those formulas. It would be easy to obtain 'Fa'. One would simply assume '- Fa', reiterate 'Fu' and '- Fu', and obtain '- - Fa' by -I. One could then obtai n 'Fa' by -E. Similarly, it would be easy to obtain '- Fa '. One would assume 'Fa', reiterate 'Fu' and '- Fu', and obtain '- Fa' by - I. The derivation is completed by following this procedure:
200
I 2 3 4 5 6 7
S5.9: DERIVATION STRATEGIES
(Vx )Fx
P A I, R A 2, R
Fu
5, VE
(3x)- Fx
('Q'xJ Fx
(3x)- Fx U
_
Fu
~ Fu Fa
A 6, R 4,R
8 9 10
- - Fa
7,8,9, - I
II
Fa
to,-E
12 13 14 15 16 17 18
- Fu
r
Fu - Fu -Fo Fa - Fa - (\fx )Fx
A 6, R 4, R 12, 13,14, -I 3, 4-11 ,3E 3, 4-15,3E 2, 16, 17, - I
Looking back at the completed derivation, it can be see n that the step at line 11 was unnecessary; '- - Fa' and '- Fa' would have served as a contradictory pair. An even greater economy is possible. Rather than finding a contrad ictory pair, one could obtain a fo rmula which would form a contradictory pair with something tha t is already on the scope line which begins at line 2. Since neither '- (3x)- Fx' nor ' - ('v'x)Fx' contains ' u', and both of (he' form ulas contradict a formula which is on the scope line which
begins at line 2, obtaining either one of them would be a sensible strategy. The shaner version of the de ri vation foll ows:
\5.9: DERIVATI01\ STRATEGIES
2 3 4
201
P
(3x)- Fx (V'x)Fx
A
(3x)- Fx u _ Fu
1, R A
5
(V'x)Fx
2,R
6 7
Fu
5, V'E
~X) FX
8 9 10 11
12
Fu
- Fu - (V'x)Fx - (Vx)Fx
A
6,R 4,R 7,8,9, - I 3, 4-10, 3E
2,2,11, - I
- (lIx)Fx
Even this shaner version of (he derivation of '- (V'x)Fx' from '( 3x)- Fx' took twelve steps. The version which started out by immediately assuming '- Fu' and obtaining '- ('Vx)Fx' by 3E took seven steps. This fact supports the suggestion made earlier in this section that immediately using 3E is usually the best strategy when one has an existentially quantified formula available. JUSt as in senrence logic one may have to do disjunction eliminations within disjunction eliminations, so in predicate logic o ne may have to do existential quantifier eliminations within ex istential quantifier eliminations. This will be necessary in deriving the conclusion from the premisses of the following argument. Clearly, the argument is quantificationally valid; if something is F and something is not F, there are at least two distinct things. (3x )Fx (3x )- Fx (3xIl3y)- x = y The derivation can be started by listing the premisses and the conclusion: 1 I (3x)Fx 2 ,(3x)- Fx Z
i (3xIl3y)-
p p x=Y
'(3x)(3y)- x = y' appears nowhere in what is available, a nd thus it might appear that no elimination rule can be used. In order to use the introduction rule, 31, one would need something like '- a::: b', and there seems to be no way of getting such a formula. Since the argument is quantificationally valid, [here has to be a way of getting the conclusion from the premisses. The thing to do in a situation lik e (his is ( 0 start by using the appropriate rule even
202
S5.9: DERIVATION STRATEGIES
though it is not dear how onc's strategy will develop. In [his case, the appropriate rule is 3E. There are (wo existentially quantified formulas, and
no apparent reason for using one before the other. As an arbitrary choice, one can start with the first premiss, assuming 'Fu' against a 'u'·barrier. The concl usion does not contain 'u', so if one can obtain it against the 'u··barrie r, it can be brought back to the primary scope line. Since it is dear t hat the second premiss is necessa ry to obtain the conclusio n, one can reiterate it and assume '- Fv' against a V-barrier. The outline of the derivation follows: 1
(3x lFx
2 3
(3x)- Fx
4 .I X y
Z
P P A 2, R A
r
U
(3 x)- Fx
: '1-=- F,
i
i (3x )( 3y)-
x=y (3x)( 3y)- x = y
(3x)(3y)- x = y
4, 5-X, 3E
1, J.-Y,3 E
What is needed at line X could be obtained by two applications of 31 if one had any negation of an identity formula. The obvious candidate is '- u = v'. This formula cannot be reiterated, nor dOts it appear as part of a formula which can be reiterated. Since th(' main operator of the needed formula is a tilde, o ne should try to obtain it by - I. The skeleton of the derivation shows this: 1 (3x)Fx
p
2 (3x l- Fx 3
P
4
5 6
ur
..
T
i,, ,
U V W
'
X y
A
(3x)- Fx
2, R
lJ' .
.
A
.1-"-= v
A
i,, III ,
i-
$
- u =v (3y)- u=y (3x)(3y)- x = y (3x)(3y)- x=y
Z (3x )(3y)- x = y
6, T, U,-l V,3(
W,31
4, 5-X, 3E
1,3-Y,3E
203
SS.lO: EXER CISES
It is easy to find a contradictory pair to take th e place of '$' and '-
¢I'. Fo r example, one can obtain ' Fu' and '- Fu' simply by reiterating ' Fu ' and '- Fv' and using =E o n ' - Fv' to obtain ' - Fu ', The derivatio n is completed using this contradictory pair:
1 (3 x )Fx 2 (3x )- Fx u Fu 3 4 (3x )- Fx , - Fv 5 6
P p
7
3, R 5, R 6,8, =E 6,7,9, - I 10,31
A
2, R A A
~;v - Fv
8
9 - Fu - u =: V 10 (3y)- u = y 11 12 (3xIl3y)- x = y (3xIl3y )- x = y 13 14 (3xIl3y)- x = y
11,31 4, 5-12, 3E
1, 3-13, 3E
S5.10: EXERCISES Fo r each o f the following argu ments, derive the conclusion fro m the premi ss(es).
