An Invitation to Formal Reasoning: The Logic of Terms

An Invitation to Formal Reasoning The logic of terms FRED SOMMERS Harry A. Wolfson Professor of Philosophy, Emeritus, ...

Author: Frederic Sommers | George Englebretsen

115 downloads 1233 Views 13MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

An Invitation to Formal Reasoning The logic of terms

FRED SOMMERS Harry A. Wolfson Professor of Philosophy, Emeritus, Brandeis University GEORGE ENGLEBRETSEN Bishop's University

Ash gate Aldershot • Burlington USA • Singapore • Sydney

© Fred Sommers and George Englebretsen 2000 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. Published by Ashgate Publishing Ltd Gower House Croft Road Aldershot Hants GUll 3HR England Ashgate Publishing Company 131 Main Street Burlington Vermont 05401 USA Ashgate website: http://www.ashgate.com

British Library Cataloguing in Publication Data Sommers, Fred, 1923An invitation to formal reasoning : the logic of terms 1. Logic 2. Reasoning 3. Language and logic I. Title II. Englebretsen, George 160 Library of Congress Control Number: 00-132808 ISBN 0 7546 1366 6

Printed and bound by Athenaeum Press, Ltd .. Gateshead, Tyne & Wear.

Contents Preface

x

Chapter 1 Reasoning 1. Introduction 2. The Form of an Argument 3. A Word About the Form of Statements 4. The Form of Singular Statements 5. Terms and Statements 6. Symbolizing Compound Statements 7. A Word About Validity 8. How Material Expressions are Meaningful 9. Terms 10. Some Terms are 'Vacuous' 11. Statement Meaning 12. Truth and Correspondence to Facts 13. Propositions 14. 'States of Affairs' 15. The facts and the FACTS 16. What Statements Denote 17. Summary and Discussion on the Meaning of Statements

1 4 4 5 7 9 11 13 13 14 17 19 20 21 22 22 23

Chapter 2 Picturing Propositions 1. State Diagrams 2. Representing Singular Propositions 3. Entailments 4. Negative Entailments 5. STATES and states 6. Positive and Negative 'Valence' 7. The Limitations of State Diagrams 8. The Statement Use of Sentences 9. Truth Relations 10. Logical Syntax

25 28 30 33 35 36 36 38 39 40

v

VI

An Invitation to Formal Reasoning 11. Term Way vs. Predicate Way 12. Some Useful Terminology 13. Subjects and Predicates

Chapter 3 The Language of Logic (I) 1. Introduction 2. Writing 'Y some X' as an Algebraic Expression 3. Affirmation (+) and Denial (-) 4. Binary and Unary Uses of a Sign 5. Positive and Negative Valence 6. Contrary Terms and Sentences 7. 'Every' 8. Why Some Equal Sentences are not Logically Equivalent 9. E-forms and A-forms 10. Transcribing Affirmative Statements 11. How to Tell the Valence ofE-form Statements 12. Negative Valence= Universal Quantity 13. The Law of Commutation in E-form 14. 'Every' in E-form Transcriptions 15. 'Isn't' 16. The General Conditions ofEquivalence 17. The General Form of Statements 18. The Logical Law of Commutation Applied to Compound Terms 19. The Logical Law of Association 20. Derivations 21. More on Regimenting Sentences 22. Uniquely Denoting Terms and Singular Statements 23. Identities Chapter 4 The Language of Logic (II) 1. Compound Statements 2. 'If... then' 3. More on Transcription

4. 'Or' 5. Representing Internal Structures 6. The General Form of Compound Statements 7. Direct Transcriptions

Contents

8. Relational Statements 9. A Word About Pairing 10. Subject/Predicate; Predicate/Subject 11. 'Dyadic Normal Fonns' 12. Commuting Relational Terms 13. Immediate Inferences from Relational Statements 14. Obversion 15. The Passive Transformation 16. Simplification 17. Pronouns and Proterms Appendix to Chapter 4 18. Bounded Denotation 19. Terms in their Contexts 20. Rules for Using Markers

vn 88 89 91 92 93 95 96 97 98 99 102 103 106

Chapter 5 Syllogistic 1. Validity 2. Inference 3. Enthymemes 4. Why REGAL Works 5. Inconsistent Conjunctions: The Tell-tale Characteristics 6. Equivalent Conjunctions 7. How This is Related to REGAL 8. Syllogisms with Singular Statements 9. The Laws ofldentity 10. Proofs ofThese Laws 11. The Matrix Method for Drawing Conclusions 12. Venn Diagrams

109 114 118 122 124 127 128 129 130 131 133 135

Chapter 6 Relational Syllogisms 1. Introduction 2. Applying the Dictum to Relational Arguments 3. Distributed Terms 4. Applying DDO 5. Indirect Proofs for Relational Arguments 6. Transforming Arguments 7. Annotating a ProofofValidity 8. Arguing with Pronominal Sentences

139 140 141 143 147 148 150 151

viii An Invitation to Formal Reasoning 9. Distributed Proterms

158

Chapter 7 Statement Logic 163 1. Introduction 165 2. Contradictions 166 3. Tautology 167 4. Inconsistent Statements 5. Contingent Statements 167 6. Direct Proofs 168 7. Rules of Statement Logic Used in Proofs 169 8. Disjunctive Normal Forms (DNF) 175 9. Inconsistency and Validity 176 10. Graphic Representation of Compound Statements 178 11. Regimenting Statements for Treeing ·183 12. Large Trees 185 13. Drawing Conclusions 192 14. Partial Disjunctions 193 15. Using the Tree Method for Annotated Proofs 199 16. Statement Logic as a Special Branch of Syllogistic Logic 201 17. Venn Diagrams for the Singleton Universe of Propositional Logic 209 Chapter 8 Modem Predicate Logic 1. Syntax 2. MPL: The Predicate Way 3. General Sentences in MPL 4. The Logical Language of MPL 5. Singular Sentences in MPL 6. How the Logical Syntax ofMPL is 'Ontologically Explicit' 7. Dyadic Normal Forms 8. Translating Pronominalizations 9. Preparing the TFL Bridge 10. Identity in MPL 11. Logical Reckoning in MPL 12. Transformation Rules 13. Rules oflnference 14. Literal Formulas 15. Reckoning in MPL