(lIxlllly)Kxy (Vy)(V x )Kx y
2 (lIxlillYIIFx " Gy) (Vx)Fx :::> (Vy)Gy
3 (Vx)(Kx
:::> Lx )
(\fx)- Lx (Vx) - Kx
4 (3x )( 3y )( 3z)Rxyz Raaa (3 x )Rxxs
5 (3x)Mx " Mk (3x )Mx == Mk
:::> Raas
204
S5.11: MORE DERIVATION STRATEGIES
6 (Vx)(Vy )(Rxy
Ryx)
:J -
R,b
- Rba 7 ("'x)( Px ; Qx) - (3x)Qx
_ l'a
8 (V'x)( Mx
Vx)
J
- ( Vx)Vx
- (' - Dyx))
1\
u~u"
Au)
i (3x)(Vy)(
1\
Wx
(3xll"yll(Wx
A
(Vy)( M y ::::> - Dyu) Ax )
1\
Ax )
A
1\
P A
(My::::> - Uyx))
(My::> - Dyx))
1,2- Y,3E
What is needed at line Y is an existentially Quantified formula. To obtain this by 31, one needs a universally quantified formula in which every occurrence of 'x' in the existentially quantified formula is either the same constant or the same parameter. A glance at what one has on line 2 shows that this term should be the parameter ·u' . T he outline of the derivation is expanded to show this: I
I (3x)((Wx
2 : X ' Y Z
u~u" 'I
Ax)
A
Au)
("YIiMy ::> - Dyx))
A 1\
(Vy)(My ::::> - Oyu)
(Vy)«Wu 1\ Au) 1\ (My::::> - Oyu)) (3x)(Vy)( Wx 1\ Ax) 1\ (My::::> - Dyx ))
(3x )( Vy)({Wx
1\
Ax)
1\
(My::::> - Dyx))
P A
X,31 I , 2-Y, 3E
Since the new formula one needs is a universally quantified formula which cannot be obtained by an elimination rule, one should use universal quantifier introduction. A 'u'-barrier is already in place, so one should introduce a 'v'-barrier in o rder to use V I. What is available at line 2 will have to be used to the right of the V-barrier, so it can be reiterated now: I 2
(3x )((Wx
Ax)
u~u " Au)
3
.
W
.
X
1\
1\
(Vy)(My ::::> - Oyx))
(Vy)( My :::0 - Dyu) V! (Wu " Au ) 1\ (\fy)(My ::::> - Dyu )
i (Wu
1\
" Au )
1\
A
Ax )
A
A 2,R
(Mv ::::> - Ovu)
(Vy)(Wu " Au ) " (My::::> - Dyu)) (3x)(Vy)(Wx 1\ Ax) 1\ (My::::> - Oyx))
Y Z (3xll"yll(Wx
P
(My::> - Dyx))
W, "I
X,31 l ,2- Y,3E
206
SS.11: MORE DERIVATION STRATEGIES
What is needed at line W is a conjunction. The left conjunct is identical to the left conju nct of the conjunction on line 3 and can be o btained by I\E. The right conjunct of what is needed can be obta ined from the right con junct of the conjunction on line 3 simply by using "tE. Accordingly, the derivation can be completed by using I\E twice and 'IE on ' ('9'y)(My ::> - Oyu)' One can then obtain the formula one needs by AI: (3x)( (Wx , Ax) , (Vy)( My ::> - Dyx))
2
u « Wu
J
1\
Au)
v (Wu
(\f y)(My ::> - Oyu))
1\
Au )
A
(\fy)(My ::> - Oyu )
A
4
Wu
5 6
(V'y)(My ::> - Dyu ) Mv ::> - Dvu
7
1\
(Wu
Au
Au )
A
8
(Vy)((Wu
9
(3x)('Vy)((Wx
1\
1\
Au ) 1\
1\
Dru»
(My::> - Dyx ))
(3x)(Vy)((Wx , Ax ) , (My::> - Dyx))
10
2,R 3,I\E 5, 'v'E
(My::> 1\
A
3,,,E
(Mv ::> - Dvu)
Ax)
P
4, 6, /\1
7, VI
8,31 1,2- 9,3E
The next derivation problem is to derive the formula on line 1 of the abov ~ d~rivation from th~ formula on line 10. On c~ again. one has an ex ist ~ntiall y quantifi~d formula and one needs an existentially quantified fo rmula . Thus, chis new deriva tion will be similar to the previous one . The outline of t he derivation reflects this: I
1
2 :
Y
Z
I'
(3x )( 'Vy)((Wx " Ax) " (:\4y ::> - Dyx)) u~YII( Wu, Au) ,
i (3x)((Wx " Ax )
(My::> - Dyu))
('Vy)(My ::> - Dyx )) (3x)« (Wx " Ax) " ('Vy)( My ::> - Dyx))
P A
1\
1,2-Y,3E
An existentially quantified formula is needed at line Y. In order to obtain this by 31, one needs a conjunctive formula, one part of which is a universally quantifi~d formula. This universally Quantified fo rmu la can be obtained only by 'VI, so it appears that the most sensible thing to do is to stan a V-barrier and cry [0 obtain che universally quantified fo rmula. The sketch of the derivation is ex pand~d appropriately:
207
§5.11: MORE D ERIVATION STRATEGIES
(3x)('Vy)( Wx
1
2
" f'1Y)( WU
3
Ax )
1\
Au)
(Vy)((Wu
4 Y Z
1\
. I'
(Wu
1\
1\
(My::::> - Dyx ))
1\
(My::::> - Dyu))
Au )
A
Au )
1\
A
(My::> - Dyu ))
(Mv ::::> - Ovu)
i (3x)(( Wx A Ax) A (Vy)( My ::> - Dyx ll (3x)((Wx A Ax) A (Vy)( M y ::> - D yx))
P A
2, R 3, VE
1, 2-Y, 3E
It seems clear now what fo rmula o ne should try CO obtain in o rder CO use 31 at line Y: it is the co njunction of 'Wu 1\ Au ' and '('1y)( My : : > - Dyu)'. The outl ine of the derivation is expanded by adding this conjunctive formula at line X and the twO conjuncts which are needed at li ne Wand lin e V: (3x)(Vy)((Wx
2 3 4
v w
Ax )
A
(My::> - Dyx) )
uf'VY)« WU 1\ Au ) 1\ (M y::::> - Dyu) ) VI I'VY)( Wu 1\ Au ) 1\ (My ::::> - 0 Yll )) , (Wu 1\ Au ) 1\ (Mv ::::> - Ovu )
i (Vy)( M y ::> . Wu (Wu
Y
(3x)( (Wx
(3x)«Wx
P A
2, R 3, 'V E
- D yu )
Au
1\
X
Z
A
1\
1\
Au) A
Ax)
1\
('1y)(M y ::::> - Oyu)
Ax) 1\
A
(Vy)(My ::> - Dyx ))
('1y)( My ::::> - Oyx ))
v, W, 1\1 X,31 1,2-Y,3 E
If one ended the V-barrier and used 'VI with the variable 'y' on the fo rmula on line 4, the result would be exactly what one has at line 2. It is clear that this st rategy would be pointless. Clearly, it is necessa ry to break up the conjunctio n on line 4. The outline of the derivation is expanded: 1 (3x)(Vy)((Wx
2 3 4
5 6
v
w X
Y Z
A
Ax )
A
(My::> - Dyx))
u ('1y)( Wu 1\ Au ) 1\ (My::::> - Oyu») v ('1y ){{Wu 1\ Au) 1\ (My ::::> - Oyu )) (Wu 1\ Au ) 1\ 1M v ::::> - Ovu ) Wu 1\ Au :\1'0' : : > - Ovu
!('1y)( My ::::>
A
Ax)
A
2,R 3, VE 4,I\E 4,I\E
- Oyu)
Wu 1\ Au (Wu 1\ Au ) 1\ ('1y)( My ::::> - Oyu) (3x)(Wx 1\ Ax ) 1\ (Vy)(My ::::> - Dyx )) (3x)IIWx
P
A
(Vy)( My ::> - Dyx ))
v, W, 1\1 X,31 1,2-Y,3E
208
SS. 11: MORE D ERIVATION SlllATIGIES
It is easy to o btai n '(\fy)(My :;, - Dyu)' by VI fro m the formula o n line 6. The problem is with the formula which is needed at line W. Thue is no rule which allows o ne to move 'Wu " Au ' from line 5 to the scope line which begins a t li ne 2. If one tries to conjoin this form ula with the formu la on line 6, to bring it over by the 'VI rule, one ends up with the fo rmula one had at li ne 2. T he form ula at line 5 is simply of no use, and it can be erased from the outline. One should use '0'1 to obtai n '(''lfy)( M y ::> - Dyu )' from ' Mv :;, - Dvu', and then turn to the pro blem of obtaining 'Wu A Au' without the complica tion of the 'v'-barrier. Th e ou tline of the derivation rcRee(s this suategy : I 2
3
(3x )(Vy)((Wx
A
Ax )
(Vy)((Wu
A
Au ) " (M y
U
v (\fy)((Wu
1\
(Wu
5
Mv ::::> - Dvu
6
Wu
X
(Wu
Z
1\
1\
::J -
D yu )
(My ::::> - D yu))
(Mv ::::> - Dvu )
, (VYIl My " - Dy")
W Y
Au )
Au)
4
1\
(My ::> - Dyx ))
A
1\ 1\
P
A 2,R 3, 'v'E 4 ,1\£
5, VI
Au Au)
1\
('v'y )(My ::::> - Dyll)
(3xll(Wx A Ax ) A (VYIlMy " - Dyx )) (3x)«Wx A Ax) A (VyilMy " - Dyx))
6, W, .... I
X,31 1, 2-Y,3E
There remains only the problem of obtaining ' Wu 1\ Au' at line W. T his problem is easi ly solved. It can be obtained from the formu la on line 2 by 'IE and I\E just as it was before, but this time the formu la is not first reiterated across a V -barrier. In us ing 'lfE here, it does not matter wha t term replaces 'y'. When the \iE step is completed. the result will be: a conjunction. The conjunct one needs, 'Wu 1\ Au', can be extracted by I\E, and it is unaffected by the choice of the term which replaces 'y', That part of the co njunctive fo rmula which is affected by this choice will not be used. The derivation is completed by a rbitrarily picking 'k' as the replacement for 'y' in the application of \iE, and going on to use I\E to extract what o ne needs:
209
§S.ll: MORE DERIVATION STRATEGIES
2
3
(3x )(V'y)( Wx
A
Ax )
A
(My ::J - Oyx) )
P
(V'y)(WU
A
Au)
A
(My ::J - Oyu) )
A
v (Vy)( WU
4
(W U
5
A
2, R
Au) " (My ::> - Dyu ))
A
Au )
A
3, VE 4,AE
(Mv ::J - Dvu )
MV::J-Dvu
6 7
(V'y)(My ::J - Dyu)
8
Wu
9
(W U
(WU
10
Au )
A
(3x)«Wx
(Mk
::J -
Dku )
7,AE
Au
A A
Au)
(3x )( Wx
11
A
5, Itl 2, ItE
A
A
Ax)
A
(V'y)(My
Ax ) A
A
::J -
Dyu )
6,8, AI
9,31
(V'y)(My ::J - Dyx ))
(V'y)(My
::J -
1,2-10,3E
OyX) )
In Chapter 4 other points about equivalent translations were made. One was the claim that the following twO fo rmulas arc Quantificationally t:4uivalt:llt: (3x )(Px " (lty)(Py :> x ~ y)) (3x)(lty)(Py = x ~ y)
'( 3x )(V'y)( Py == x = yl' was derived from '( 3x HPx A (V'y)(Py ::J X = y))' in §5.8, with IllOSt of the justifications left to be supplied as an exercise. Sillce these formulas are quantificatiooally equivalent, o ne can derive '(3x)(Px " (lty)( Py :> x = y))' from '(3x)(lty)(Py = x = Yl'. The standard procedure for dea ling with an e xistentia lly Quantified fo rmula is followed in starring the derivauon: I 1 (3x)(lty)(Py 2 : Y Z
I·
= x = y)
P
=y)
A
u~ y)(Py . u
i (3x)(Px
" (lty)(Py :> x = y) )
(3x)(Px" (lty)(Py :> x ~ y))
1,2-Y,3 E
To obtain the formula 00 line Y, one: should use 31 on a conjunction. Since a 'u'-barrier is in place, 'u' seems like a reasonable choice for the term which is to be replaced by the variable when 3 1 is used. The outline o f the derivation is expanded accordingly:
210
55.1 1: MORE DERlvAnON STRATEGIES
1 I (3x)(Vy)(Py = x = y) 2: u~ YIIPy , ,=y)
v i
P A
i Pu
w
(Vyil Py :>
X
Pu
1\
,= y)
(Vy)( Py ~
?