213 214 215 216 220 222 224 227 229 230 232 233 235 237 240

Contents

16. Canonical Normal Forms (CNF) 17. Indirect Proofs in MPL 18. Relational MPL Arguments 19. Identity Arguments in MPL

IX

241 242 244 249

Rules, Laws and Principles

253

A Note on Further Reading

259

Preface

It seems to be a fairly widely held belief among contemporary teachers of logic that one must introduce logic via the propositional, and then predicate, calculus. In particular, one would not, even if he or she believed otherwise, properly or fairly serve novice students by offering them instead something like syllogistic logic. Nonetheless, we intend to do just that here: introduce the subject of formal logic by way of a system that is 'like syllogistic logic'. Our system, like old-fashioned, traditional syllogistic, is a term logic. Our version of logic ('term-functor logic', TFL) shares with Aristotle's syllogistic the insight that the logical forms of statements that are involved in inferences as premises or conclusions can be construed as the result of connecting pairs of terms by means of a logical copula (functor). This insight contrasts markedly with that which informs today's standard formal logic ('modern predicate logic', MPL). That version of logic is due to the work of the great nineteenth century innovator in logic, Gottlob Frege. His insight concerning the logical form of statements was inspired by the language of mathematics. It construes the logical form of statements as the result of functions (incomplete expressions like 'the square root of. .. ' or ' .. .loves ... ') being completed by the insertion of the appropriate arguments (name-like expressions such as '2'or 'Romeo' and 'Juliet'). This difference between TFL and MPL is important because formal logic takes the validity or invalidity of inferences to depend completely on the forms of the statements making up those inferences. Formal logic rests on a theory of logical form (syntax). A second important difference between TFL and MPL is this. Most of the time when inferences are made we need to pay attention to the forms of the statements involved. But sometimes, especially when most or all of those statements are compounds of simpler statements, we can ignore the particular forms of the statements and concentrate instead on the arrangements of simple statements used to form the compounds and, ultimately, the inference itself. The calculus ofMPL which accounts for these kinds of inferences is called 'propositional'. Modern logicians take the logic of unanalyzed statements, the propositional calculus, to be the foundation of all of MPL. This is why

X

Preface

Xl

teachers today begin the introduction of fonnal logic with the propositional calculus. When Aristotle invented syllogistic, indeed the whole field offormal logic, in the fourth century B.C. he dealt first and foremost with the logic of terms, the logic of inferences that depend for their validity on the arrangement ofterms within their statements. The logic of propositions was only developed later by Stoic logicians, and then, many centuries later, again by Frege. Following an inspiration by the great seventeenth century polymath Leibniz, TFL incorporates the logic of propositions into the logic of terms by construing entire statements, or propositions, as themselves nothing more than complex terms. So, where MPL sees propositional logic as 'foundational', or primary logic, TFL takes the logic of terms as primary. As it happens, some version ofTFL, either Aristotle's syllogistic or, later, the Scholastic logicians' revised traditional syllogistic, dominated the field offonnallogic until the end of the nineteenth century. Yet even by the beginning of that century logicians had come to agree that traditional syllogistic logic was inadequate for the analysis of a wide variety of inferences. When Frege built MPL he offered logicians a system of logic far more powerful than any system that had gone before it. The power of MPL (its ability to offer analyses of a wide variety of kinds of inference) coupled with Frege's claim that the logic could serve as the foundation of mathematics (by the late nineteenth century mathematicians had become quite worried about the foundations of their field), insured that it would displace the old logic in short order. Today the hegemony ofMPL is almost complete. Still, there is a price to be paid. MPL is indeed powerful, but it is not simple and the logical forms which it ascribes to statements are remote from their natural language forms. Traditional formal logic lacked the scope enjoyed by MPL by not being able to analyze a number of types of inference. Yet it did at least enjoy the double advantage of (i) being simple to learn and use and (ii) construing the logical forms of statements as close to their natural language forms. Clearly a system of fonnal logic which has the power of MPL and the simplicity and naturalness of traditional logic would provide the best ofboth logical worlds. Beginning in the late 1960s Fred Sommers set himself the task of developing a system offormallogic (viz., TFL) that was powerful, natural and simple. The challenge faced by Sommers in accomplishing this was threefold. The first was to extend the power of term logic by incorporating into it the kinds of inferences beyond the powers oftraditionallogic. Those inferences were of three types: inferences involving statements with relational expressions, inferences involving statements with singular terms, and

xii An Invitation to Formal Reasoning inferences involving unanalyzed statements. The second challenge was to offer a theory oflogical form, or syntax, that was natural in the way that the syntax ofMPL was not. The third challenge was to provide a symbolic algorithm (a system of symbols along with rules for manipulating them) much simpler than the one employed by MPL (viz., 'the first-order predicate calculus with identity'). During the past three decades Sommers has perfected just such a system of formal logic. TFL is at least as powerful as MPL, and it is far simpler and more natural. The most important factor accounting for the difficulty in learning and using MPL is its theory of logical form. By requiring statements to be analyzed as functions completed by arguments it achieves its great power, construing singular, general, relational, and compound sentences in a uniform manner. Predicates (like 'is wise' or 'runs'), quantifiers (like 'some' and 'every'), relational expressions (like 'loves' or 'taught'), and 'sentential connectives' (like 'and', 'only if, or 'not') are all taken to be function expressions. Proper names ('Socrates', 'Romeo'), personal pronouns ('it', 'they', 'he', 'her', etc.), and entire sentences are all taken to be arguments. Thus the following sentences can be given a uniform function/argument(s) analysis. (1) (2) (3) (4)

Socrates is wise Some philosopher is wise Romeo loves Juliet It is cold and it is wet