u ~ y)
Y (3xlIPx A (VyilPy :> X = y)) Z (3xlIPx A (VyilPy :> X = y))
V, W, AI
X, 31 1, 2-Y,3E
There a(e now two problems to be addressed. Approaching one problem at a ti me , one can deal first with [he matte r of obtaining 'PIl'. One could get 'Pu' by using vE and then ~E on the formula on line 2, provided one had ' u = u', This is an occasi on on which the rule of =1 is useful; one can simply write 'u = u' and justify [his Step by =1. The Dudi ne of the derivation is developed to show how the first problem is solved:
(3x )(VyIlPy •
2 3 4 5
w
~:~(PY •
X
= y)
u = y)
Pu '" u = u p,
3, 4,=E
(VYIlPy :> ,= y)
Pu 1\ (\I'y)(Py ::J u = y) (3 x){Px A (Vy )( Py ::l X = y)) Y Z (3xllPx A (VYIlPy :> X = y))
X
P A =1 2, 'IE
5, W,,,I X,31 1, 2-Y, 3E
The mhcr fo rmula which is needed is '(Vy)( Py => u = yl' . It is a universally quantified formula which cannot be obtained by an elimination rule, One should, therefore, introduce a barrier with a parameter which has not been used, and try to obtain ' (\iy)(Py ::l u = y)' by \i l. 'Pv ::l u = v' against a 'v' -barrier would serve, To obtain this formula, one should assume ' Pv' and try to obtain 'u = v', The outline of the derivation is amended to show this:
211
S5.11: MORE D ERIVATION STRATEGIES
1
13x )("y)( Py
= x = y)
P
2
l"y)(Py
= u = y)
A
3
u=u
:I
4
Pu:;;; u =u
2, "E
5
Pu
3, 4,=E
6
v,
U
I
V
W
~
A
I u=v PV:J u=v
6- U, :::::ll
l" y)(Py => u = y)
X
Pu " (Vy)( Py :J u = y )
Y
{3x )( Px "(Vy)( Py::> x=y ))
Z
(3x )( Px " (V'y )(Py :J
X
= y))
V, VI 5, W, ....I X, 31 I , 2- Y, 3E
The obvious wa y of o bta ining ' u = v' is from the form ula on line 2. Once this is rei terated, an application of VE yi elds ' Pv "" U = v' which, together with 'Pv', allows one to obtain 'u = v' by i!!E. Thus, the derivation can be compl eted: 1 2
= x = y) l"y)( Py = u = y)
13x )( "y)( Py u
3
u=u
4
Pu
5
Pu
6
v
A =1
= u= u
2, VE 3, 4, :sE
7
8 9 10
P
Pv
~
A
l" y)(Py • u = y) Pv:;;; u =v
2,R
u= v
6,8,:o:E
~
u= v
6- 9, :::::lI
11
l"y)(Py => u = y)
12 1)
Pu " (Vy)(Py ::> u = y)
14
(3x)( Px (3x)( Px
A
A
7, VE
(Vy)( Py :::::l
X
= y))
(V'y)(Py ::> x = y ))
10, VI 5, I I , A I 12, 31 1,2- 13,3E
Ano ther problem which requires some tho ught is the derivati o n of the conclusion from the premiss of the fo ll owing argument: (Vx )( Fx v Gx ) (\fx )Fx v (3x)Gx
S5.11: MORE DERIVATION STRATEGIES
212
looking at the problem should make it clear that neither of the disjuncts which make up the conclusion can be derived from the premiss; from the claim that everything is either F or G, onc can infer neither that everything is F nor (hat something is G. Trying to obtain one of the disjuncts and using v i to get the concl usion, therefore, would be a hoptless suatcgy. What onc must do here is go immediately to an indirect proof. The derivation is started accordingly:
JUSt
1
I (' Rkcl Rkc ::> ('o'x)Rxk (V'x)(Rkx ::> Rck )
2 (lI xl- Kx - (3 x)Kx
3 (3x)Fx v (3x)G x (3x)( Fx v C x)
216
§5.12: EXERCISES
4 (Vx)(Lx v Ma l - (3x)Mx ::> (3x )Lx
5 ( 3x )( Fx v Gx ) {3x )Fx v (3x)Gx
6 ( Vx)Fx " (V'x )Gx (Vx)( Fx
1\
Gx )
7 (Vx)- Rxx (lIx)( lIy )(( 3z )( Rxz
A
Rzy ) ::> Rxy )
(lI x)( lIy )( Rxy ::> - Ryx ) 8 (V x)(Ax ::> Bx ) (\f x )- (Sx vex )
- (3x)Ax
9 (Vx)(Gx ::> (3y )Rx y) (lIx)(lIy)( Ryx ::> Ryy ) (Vx)( Gx ::> Rxx )
10 (lIx)( Mxa ::> Mxb) (3x )(Mxa v Mxc) (3x )( Mxb v Mxc ) 11 ('Vx)Fx v ( Vx )- Fx (3x )Fx :::> ( \fx lFx
12 d=e ::> (3 x )( Fx " - Fx ) - d=e
13 (Vx)( Vy )«(( Fx " Fy)" (- Gx " - Gy) ) ::> x= y) - a= b
Fa /\ Fb Ga v Gb 14 i:k v i=m - k=m
(3x )- i=x 15 (3x )( Mx A (3y)- Qxy ) (lIx)( Mx ::> (lIy )(Py ::> Qxy)) - ( Vx )Px
S5. 13:
CATEGORICAL DERIVATIONS IN PREDICATE LOGIC
217
16 (\fx)(Ax ~ (3y)(Dy A Cxy)) (IIx){Ax => (IIy){ Dy => - Cxy) ) (3x)Ax
::J -
(V'x)Fx
17 (Vx)( Fx A GX) - (V'x)Rx (3x)- Fx
V
Rx)
::J
(3x)- Gx
18 (3x)( Fx V Gx) (Vx)(Hx ::> Ix) (3x)Fx
::J
(3x)Hx
(3x)lx
19 (3x)Fx => (3x)Gx (IIx){3y){ Fx => Gy)
20 (V'x)- Fx (V'x)- Gx - (Fa
V
Ga)
S5.13: CATEGORICAL DERIVATIONS IN PREDICATE LOGIC Categorical derivations in predicate logic are very similar to categorical derivations in sentence logic. They are used to derive Quantificational truths and they almost always begin with an assumption. That everything is either F or not F is intuitively obvious, and it should be possible to produce a categorical derivation of '(V'x)(Fx v - Fx )'. To obtain this formula, one needs 'Fu v - Fu' against a 'u'-barrier, which means that the problem is no more difficult than a categorical derivation in sentence logic. The derivation follows: u
2
J 4 5
6 7 8 9
- (Fu v - Fu)
t
Fu v - Fu - (Fu v - Fu ) - Fu Fu v - Fu
- - (Fu v - Fu) Fu v - Fu (V'x)(Fx v - Fx )
A A 2, vi I, R 2,3,4, -I 5, vI 1, I, 6, -I
7,-E 8, II I
S5. 13:
218
CATEGORICAL OERIV ATlONS IN PREDICATE LOGIC
In predicate logic, juSt as in sentence logic, indirect proof is often very useful in producing ca tegorical derivations. Indirect proof is used again to derive '(\lxIl3y )Rxy v (3xll\ly)- Rxy',
- (l\lxIl3y)Rxy v 13xll\ly)- Rxy ) u - 13y )Ruy
2
v
3 4 5 6
~"'
13y)Ruy - 13y)Ruy
3,31 2,R 3, 4,5,-1 6, VI
- Ruv
7
(' 13y)Gy ) 8 l"xllFx => Gx) v 13y)Fy 9 l"xll Fx => 13yllx = y
A
Fy))
10 ('u
p
i ¢u
q
A
A
o/u
(11,),
m,R n, 'iE
0,
"E
p, VI
Identity introduction (See p. 190): A rule, =1 , which allows one to write '~ =~' anywhere in a derivation with no appeal to a line number. An example of =1 follows: m ! p=p~4>'t
n
! p= p
o i 4>'t
m, n, =.IE
Idencity elimination (See p. 191): A rule, =E, which allows one to replace any term, ''1\', in a formula such as '4l't 1', with another term, ''1 2', provided that an identity formula, ''1 t ='1 2', is already on the current scope line. Every use of identity elimination requires an appt'aJ to exactly twO lines. An example o f =E follows: m
! (3x)~ac
n
i c=d
o
i (3x)cDad
m, n,=E
CHAPTER SIX
MISCELLANEOUS MATTERS
S6.1: THE SCOPE OF THE "TEXT Essentials of Symbolic Logic was written to provide students with a short introd uction [Q the vast and sometimes bewildering fiel d of symbolic logic. Those students who pursue: the: subject furthe r will find some unfamiliar vocabulary, conventions and symbols. This cha ptn is provided to mah things a little easier for such students. Those students who approach metameory will find several unfamiliar symbols. Any decent discussion of met3theory will explain the meaning of such symbols as they are inuoduced. so thq will not be d iscussed here. Some systems of derivations use very few rules and ma ke appeals to axioms. The Elements. Euclid's study of geomeuy, provides the most famous examples of proofs which appeal to axioms. The system of derivations used in this book is called a system of 'natural deduction' beca use it uses rules without any appeals to axioms. In this respect, it resembles everyday reasoning. Some natural deduction systems use scope lines and make frequent use of reiteration, while others do not. Some of these variations are discussed in this chapter. However, the student should be aware that Essentials of Symbolic Logic is a shon introductory text, and there is much which is not even mentioned.
223
224
S6.2:
S 6.3: THE N AMES OF THE RUL ES
EXISTENCE AND TRANSLATION PROBLEMS
In S4 .1 3. it was pointed out that ' Mary is in te rror o f werewolves' could not be adequately translated using either the universal or existential quantifier, since such u anslations would either be triviall y true or assert the exiStence of werewolves. There. the prob lem was avoided by the simple expedient o f using the one-place predicate, 'Tm', [0 translate the entire sentence. There are other problems about existence in the discussion of dc:finite descriptions in S4.12 and the rule of existential qua ntifier introduction in SS.2. In general, there is no problem in using the existential quantifie r to translate expressions such as 'the first man to climb Mount Eve rest'. However, it is possible to use a definite description in a sentence such as 'The dog which is smarter than Einstein is old.' There is no such dog and it would be misleading to assert the existence of such a dog. Problems of this SOrt have concerned logicians for years; the re are serious questions which can be addressed in the logic of fiction and other areas. These topics cannOt be addressed in a book of this SOrt. 'The dog which is smarter than Einstein is old' can be translated as: '(3xj(Sxe 1\ (Dx 1\ Ox»'. To deny that the re is such a dog one can write: '- (3x)(Sxe 1\ (Dx 1\ Ox)),. The rule of existential Quantifier introduction can lead to simi lar puzzles. From 'John is a mief', one can de rive '(3x)Tx'. Howeve r, from ' Atlantis is a lost comine nt', it would be wrong the assert the exis tence o f a lost continent: '(3x)(Cx 1\ Lx)' . The easiest way ro avoid such problems is to avoid using the names of non-existent people and things. Not using such names, of course, makes d ea ling with fictional people and things impossible. One final problem: from 'John read a big book cardully', one cannOI derive 'J ohn read a big book', 'John read a book carefu lly' or even 'John read a book', almough these are intuitively correct inferences. The predicate logic discussed in this text cannot deal wdl with modifiers, and is unable to reflect the connections between reading carefully and reading, or between big books and books. To deal with inferences of this sort, one needs a more advanced system of logic. A short introductory text can only point out a few such limitations and the need for a more sophisticated predica te logic.
S6.3:
THE NAMES OF THE RULES
There is a minor problem with the names of some of th e rules in Essentials of Symbolic Logic and in othe r texts. For example, strictly speaking, conjum::tion intnx..lUl.:tion docs not introduce a conjunction; it introduces the caret, and should be called 'caret introduction'. Since using this terminology would be eccentric, it is not used in this book. The following list provides rule names in English which could be used to avoid a rather pedantic objecti on.
S6.4: UNFAMILIAR TERMINOLOGY AND SYMBOLS
225
1 '-E' could be called 'tilde elimination'.
2 '-I' could be called 'tilde introduction'.
3 '"E' could be called 'caret elimination'. 4 '"I' could be called 'caret introduction'. 5 'vE' could be called 'vel elimination'. 6 'vI' could be called 'vel introduction '. 7 '=E' could be ca lled 'identity symbol elimination'.
8 '=1' could be called 'identity symbol introduction'.
S6.4: UNFAMILIAR lERMINOLOGY AND SYMBOLS For the most part, this book has avoided eccentric terminology and symbols. However, many authors use differe nt names for rules which operate in much the same way as the rules of this book, and use symbols which are slightly different from the symbols used in this book . A few of the more common variations in the names of rules are listed below:
1 'Conjunction elimination' is called 'simplification', or 'Simp'.
2 'Conjuncti on introduction' is ca lled 'conjunction', or ' Conj '. 3 'Negation elimination' is called 'double negation', or ' ON'. 4 'Negation introduction' is abbreviated as 'NT'.
5 'Disjunction eli mination' is called 'constructive dilemma', o r 'CD'. 6 'Disjunction introduction' is called 'addition', or 'Add'. 7 'Horseshoe elimination' is called 'modus ponens', or 'MP'.
8 'Horseshoe introduction' is called 'conditional proof', o r 'CP'. 9 'Triplebar elimina tion' is call ed 'biconditional elimination', or ' BE'.
to
'Triplebar introduction' is called 'biconditi onal introduction', or 'BI'.
11 ' Indirect proof' is often called 'reductio ad absurdum', or 'RAA' .
12 'Universa l quantifier introduction' is ca lled 'universal generalization', o r 'UG'.
13 'Universal quantifier elimination' is called 'u niversa l instantiation', o r 'VI'. 14 'Existential quantifier introduction' is called 'existential generalization', or 'EG'. 15 'Existential quantifier elimination' is called 'existential instantiation', or 'EI'. Some more examples of variations in symbols and terminology follow.