Symbolically, predicates are symbolized by uppercase letters, proper names by appropriate lowercase initials, pronouns by lowercase letters at the end of the alphabet, unanalyzed propositions by lowercase letters near the middle of the alphabet, and quantifiers by special symbols incorporating the pronouns for which those quantifiers serve as grammatical antecedents. Function expressions are written to the left of their arguments. Finally, parentheses are used as punctuations to ease the reading of formulas. The sentences above are usually formulated by 'translating' them into the standard symbolic notation. Thus:

(1.1) Ws (2.1) (Ex)(Px & Wx)

(3.1) Lrj

Preface

xiii

(4.1) c & w

These formulas are unnatural, and the more complex a statement is the farther its logical form is from its natural language form. A sentence such as 'Every dog has a master' is first paraphrased as 'Each thing, call it x, is such that if it is a dog then there exists at least one thing, call it y, such that it is a master and x has it'. This is finally formulated as: (x)(Dx => (Ey)(My & Hxy)). Almost any teacher of MPL today will admit that the most difficult thing students must learn is this process oftranslation. Simple English sentences are paraphrased into sentences saturated with pronouns and sentential connectives, which had no place in the original. TFL requires only the minimum of'regimentation' (paraphrasing into a standard pattern) before symbolization. Translation is replaced by 'transcription'. This is because the TFL syntax of pairs of connected terms is close to the grammatical form of most natural language statements. The symbolic language of TFL is exceptionally easy to learn. All simple terms, singular, general, relational, are marked by upper-case letters. Unanalyzed propositions are symbolized by lowercase letters. All terms are either simple or complex. All complex terms are pairs of connected terms. All unanalyzed statements are complex terms. All terms are either positive or negative. All statements are affirmed or denied. All term-pairs are connected by positive or negative functors. Plus and minus signs(+/-) are used for all of these. As in arithmetic or algebra, positive signs are often suppressed (compare: '+3+(+4)=+7', read 'positive 3 added to positive 4 equal positive 7', which is normally written as '3+4=7', and read as '3 plus 4 equals 7'). Examples of positive/negative simple terms are 'wise/non-wise' (written: '+W/-W'), 'happy/unhappy' ('+H/-H'), and 'massive/massless' ('+M/-M'). Connective . of pIuses and mmuses . (vtz., . ' +... + ' , ' +... -' , ' -... - ' , an d ' functors are patrs ... +'). The first ofthese indicates the quantity(+ for 'some', 'at least one', etc.; - for 'all', 'every', etc.). The second part of the connective functor indicates the copula (e.g., 'is', 'are', 'was', 'isn't', 'ain't'). Singular terms are marked with an asterisk,*. They have 'wild' quantity; they are indifferently + or-, (written '±'). Parentheses are used to group pairs of connected terms. Our sample sentences above would be formulated in TFL as follows. (1.2) ±S*+W (2.2) +P+W (3.2) ±R*+(L±J*)

xiv An Invitation to Formal Reasoning (4.2) +c+w Note that unanalyzed statements (as in 4) are connected by the same functors as other term pairs. This is because such functors only represent relations with given formal features. Thus, for example, the+ ... + functor is symmetric, but not reflexive or transitive. These are just the features that guarantee the validity of such inferences as 'Some philosopher is wise, therefore some wise (person) is a philosopher' and 'It is cold and it is wet, so it is wet and it is cold'. From the point of view of 'formal' logic, only these formal features are of interest. Another source ofdifficulty for beginning students ofMPL is the large variety of rules required to adequately construct proofs of valid inferences. In addition to rules for the propositional calculus, there are rules for eliminating and for introducing each ofthe quantifiers and for manipulating identities. The relative naturalness of TFL' s syntax has already given it a degree of simplicity, which is now augmented by its algorithm for proofs. Since all formative expressions are plus or minus signs, it is easy to show that proof amounts to addition and subtraction (this turns out to be the 'cancelling of middle terms' familiar in traditional syllogistic). The present text book is intended as a tool for the introduction ofTFL to the beginning student of logic. It also includes a final chapter introducing standard MPL. One of the important advantages of coming to formal logic through TFL is that it makes the subsequent learning ofMPL so much easier. For TFL provides 'bridging formulas' that ease the usually difficult translation process that takes natural language statements into MPL formulas. The text contains several exercise sections and a summary of the main rules, laws and principles ofTFL. It is designed so that it could be used for self-teaching. But it is also designed to be used in classrooms as an introductory text for a onesemester course in formal logic. For those going on to do more mathematical logic it is an appropriate and (because of the bridging formulas) useful first text. For the more philosophically oriented it contains extensive discussions of important issues at the intersections of semantics, metaphysics, epistemology and logic. There has been much enthusiasm is recent years for either the replacement or supplementation of courses in formal logic with courses in informal logic. Much of this enthusiasm is due to disenchantment with MPL, which is seen as remote from the ways in which we naturally and ordinarily use our reason and language. In addition, as students arrive at colleges and universities in larger numbers, with a greater variety of

Preface

xv

educational backgrounds and abilities, more of them are enrolled in introductory logic courses. The rigours of standard mathematical logic are often beyond the capabilities or interests of many such students. Thus there is often pressure to 'soften' the blow. We are convinced that mathematical logic ought to be taught, and taught with the appropriate high degree of formal rigour. We also believe that those who seek an account of reason which is more natural and simpler than the one embodied in MPL are right to do so. But one need not abandon formal logic to achieve this end. TFL represents a system of logic which is at once formal, rigorous, powerful, effective, natural and simple. Just as Sommers is the author of TFL, he is the true author of this text. It is a sign not only of his innate generosity but of his deep conviction that logic is more important than any logician that he freely shares his ideas with no concern for personal renown and little sense of proprietorship. Sommers began work on the text in the early 1980s and has thoroughly revised it several times in light of critical suggestions by myself and others, and as the result of his use of the material in the teaching of introductory logic courses over several years at Brandeis University. With less patience and judgment, I have urged completion and publication from the beginning. I have taught MPL for three decades. I have also used much of the material here in the teaching of introductory logic courses during the past few years. I have made use of that experience as well as the results of my own research in logic over the past quarter century to make some minor modifications and additions to the text. There are several people who have been instrumental over the past several years in helping to clarify the ideas in this book, offering critical commentaries, providing useful suggestions, or patiently listening to one or both of the authors go on and on about terms. In addition to the many students who have served as guinea pigs through those years, particular mention must be made of Michael Pakaluk, Graeme Hunter, Thomas Hood, Aris Noah, Philip Peterson, Lome Szabolcsi, George Kennard, William Purdy, Wallace Murphree and David Kelley. Note for Instructors: Some of the material presented in this text deals either with semantic issues or philosophical issues often deemed beyond the scope of a purely technical course in symbolic logic. The instructor who wishes to present a streamlined