226
S6.4: UNFAMILIAR TERMINOLOGY AND SYMBOLS
Some authors use a system of ranking which reduces the need for brackets. For example, one can stipulate that '::J' outranks ',,'. Given this stipulation, ' A :::> 8 "C' is nor ambiguous; it is to be read as 'A ::> (8 1\ q'. Fortunately, this practice seem to be falling into disuse . Another pracria which students may come across is the use of different kinds o f brackets. '( - (A v - (8 ::::> C)) '" 0) v A' is sometimes written as '{- (A v - (8 ::J C)] ;;;: O} v A'. Students can decide for themselves whether this increases clarity. Many authors use a few lower-case letters and subscripts for sentence letters rather than the upper-case letters used in this book . Some authors refer to sentence letters as ' propositional variables'. Rather than use 'T's and 'F's in truth tables, some authors use 'l's and 'D's o r 'T's and '1.'s, Neither of these practices should confuse students. Rather than use '-', some authors use '.,'. Both '&' and ' . ' are often used instead of ' A', '=>' and '-+' are often used instead of ':J', and .~' and 'H' are often used instead of '!;'. In predicate logic, many authors use '(3x)' and '( x)" rather than '(3x)' and '('ix)'. An open sentence is sometimes said to express a 'propositional fun ction', and many authors use 'propositional function' in t he way that 'open sentence' is used in this book. When a variable in an open sentence is not linked to a quantifier, it is often called a 'free variable'. When a variable is linked to a quantifier, it is often called a 'bound variable'. Some derivation systems use free variables in much the way that parameters are used in derivations in this book. A notation which is very different from that of Essentials of Symbolic Logic is Polish notation , It will strike most students as very odd, but it does have the advantage that all formulas of sentence logic can be produced on a standard typewriter. Polish notation uses lower-case letters for sentence letters and requires no brackets. A few examples follow: '- P' is written as 'Np', 'P :J Q' is written as ·Cpq'. 'P A Q' is written as 'Kpq', 'P v Q' is written as 'A pq'. ' P == Q' is written as 'Epq'. 'P :J (Q :J R)' is written as 'CpCq r'. '(P :J Q) :J R' is written as 'CCpqr'. As might be guessed from the last twO examples, the first upper-case letter of a formula is the main operator. Any good discllssion of Polish notation explains how the position of the upper-case letters makes brackets unnecessary. The formulas of predicate logic can also be written in Polish notation, but a standard typewriter is inadequate for this. A variation of Po lish notalion is sometimes used in electronic calculators.
S6.4: Ur-.'FAMllLAR TERMINOLOGY AND SYMBOLS
227
In Chapter 2, truth-functionally true, truth-functionally false and truthfunctionally indeterminate formulas were discussed, as were truthfuncti onally equivalent formu las. In Chapter 4, analogous concepts in predicate logic were introduced: quantificational truth, quantifica tional falseh ood, quantificational indeterminacy and quantificational equiva lence. Both truth-functional truths and quantificational truths are often ca lled 'theorems', 'tautologies' or 'valid formulas'. Sometimes they are ca lled 'logical') 'ana lytic' o r 'necessary' truths. Both truth-functional falsehoods and quantificati onal falsehoods are sometimes said to be 'logically', 'analytically' or 'necessarily' false. BOlh truth-functionally indeterminate and quantificationally indeterminate formulas are often called 'contingent' or 'logically contingent'. Similarly, the term 'valid argument' is often used to describe both truth-functionally valid arguments and quantificationally valid arguments. The student should be aware that there is conside rable variation in the way that the terms mentio ned in this paragraph are used . For example, some authors usc 'analytic' and 'necessary' as synonyms, while others usc the different terms to make what they conside r to be an important distinction . Necessity and contingency arc nmions dealt with by modal logic, a topic which has not been addressed in this book. Students who continue in logic will come across [he three symbols: '0', '0' and '-3'. 'A ::J A' is necessarily true, and this necessary truth can be expressed in terms of modal logic as '0 (A ~ A)' . ' 0 (A :::> A)' can also be expressed as 'A ---3 A' which is rea d as 'A strictly implies A.' Similarly, the necessary falsehood of 'A 1\ - A' can be expressed as '0 - (A 1\ - A)'. What is necessa rily false is not possibly true. Acco rdingly, this same necessary falsehood can be expressed in terms of '0', the possibility symbol: '- 0 (A 1\ - A)'. Besides being necessarily true or necessarily false, a statement can also be contingent, meaning that it is possibly true and possibly false. The mark of a contingent statement is that its truth value cannot be determined by logical considerations alone. For example, a formula such as 'A v B' is conti ngent; it is possibly true and possibly false. In the symbols of modal logic, these twO possibilities can be expressed as '0 (A v B) 1\ 0 - (A v B)" Since '0 - (A v B), is true, '0 (A v B)' is false. Turning to predicate logic, '(Vx){Fx v - Fx )' is necessari ly true, and this can be expressed as '0 (V'x)(Fx v - Fx)' or as '- 0 - (V'x)(Fx v - Fx )'. Similarly, the necessary truth of '( V'x )( Fx :::> Fx), can be expressed as '0 ('v'x)(Fx :::> Fx)'. The necessary falsity of '( 3x )( Fx 1\ - Fx )' can be expressed as '0 - (3x)(Fx 1\ - Fx)' or as 0 (3x)(Fx 1\ - Fx)'. The contingency of '(3xlFx ' can be expressed in terms of modal logic as '0 (3x lFx 1\ 0 - (3x)Fx'.
S6.5: AOOmONAL DERIVATION RULES
228
S6.5: ADDITIONAL DERIVATION RULES Systems of logic frequ ently use morc rules than the elimination and introduction rules of this book. Using more rules allows fo r shorter derivations, but requires that one: remember morc rules. The extra rules were avoided in this book for three reasons. First, the system o f having a single: inuoduction rule and a single elimination rule for each sentence operator is straightforward and easily understood. Second, the set of rults presented in this book is a minimal set; each rule is necessary and everything [hat can be done with additional derivation rules can be done without them. Third, all the fUltS of this book arc: presented with what is intended (Q be: a justification for their use. To learn a large number of rules which are to be accepted on faith does nothing to increase one's understanding. It is possible in principle to make all derivations extremely short. For every derivation, ant would need a rule allowing one to derive the condusion from the premiss or premisses. Obviously, it would be impossible in practice to formulate a complete set of such rules. Between this ridiculous extreme and the minimal set of rules in this book, it is possible to make derivations shorter by adding a few rules which are commonly used. tn this section, three additional rules are discussed. It should be clear that anything which can be derived with these rules can be derived using the rules of Chapter 3.