xvi An Invitation to Formal Reasoning approach can safely ignore several sections of the text devoted to those less technical topics. For such a course, we would recommend the omission of the following sections: Chapter 1, sections 8 through 17; Chapter 2, sections 1 through 7; all of Chapter 4; Chapter 6, sections 8 and 9 are optional; Chapter 7, sections 16 and 17.

George Englebretsen Lennoxville, Quebec

1 Reasoning

1. Introduction A normal adult possesses information stored in memory in the form of statements like 'Socrates taught Plato', 'Frenchmen eat frog legs', 'my brother is taller than I am' and so forth. Some of the statements in our memory are false but most are true. In any case our ability to retrieve information from the stock of statements we believe to be true is useful to us in countless ways. A good memory is a distinct advantage in life. But just as important is our ability to reason with the information we have. We reason by using one or more of the stored statements as premises to derive another statement, a conclusion, which may not previously have been thought of but which may now be added to the store of information in our possession. Logic is the science that studies reasoning. It shows how to reason well and how to distinguish bad reasoning from good reasoning. A unit of reasoning is called an argument or inference. An argument consists of one or more premises together with the conclusion that has been drawn from them. Any argument is either valid or invalid. When an argument is valid, its conclusion is said to follow from or to be entailed by its premises. As an example of arguing from a single premise to a conclusion, suppose that, knowing of your interest in women's achievements, a friend asks you whether any woman has been a British Prime Minister. Let us say that your memory contains the statement (S 1) Some Prime Minister was a woman. Applying your reasoning capability you will take S 1 as a premise and immediately (perhaps even automatically and unconsciously) derive the conclusion: (S2) Some woman was a Prime Minister.

1

2

An Invitation to Formal Reasoning

Let us call your argument 'A 1': AI:

S 1. Some Prime Minister was a woman. I S2. some woman was a Prime Minister.

(The forward stroke sign should be read as 'therefore' or 'hence'.) You offer S2 to your friend as the answer to his question. Note that S2 may not actually have been in your memory. However, since S2 is entailed by Sl, you can now add it to your store of information. Al is an example of immediate inference. In immediate inference the conclusion is drawn from a single premise. Reasoning like this takes place very quickly, and usually without the conscious application of a technique for deriving conclusions from premises. But a great deal of reasoning is done carefully and reflectively and in many cases by deliberately using a method that has to be learned. Consider an example taken from a book on logic written by Lewis Carroll, the author of Alice in Wonderland. Carroll asks the reader to draw a conclusion from the following premises: (1) Babies are illogical.

(2) Nobody is despised who can manage a crocodile. (3) Illogical persons are despised. Here one must reflect a bit before coming up with the conclusion Carroll has in mind: (4) No baby can manage a crocodile. The whole sequence of four statements is an argument. The first three statements are its premises. The fourth is its conclusion. The next Carroll example is more complicated; in solving it we are well advised not to rely on our unaided wits; it is the sort of problem that is best approached with a logical method or technique for solving just this sort of problem. (1) Everything not absolutely ugly, may be kept in a drawing-room. (2) Nothing that is encrusted with salt is ever quite dry. (3) Nothing should be kept in a drawing room unless it is free from damp. (4) Bathing-machines are always kept near the sea. (5) Nothing that is made of mother-of-pearl can be absolutely ugly.

Reasoning

3

(6) Whatever is kept near the sea gets encrusted with salt. Here mere reflection may get one confused and we are better off not relying on our wits but on a mechanical procedure for drawing conclusions from premises. We will later learn how to do this example by representing the six premises as algebraic expressions that can be added like numbers to derive the conclusion in a mechanical way. By using this technique you will be able to do examples that look fairly complicated very quickly and surely. When one applies the algebraic method to the six premises given by Carroll, one derives the conclusion: (7) No bathing-machine is made of mother of pearl. (This too is Carroll's conclusion; he arrives at it by using a method quite similar to ours.) For the moment we have no method for drawing conclusions from premises. So we shall leave Carroll's droll arguments for later consideration. In learning logic, as in other fields of exact knowledge, we are better advised to begin by attending first to simple easy-to-follow examples. So let us look again at the simple example of reasoning, AI, where we moved from SI as premise to S2 as the conclusion. AI is a typical example of how we use a truth that we have stored in memory to derive a new truth that we have not (yet) stored. Now it may seem that the move from S I to S2 is trivial. In fact, even in so simple a case as AI, the practical value of being able to infer a new truth from the truths we have at our immediate disposal is enormous. Thus suppose you did not know whether either S I or S2 is true. To find out about S I one need only take a casual glance at the biographies of the British Prime Ministers. There are only nine of these and the biographies are publicly available. Anyone taking the trouble to do this will quickly discover that at least one of them (viz., Margaret Thatcher) is a woman. Having learned that S I is true we should now infer the truth of S2 as well. In this way we get to know about S2 indirectly. We get to S2 by deriving it from Sl. But suppose we were somehow incapal1le of reasoning in the manner of AI. We should then be forced to approach the question of the truth of S2 directly, in the same way we learned about the truth Sl. A direct approach would require us to examine the biographies of all woman to see whether any woman was a Prime Minister. Of course this is a practical impossibility. In effect, if we were unable to reason in the manner of AI, we could not arrive at the truth of S2 at all. Clearly the only sensible and practical way ofleaming

4


that S2 is true is indirectly: by way of inferring S2 from S 1. And in fact that is how we do it: S 1 is our starting point and anyone who knows that S1 is true will unconsciously and immediately infer S2 from S 1.

2. The Form of an Argument

S 1 and S2 are converses of one another. Our confidence in the move from S 1 to its converse, S2, is due to the confidence we have in the general pattern of reasoning where we move from one statement taken as premise to its converse. Conversion is a valid form or pattern of reasoning. Let us call this pattern F 1.

F1:

someXisaY I some Y is an X

(Here again, the stroke sign is read as 'hence' or 'therefore.') F1 is also the pattern of the following argument: A2: S3 Some member of the Armed Services Committee is a Southerner. I S4 Some Southerner is a member of the Armed Services Committee. A2, like A1, is of form F1 and we have confidence in any argument of that form. F 1 is an abstract pattern of reasoning and any argument that fits this form is called an instance of this pattern. Thus A1 and A2 are instances ofF 1 and so is A3: A3

some farmer is a noncitizen I some noncitizen is a farmer

3. A Word About the Form of Statements

Every argument consists of two or more statements (a conclusion and one or more premises). Each statement within the argument has a form and the argument as a whole has a form. To reveal the form of a statement we simply replace its terms by 'place holder' letters like 'X' and 'Y'. A place holder letter does not stand for a term; it merely occupies the places that a term or

Reasoning

5

letter that stands for a term occupies. For example, by putting 'X' in place of 'ape' and 'Y' in place of 'genius' we show that 'some ape is a genius' has the form 'some X is a Y'. If we do this systematically to each statement of an argument, the form of the whole argument stands revealed. For example, the arguments Al, A2, and A3 are then revealed as all being instances of the argument form F 1. The form of the following argument every cat is a feline no feline is a herbivore I no cat is a herbivore is: every X is a Y no Yis aZ I no X is aZ Another instance of an argument of this form is every Greek is a philosopher no philosopher is a vampire I no Greek is a vampire

4. The Form of Singular Statements

In a statement like 'A president of the United States slept here' the expression 'president of the United States' is being used as a general term. There are many presidents. And when 'president' is used as a general term, it may denote many individuals. But in some uses it denotes no more than one individual. So used 'president' is a uniquely denoting term (UDT). An example is 'President' as it occurs in 'The President is tired'. Proper names are almost always used in a uniquely denoting way. Thus 'Garbo' in 'Garbo was lonely' is a UDT. On the other hand, it is not a UDT in 'Roseanne Barr is no Garbo'. We refer to UDTs as singular terms. A statement whose subject term is singular (e.g., 'Garbo is beautiful', 'The President is tired') is called a

6


singular statement. Statements with general terms in subject position (e.g., 'some presidents are funny') are general statements. In a statement like 'Mark Twain is Samuel Clemens' both terms are singular. Such statements are called 'identities'. The difference between a singular and a general statement is semantic; it lies solely in the difference in the terms and not in the forms of the statements. Consider the two sentences i) Garbo was laughing. ii) Children were laughing. Though (i) is singular and (ii) is general, they have the same form. 'Garbo was laughing' has the form 'Some X* is Y' while 'Children were laughing' has the form 'some X is Y'. (We mark UDT occurrence by affixing a star to ·the letter.) From a strictly logical point of view 'Garbo was laughing' should be 'some Garbo was laughing'. In practice that is not done. For we know that whenever 'some Garbo is P' is true, 'every Garbo is P' will also be true (there being only one person who is Garbo). Since 'some Garbo is P' entails 'every Garbo is P' we do not bother to use either 'some ' or 'every' before 'Garbo'. Generally, whenever N* is a proper name, we use the form 'N* is P' and not 'some N* is P'. Nevertheless, for the purpose of seeing how singular sentences function inside of arguments, their form must be made explicit. As speakers ofEnglish we are content to say 'Garbo is laughing'; as logicians we need to represent this as 'some Garbo* is laughing' a statement that entails 'every Garbo* is laughing'. ****************************************************************** Exercises:

I. What is the form of the following arguments? (use X,Y Z) 1. All geographers are patriots. /All patriots are geographers. 2. No geographers are logicians. /No logicians are geographers. 3. Some nonvoters are citizens. /Some citizens are nonvoters. 4. All citizens are patriots, Some natives are citizens. /Some natives are patriots. 5. Bill Clinton is the President. I The President is Bill Clinton. (hint: The premise has the form 'some X* is Y*'.)

Reasoning

7

6. Only Garbo is laughing. (hint: Only X is Y can be construed as 'no non-X is Y'. Remember too that 'Garbo' is a UDT.) II. Give two instances for each of the following argument forms. 1. no X is Y I no Y is X 2. some X is Y I some Y is X 3. all X are Y, no Yare Z I no X are Z 4. every Y is non-X* I no non-X* is Y ******************************************************************

5. Terms and Statements Every statement consists of two kinds of components. Elementary statements consist of terms and expressions that join them, called term connectives. For example, 'some women are farmers' consists of the terms 'women' and 'farmer' and the term connective 'some ... are .. .'. Compound statements consist of component statements and expressions that join them called statement connectives. For example the compound statement 'if every person is mortal then some accidents will be fatal' has as its components the two elementary statements, 'every person is mortal' and 'some accidents will be fatal' joined by the statement connective 'if.. then .. '. The terms of an elementary statement are its 'material' components; they carry its matter or content. The term connective is the 'formative' component; it determines the form of the statement. An elementary· statement is either universal or particular in form. For example, 'every logician is a charmer', is universal, being ofthe form 'every X is Y'. 'Some logician is a charmer', which is ofthe form 'some X is Y', is a particular statement. In a compound statement, the component statements are the material elements and the statement connective that joins them is the formative element. Statements of the form 'ifx then y' are called 'conditionals'; those of the form 'x andy' are called 'conjunctions'. (The distinction between material and formative components was first drawn by medieval logicians; they called material expressions 'categorematic', contrasting them to the formative expressions, which they called 'syncategorematic'. Medieval logicians also used the vowels 'a' and 'i' to represent the term connectives in universal and particular statements. Thus a