Hypo thetical Syllogi sm is usually abbreviated as 'HS'. In this derivation, n' is derived from '$ :::) IV' and 'IV :::) n' using this rule:
'$ J
p P 1,2, HS 2 Disjunctive Syllogism is usually abbreviated as 'OS'. This rule is used in the next derivation to obtai n 'cJ>' from '$ v IV' and '- IV'. I 2
cJ> vlV -
3 'V
p P 1,2, OS
3 Modus ToJJens is usually abbreviated as 'MT'. In the next derivation, '- cJ>' is derived from 'cJ> :::) 1jI' and '- IV' using this rule. 1 ¢:JIV
p
2 3
P 1,2,MT
- 'V -
S6.6:
229
REPlACEMENT R ULES
56.6: REPLACEMENT RULES As well as the additional deri vation rules, some systems of logic have replacement rules, rules which allow one to replace part of a formula with something which is logicall y equivalent, A typica l example is double negation, abbreviated as 'ON', This rule is used in the following deri vation:
A co --8 A 3 Aco8 4 8
P P 1, ON 2, 3, ::JE
2
As with the additional deriva tion rules, anything deri ved with replacement fules can be derived using the rules of Chapter 3 and Chapter 5, Four dots are usually used to signify that replacement is permissible. Thus, the rule of double negation is usually stated as '- - ¢I :: $ '. What is o n the left of the four dots can always be replaced by wha t is on the right, and vice versa. It shou ld be clear that, for every replacement rule, there is a co rresponding biconditional formula which is truth-functionally or Quantificationally tr ue. For example, the truth-functionally true formula, '- - ¢I ~ $ ', corresponds to the rule of double negation. A few of the more common replacement rules are listed below: 1 Double Negation (ON) - - 41: :41 2 Commutati on (Comm) ¢I v W::\jfV¢l ¢I
1\
\11 :: \II
1\
¢I
3 De Morgan (De M ) - (¢I v \II ) :: - $ 1\ - '4' _ (¢I 1\ \If) :: - ¢I v - 'If
4 Distribution (Dist) ¢I 1\ ('4' v 0 ) :: (41 41 v (\II
1\
1\
\11 ) v (41
0 ) :: (41 v \11 )
1\
1\ 0 ) (41 v 0 )
5 Contra position (Cont) 41 :::>\11 ::-\11:::>-41 6 Association (Ass ) (¢I v \11) v 0 :: $ v (\If v 0 ) (¢I 1\ \11) 1\ 0 :: ¢ 1\ ("IV 1\ 0 )
7 Exportation (Exp) 1 A 'III co 0 :: co I'll co 0 1
230
S6.7: AVOIDING REITERATION
8 Implication (Imp) ¢! :::>\fI::- ¢l V Ij!
9 Equivalence (Equiv) ¢l ::: "':: (41 :::> IV)
1\
(W :::> ¢l)
10 Q uantificational Equivalence (QE) ('1X ) X :: (3x)x :: - ('1. ). :: - (3. ). ::
(3X)('1 x )(3.)('1. )-
X X . .
S6.7: AVOIDING REITERATION Some: systems of logic do nOt req uire reiteration. In these systems, o ne can appeal to any formula which can Ix reiurated according to the r ules of C hapter 3 and Chapter 5. Using reiteration makes stati ng the rules simpler,
and many students fi nd it less confusing when appeals to rules such as 1\1 and : : :I E al ways in volve form ulas on lhe current scope line. An examplt of how reiteration is avoided in some systcrns follows. It should be noted that "I', ::lE', ,,[' and vI' are not the regular rules of Essentials of Symbolic Logic in [hat each of them allows appeals I
2
to
other scope lines.
D A8 N co>C
P P P P
3 B ::> M 4 K
10
N K AN C 8 K A (8 Kv R
II
(K
5 6 7
8 9
12
N
CO>
f\
A
CO>
4,5,,,r 2,5, ::::>E' 1, "E' 3,4, " I'
M)
4,vr
(8 :::> M )l
11K A (8
CO>
f\
(K v R)
Mil A (K v R»
9, 10, AI
5-11, :::>1
Rather than having for mulas on diffe rent scope lines, some systems of logic have all fo rmulas in a numbered list and use lines at the side to hep track of assumptions. Very often these systems mark assumptions with a note exp laining why the assumption is made . In the following derivation, so me
§6.8: JOINT DENIAL AND mE SHEFFER STROKE
231
of the rules discussed in §6.4 and S6.5 are used in a derivation which does not use secondary scope lines. p
-Qv-R
p
2 Q" R
3 (- 5
A
-
L) " Q
P
4 M
A (CP)
Q
A (NI)
5
6 R
2,5, MP
7 - R
1,5, OS
8 R
A
-
6,7, Conj
R
9 - Q
5-8, NI
10 - (- 5
A
-
L)
IlM::::>-(-SA-L)
S6.8:
3,9,MT 4-10, CP
JOINT DENIAL AND THE SHEFFER STROKE
In S2.t8, it was demonstrated that two sentence operators could do the work of the five discussed in Chapter 2. In particular, it was shown that the tilde and the horseshoe would serve to do anything done by the entire set of five operators. A major theoretical interest of logic ians and computer scientists is the possibility of reducing the number of operators. There are twO well-known ways of reducing the five sentence operators to a single operator: joint denial and alternative denial. The joint denial of '¢I' and '\11' is the claim that '¢I' and 'w' are both false; that neither .~' nor ''+I' is true. Sometimes joint denial is simply known as 'nor'. The symbol for joint denial is a downward-pointing arrow, '+'. The truth-table definition of the joint denial symbol follows: ~
IV
~"''+I
T T
F
T F
F
F T
F
F F
T
232
S6.8:
JOINT DENIAL AND THE SHEFFER STROKE
Since '41 + lV' is tru e only when flanked by two false form ulas, one can use '.J,.' to express negat ion. The truth table fo r '- $ ' usi ng the joint denial symbol follows: 4>
- cD
¢q· 41
T
F
F
F
T
T
The (ruth table for '41 :::> lV' using the joint denial symbol follows. The correctness of each column on this table can be checked against t he truthtable definition of joint denial. IjI
::> IjI
T T T F F T
T
F F F
F T T
F F
11 .1jI1. "'I. 11 ."'1."'1
T 5
F
T
T
F
F F
T T
3
F F F T 4
F T
F F
2
Since :mything done with 'v', ',,' and 'ii' ca n be done using JUS[ the tilde and the horseshoe. It follows that anything done with 'v', '1\' and ':i' can be dom~
using JUSt joint denial. Enthusiastic readers with lots of s(';rap paper may care to develop joint denial equivalents of 'v', ',,' and '!If'. Alternative denia l is best known by its symbol, the Shdfer stroke, ~'. Computer people sometimes use the expression ' nand' to mean 'nOt both' and the Sheffer stroke represents this. The truth-table definition of the Sheffer stroke follows: 0'. Whenever a complex formu la is broken up into its components, it is marked with a check mark, './', to show that it needs no further attention. As mentioned, '{C, - R, C :> RJ' is truth-functionally inconsistent. On the following tree, all the mem~rs of this set are placed on the 'trunk', Conditional decomposition is used to form two branch~s (rom 'C ::l R'. When this is done, './' is plac~d at lin~ 3 to show that it n~~ds no further aUentioll.
C
Set
Memb~r
2
-R
Set
Memb~r
3
C::lR ......
Set Membe r
~ 4
- C
R
X
X
3, ::lD
One can now trac~ a path from ~ach of th~ formu las on line 4 to the top of the tree. Th~ ch~cked line 3 can be ignor~d, since mher lines have replaced it. Both of the formulas on lin~ 4 are marked with a cross, ' X'. These marks indicate that both branches are d osed. In a dosed branch, there is no path which does not includ~ a contradictory pair. O n m~ left branch, this pair is 'C' on line 1 and ' - C' on lin~ 4. On the right branch, the pair is '- R' on line 2 and 'R' on line 4. What this shows is that the re is no assignment of mIth va lues which can make all the members of '(C, - R. C => R)' uue. In tracing a path from the lower end of a branch to th~ top of a tree, the only formulas which n~ed to b~ consid~r~d ar~ atomic formu las and the negations of atomic fo rmulas. If the tree has been compl~ted properly, all other formulas will be marked with a ' ....... , indicating that they have been replaced by simpler formulas.