8


statement of form 'every X is Y' was represented as 'YaX' while 'some X is Y' was represented as 'YiX' .) In the case of some arguments, called propositional arguments, the material elements are whole statements and the formative elements are 'statement connectives'. In dealing with propositional arguments we may ignore the internal form and content of the component statements. The following is an example of a propositional argument: A4: some roses are red and no violets are yellow I no violets are yellow and some roses are red Let 'p' stand for 'some roses are red' and let 'q' stand for 'no violets are yellow'. Herethestatements, 'p' and 'q', arethematerialelements. They are joined by the statement connective 'and'. Al may be represented as 'p and q lq and p'. We call 'p' and 'q' 'statement letters' . Unlike term letters, which are formulated using upper case letters, statement letters make use of lower case. The form of A4 is F4:

xandy ly andx

where x and y stand for any two statements. Clearly any propositional argument of the form F4 is valid no matter what statements we substitute for x andy. Thus the following instance ofF4 is valid: A5: some barbers are not Greeks and some Greeks are not barbers I some Greeks are not barbers and some barbers are not Greeks Let 'r' be the statement 'some barbers are not Greeks' and 's' be the statement 'some Greeks are not barbers'. Then we may represent A5 as 'r and s I s and r'. The statements in A5 are different in form and content from the statements in A4. But that does not matter since those differences play no part in the argument which is concerned simply with the move from a conjunction of the form 'x and y' to one of the form 'y and x'. Paying no attention to the internal form of the statements involved, we recognize that A4 and A5 are

Reasoning

9

instances of the same general form: x andy I y and x. And any instance ofF4 is valid. Another example of an argument whose validity is not due to the internal form of its component statements is A6:

if there is smoke then there is fire

I if there is no fire then there is no smoke A6 is an instance of the general form: F2: ifx then y I if not y then not x Here too we may replace x and y by any two statements of whatever internal form and content and the result will be a valid argument. Statements of form 'x andy' and 'ifx then y' are called compound statements since they contain two or more component statements joined together by 'and' or 'if... then' or some other formative statement connective. Arguments involving compound statements are the subject of a special branch of logic called Statement Logic (also called Propositional Logic). When the components statements are joined by 'and', the compound statement is called a conjunction and the two components are called 'conjuncts'; an example is 'roses are red and violets are blue'. Suppose that 'p' and 'q' stand for the respective conjuncts. Then one common way to write 'roses are red and violets are blue' in logical language is to use a symbol to stand for the English word 'and'. Thus we may use '&' to represent 'and' and then represent the conjunction as 'p&q'. But another way is to use algebraic operators like '+' for the statement connective. We should then represent the conjunction as 'p+q'. The algebraic way is called a 'transcription' of the English sentence. We shall later find that the algebraic way oftranscribing English sentences makes it especially easy to 'reckon' with them logically.

6. Symbolizing Compound Statements We have been using upper case letters to stand for terms. We shall always use lower cases letters to stand for whole statement. For example, we may let 'p' stands for 'Socrates was executed', 'q' for 'Plato died in his sleep' and 'r' for

10


'Aristotle went into exile'. We may then form compound statements which have the elementary statements 'p', 'q' and 'r' as components. For example, we may use 'p', 'q' and 'r' to form such compound statements as 'p and q' (symbolically written 'p & q'), 'ifp then q' (which we symbolize as 'p => q'), 'p or q' (symbolized as 'p v q', 'We then read 'p & (q v r)' as 'p and q orr' and we write 'ifp then (q and r)' as 'p => (q & r)'. In symbolically representing any compound statement we adhere to the convention of using lower case letters to represent the component statements.

******************************************************************* Exercises: Using the symbols'-' for 'not,'&' for 'and','=>' for 'ifthen", 'v' for 'or' we represent some common propositional statements thus: notp pandq ifp then q p orq rand (p orq) (randp)orq ifp and q then r if not p then q

-p p&q p=>q pvq r&(pvq) (r&p)vq (p&q)=>r -p ;:) q

Using the above symbols for the statement connectives, represent the following compound statements in the language of 'symbolic logic':

1. not-p orr 2. ifp then not-r 3. p ornot-p 4. q and (ifr then s) 5. s or not(q and r) 6. if not either p or q then r 7. p or not (q and r) 8. not (p and (q or not-r)) 9. (not-p and q) or not-r 10. p and (a and (rands))

Reasoning

11

11. if neither p nor q then not r 12. not (if p then not q) and either p or r

****************************************************************** 7. A Word About Validity An argument whose conclusion follows from its premises is called valid. When an argument is valid and its premises are true its conclusion must also be true. Whether an argument is valid or not, depends on its form. Most forms of argument are invalid. When an argument form is valid, none ofits instances have true premises and a false conclusion. When an argument form is invalid, it will be possible to find an argument of that form that has true premises but a false conclusion. An argument whose premises are true and whose conclusion is false is clearly invalid. Producing one invalid instance shows that all instances of that form are invalid. F1, above, is an example of a valid argument form: it has no invalid instances. But consider F3:

some X is not a Y

I some Y is not an X Now it might seem to us that F3 is a valid argument form. But if it is a valid argument form then we should never be able to find an argument of form F3 whose premise is true but whose conclusion is false. If we can find but a single invalid instance ofF3, that would be conclusive evidence that all arguments of form F3 are invalid. So we look for an invalid instance and after some thought we could come up with the following argument: A7:

some horse isn't a colt

I some colt isn't a horse The conclusion of A7 is false even though its premise is true. So A7 is invalid. But A7 is an instance of F3. And this shows that F3 is an invalid argument form. We call A7 a 'refuting instance' ofF3. Once we find a single refuting instance of an argument form we immediately lose confidence in all of its instances. For example, the following argument, A8, is also invalid:

12

AS:

An Invitation to Formal Reasoning some Greek is not a philosopher I some philosopher is not a Greek

Admittedly, AS looks like a good argument. But appearances are deceptive; the conclusion of AS does not follow from its premise. It is after all logically possible for its premise to be true and its conclusion false. For we may imagine a period in history when no one but a Greek is a philosopher, in which case 'some philosopher is not a Greek' is false even though 'some Greek is not a philosopher' is true. But we need not bother to imagine this. For we know that AS is invalid since it is an instance ofF3 and the validity ofF3 was refuted by A7. Thus F3 and Fl are different. Unlike F3, Fl is a valid form of argument and all of its instances are valid. And this means that given any two statements of form 'some X is a Y' and 'some Y is an X', we may be sure that if one of them is true so is the other. In other words we can never find an instance Fl whose premise is true and whose conclusion is false. Let us look also at a third argument form: F5:

no X is a Y /no Yis an X

F5, like Fl, is a valid argument form: no matter what terms we choose for X and Y, no matter what situations we imagine, we shall never find a refuting instance of form F3. In this respect F5 is like Fl and unlike F3. But now the question arises: what makes us so sure that Fl and F5 are valid forms? How do we know that we could 'never' find refuting instances for Fl or F5? After all, F3 also looked like a good way to reason yet we found it to be invalid. Why should we have more confidence in Fl and F5? We here touch on some fundamental issues in logic. Some statements entail one another, so that if one is true the other must also be true. What are statements, and how are they tied in this way? One approach to logic is by way of the concept of truth. 'What is truth?' When a statement is true, what is it about the world that makes it true? To answer such questions we must explain how the terms in a statement are related to things in the world and how the statement itself is related to the world as a whole.

Reasoning

13

8. How Material Expressions are Meaningful A statement that has whole statements (represented by lower case letters) as its components is compound and these component statements are its material elements. The material elements of an elementary statement are its terms. Terms and statements are the two basic kinds of material expressions. A material expression (a term or a statement) is meaningful in three ways: 1. It expresses a sense or characterization. 2. It denotes something to which the characterization applies. 3. It signifies a characteristic, property or attribute.

9. Terms We first discuss terms. (1) The term 'farmer', for example, expresses the description or characterization, BEING A FARMER. We call BEING A FARMER the sense or expressive meaning of 'farmer'. {2) 'In a sentence like 'a farmer was going into the bam' the term 'farmer' denotes an individual to whom the characterization BEING A FARMER applies. (3) 'Farmer' signifies the characteristic or attribute, being a farmer, that any farmer possesses. The attribute signified by a term is called its 'significance'. For example, being wise or wisdom is the significance of 'wise'. Note the distinction between the characterization that describes a thing and the characteristic that the thing itself possesses when we correctly describe it. We adopt the practice of writing the characterization in upper case letters and the characteristic in lower case letters. We say that a term expresses a characterization and that it signifies a characteristic. For example, the term 'wise', will be said to express the characterization BEING WISE and to signify the corresponding characteristic of being wise or wisdom. The characterization BEING WISE is said to be 'true of any individual that possesses the characteristic ofwisdom. A term denotes a thing only if the characterization it expresses is true of that thing. In the statement 'someone wise advised me' the term 'wise', which expresses BEING WISE, denotes an individual that possesses the characteristic (being wise, wisdom) that 'wise' signifies. The characterization, BEING WISE, and the characteristic, being wise, correspond to one another.

14


Let '[wise]' represent the characterization BEING WISE and let '<wise>' represent the characteristic of wisdom that any wise person possesses. Then 'wise' expresses [wise] and signifies <wise> and what 'wise' expresses corresponds to what it signifies. Also [wise] correctly describes (is true of) whoever is wise. And generally, if'#' is a term, the characteristic ,, that a #-thing possesses is said to correspond to the characterization, [#], that characterizes (is true of) the #-thing.

10. Some Terms are 'Vacuous' A term like 'mermaid', 'flying saucer' or 'woman who will love living on Pluto' does not fail to express a characterization. But it may fail to denote. Terms that fail to denote are called vacuous. Consider the term 'mermaid! in the statement 'a mermaid lives in the bay'. The sense or expressive meaning of 'mermaid' is BEING A MERMAID. The characterization expressed does not characterize anyone since no one possesses the characteristic of being a mermaid. Thus 'mermaid' fails to denote anyone or anything. Proper name terms are usually not vacuous. But some do express characterizations that characterize no one. There is an ancient tradition that a hero called Theseus founded the city of Athens. Suppose that Theseus was only a legendary figure and that no such person as Theseus ever existed. In that case no one ever possessed the characteristic of being Theseus and 'Theseus' fails to denote anyone. All the same, 'Theseus' has a sense; it expresses the characterization of BEING THESEUS, a characterization that is not true of anyone but which nevertheless is the expressive meaning of 'Theseus'. A vacuous term such as 'Theseus' or 'mermaid' is analogous to a statue of Theseus or a picture of a mermaid. The term 'mermaid' denotes no one. Similarly, the picture does not portray a mermaid (in the sense of 'portray' that a photo taken of a bridesmaid at my cousin's wedding is a portait of the bridesmaid). Nevertheless, the mermaid picture (like the term 'mermaid') is representational. We all understand what the picture means; as it were, the picture 'expresses'BEING A MERMAID. Similarly, the expressive meaning of the term 'mermaid' is understood by most speakers of English. In that sense what 'mermaid' means is something public and objective. We all 'grasp' it. (Considered as an object of understanding that we all grasp, the sense of a term is called a 'concept'.) We may put the matter