S6.9:
235
CONSISTENCY AND TRUTH TREES
All of the decomposition rules needed for trees in sentence logic are illustrated below. It can be seen that many of these rules resemble the replacement rules discussed in S 6.6. vD
=D
-AD
41='11'/
- (¢ /\ "') ,/
A
A
-
'II
-ljI
-
-.D - (41
;jI!
'11 ) ,/
A
-'II
'II
-'II
-
-vD
--D
- ( '" 'II) ./
- ( v ljI) ./
--41'/
-
-'I'
- ljI
S6.9:
236
CONSISTENCY AND
TRUTH TREES
If '(A => B) /I, (8 ::> C)' were put o n a (ruth table, there would be at least one 'r under the main operator. Th e (act that '(A::> B) 1\ (8 :::::l C)' is not truth·funccionally false can be shown o n a tree. Set Member
(A ::> B) " (8 ::> C) ./
2
A::> 8..1
1. I\ D
3
B ::>C./
1, AD
~ -A
4
~ 5
2, =>D
B
~
- B
C
- B
C
'"
'"
X
'"
3, ::>D
It should be noted that the conditional decomposition of line 3 appears
twice at line 5, on both of the branches formed at line 4. This is to ensure that all necessary branches are included in the tree. All of the open branches are marked with './' JUSt as the closed branch is ma rked with ·X'. '(A :::l BI /I, (8 ::::> C)' is not truth-functionally false because there is at least one open branch . This means that there is at least one assignment of truth values on which the formula is true. A truth·functionally valid argument is one in which there is no row of a truth table on which all the premisses are true and on which the concl usion is false. PUt in terms of truth· functional consistency. a set consisting o f all the premisses and the negation o f the conclusion of any trtlth·functionally valid argument is bound [Q be inconsistent. The following argument is truth-functionally valid: P => (Q
P
Q
A
R)
237
S6.9: CONSISTENCY AND TRurn TREES Th~
truth
u uth·functional validity of
th~ a rgum~nt
is shown on
th~
following
tr~~.
P :> IQ A R) ./
Pr~miss
2
P
Pr~m i ss
3
-Q
N ~ga t~d
---------------
4
- P
Q AR./
1. ::J O
5
)(
Q
4..... 0
R
4..... 0
6
Conclusion
)(
Since all branch~s close. the argument is valid. By contrast. (h~ following argument is uuth-functionally inva lid:
p :> IQ - P
A
R)
-Q As before. the premisses are
foll ow~d
by
th~
negation of
[h~
conclusion.
P :> IQ A R) ./
Premiss
2
- P
Pr~miss
3
--Q./
Negat~d
4
Q
3.--0
---------------
5
- p
Q A R ./
1. ::J D
6
./
Q
5. ,, 0
R
5. .... 0
7
Conclusion
./
Since there is at least one open branch. th~ argument is invalid. In th~ discussion of pr e dicat~ logic and interp r~tarions in S4.17, it was shown how universa lly Quantified formulas ar~ r~plac~d by conjunctions and existentially Quantified fo rmu las are replaced by disjunctions. This suggests that u~ es can be us~d to ass~ss quantified formulas within
238
S6.9: CONSISTENCY AND TRUTH TREES
interpretations. In some cases - but only in some cases - trees can be used to
show quantificational validity and consistency, and even
to
determine the
necessary domains, However, such trees can rapidly expand to an unwieldy siu. In many cases, utes expand to an infinite size. Using trees to deal with interpretations is a complex procedure, involving many issues. A serious discussion o f these: matters is beyond the: scope of this book. Even when onc is dealing with sentence: logic, trees can grow to unmanageable proportions. It is easy to run Out of room on the page or else: be: reduced to writing in such small characters and symbols that the tree:
becomes unreadable. Trees have been me ntioned in this chapter so that students will not be completely bewildered when they see one. As a practica l technique for dealing with such things as arguments and interpretations, the method of using trees has serious deficiencies.
Page 239
Index Technical Terms When a technical term is first introduced, it is printed in bold face. The following pages contain such occurrences of technical terms. Terms which are mentioned only in Chapter 6 are not included. A antecedent 18 argument 39 assign a truth value 8 assumption 80 atomic formula in predicate logic 143 atomic formula in sentence logic 12 C caret 10 categorical derivation 123 charity, principle of 47 closed sentence 140 conclusion 39 conclusion-indicators 48 conjunction 15 conjunction elimination 59 conjunction introduction 57 conjuncts 15 consequent 18 constant 134 Start of Citation[PU]Broadview Press[/PU][DP]1999[/DP]End of Citation
Page 240
D deductive 39 definite description 153 derivation 55 disjunction 17 disjunction elimination 95 disjunction introduction 62 disjuncts 17 domain 155 domain of the interpretation 166 E elements 156 elimination rule 56 empty set 166 exclusive 17 existential quantifier 140 existential quantifier elimination 183 existential quantifier introduction 178 extensions of the predicates 166 H horseshoe 10 horseshoe elimination 64 horseshoe introduction 84 I identity 150 identity elimination 191 identity introduction 190 inclusive 17 indirect proof 115
individual 134 interpretation 166 introduction rule 56 invalid 39 L list notation 156 M main operator 13 metalanguage 11 metatheory 12 multi-place predicate 136 Start of Citation[PU]Broadview Press[/PU][DP]1999[/DP]End of Citation
Page 241
N negation 12 negation elimination 61 negation introduction 91 O object language 11 one-place predicate 135 open sentence 140 ordered pairs 167 ordered triples 167 P parameter barrier 184 parameters 183 predicate 134 premisses 39 premiss-indicators 48 primary scope line 56 principle of charity 47 property notation 156 Q quantificational invalidity 166 quantificationally equivalent 142 quantificationally false 165 quantificationally indeterminate 165 quantificationally true 165 quantificationally valid 161 quantifier 140 R reflexive 218
reiteration 82 S scope line 55 scope of a quantifier 141 secondary scope line 80 sets 156 subderivations 81 symmetric 219 Start of Citation[PU]Broadview Press[/PU][DP]1999[/DP]End of Citation
Page 242
T tilde 10 transitive 219 triplebar 10 triplebar elimination 66 triplebar introduction 88 truth-functionally equivalent 35 truth-functionally false 33 truth-functionally indeterminate 34 truth-functionally invalid 40 truth-functionally true 32 truth-functional operator 8 truth-functional validity 40 truth functions 8 truth-preserving 40 truth table 26 truth value 7 U universal quantifier 142 universal quantifier elimination 181 universal quantifier introduction 186 V valid 39 variable 140 vel 10 W well-formed formulas 12 WFF 12 Start of Citation[PU]Broadview Press[/PU][DP]1999[/DP]End of Citation