Reasoning

15

this way: The characterization expressed by any term (even one that is vacuous) exists as a 'concept'. Thus every meaningful term expresses a sense (or concept) and no term is expressively vacuous. (This doctrine, that concepts exist, is called Conceptual Realism.) No meaningful term is expressively vacuous. What about signification? Can a term be vacuous by signifying nothing? Some persons are kind, others are cruel. But nobody is perfect. So there is kindness and there is cruelty but no perfection. If nobody is perfect, nothing possesses the characteristic of being perfect. What is the status of a characteristic like that nobody possesses? There is an ancient dispute about characteristics that nothing possesses. Some philosophers and logicians, called Platonic Realists, follow Plato in holding that a characteristic <X> exists even if there are no X -things so that nothing or no one possesses <X>. Others (among them, the authors of this text who are Conceptual Realists but not Platonic Realists) deny this. They hold that a term like 'mermaid' or 'perfect' expresses a characterization but it fails to signify any characteristic. For there is no such thing as perfection and no such characteristic as being a mermaid. (On the other hand, there are such things as BEING A MERMAID and BEING PERFECT. For example, these meanings are expressed by the two terms in 'no mermaid is perfect'.) According to this view (which we shall 'officially' adopt) a term, 'T', will always express a sense or characterization, [T], but ifthere are noT-things, then there will be no characteristic for 'T' to signify. Thus 'mermaid' is expressively meaningful; it expresses BEING A MERMAID but it lacks both denotation and significance. We noted earlier that when a term lacks denotation we call it 'vacuous'. The nonPlatonist philosopher believes that 'mermaid' is doubly vacuous: not only does it fail to denote, it also fails to signify. Using the square brackets for the characterizations and the angle brackets for the characteristics, we summarize the above account of the meaning of terms. If 'T' is a term, then 1. 'T' expresses [T] or BEING T. If there are T -things then 2. 'T' denotes T things, 3. 'T' signifies , the attribute ofT-ness or being T, 4. [T] corresponds to . 5. [T] characterizes (is true of) some T thing If nothing is T, then 'T' is doubly vacuous because

16

An Invitation to Formal Reasoning 6. 'T' fails to denote aT thing. 7. 'T' fails to signify, (there being no such thing as ).

******************************************************************* Exercises: I. Give the sense and significance of the following terms: Examples: 'pious (person)' sense: BEING (A) PIOUS (PERSON), significance: being pious, piety 'logician' sense: BEING A LOGICIAN significance: being a logician 1. bridesmaid 2. red (thing) 3. unmarried ll. Repeat example I, using the bracket notation for the sense and significance of terms. Example: 'pious' sense: [pious] significance:

Ill. Which of the following terms lacks denotation and significance? bridesmaid mermaid sea cow sea squirrel

N. To what attributes, if any, do the following characterizations correpond? BEING A BACHELOR BEING A MARRIED BACHELOR BEING A MERMAID

******************************************************************

Reasoning

17

11. Statement Meaning Statements and terms are the two basic kinds of material expressions. Terms are used for characterizing things in the world. Statements are used for characterizing the world itself. We use lower case letters to represent statements. Like a term, a statement, 's', has three modes of meaning: (1) it expresses a sense or characterization, [s]. (What a statement expresses is called a proposition); (2) it denotes the world characterized by the proposition it expresses and (3) it signifies a characteristic of the world. (The characteristic, <s>, signified by a statement is called afact.) By definition, a statement is an utterance that is being used for saying something. And 'what is said' or expressed is a proposition. The proposition expressed may or may not characterize the world. If it does, the proposition is called true. If it does not, the proposition is false. Calling a statement true is a convenient shorthand way of saying that it expresses a true proposition. Not all utterances are statements but no utterance that is a statement can fail to express a proposition. Since every statement expresses a proposition that is either true or false, every statement is itself said to be true or false. How is the world characterized? What are its characteristics? We may think of the world as a collection or totality of things. (Here 'thing' is used in its widest sense to apply to whatever may be said to be present in the world including such things as London, The President of the United States, snow, hurricanes, pollution, wisdom, democracies and friendship.) Any totality, be it large or small, finite or infinite, is basically characterized by what is present in it and by what is absent from it. We may speak of such characterizations as 'existential'. Consider the collection of things now lying on your desk. Assume that the constituents of this little totality include a pen, ink, a lamp, a notebook and nothing else. Among the infinity of things not in this totality are horses, mermaids, envelopes, screwdrivers, etc. Suppose we call a totality ' {Q} ish' if it has a Q thing as a constituent and 'un {Q} ish' if it has no Q constituent. We may then existentially characterize the totality of things on your desk by saying that it is {pen} ish, {ink} ish, {lamp} ish and {notebook} ish. But negative characterizations are also true of it: for example, the little totality can be characteritzed negatively by saying that it is un {horse} ish, un {screwdriver} ish, etc. Any statement is a truth claim. Looking into a drawer I say 'there is no screwdriver', thereby claiming that the little totality under consideration, whose constituents are the objects in the drawer, is un {srewdriver} ish. We call

18


the totality of things under consideration when a given statement is made or when a given argument is presented 'the domain of the claim' (DC). Logicians sometimes refer to the DC as 'the universe of discourse'. Very often the totality under consideration is the whole world. For example, in asserting 'some women are farmers' one expresses the characterization SOME WOMEN BEING FARMERS or BEING {WOMAN FARMER}ISH, claiming (correctly) that this characterizes the contemporaneous world. In what follows we shall assume that the DC of a statement is the world. Here are some existential characterizations (propositions) that are true of the world at this time: BEING {WOMAN FARMER} ISH (that some women are farmers) BEING {ELK} ISH (that there are elks) BEING UN{ELF}ISH (that there are no elves) BEING UN{MERMAID}ISH (that there are no mermaids)

If we use the bracket notation to represent what a statement expresses, then a statement, 's', expresses the proposition [s]. For example, 'some women are farmers' expresses the proposition [some women are farmers] and 'there are no mermaids' expresses [there are no mermaids]. Using the convention of upper case letters for representing characterizations, we can also write 'BEING {ELK}ISH' as 'THE EXISTENCE OF ELKS' and 'BEING {MERMAID} ISH' as 'THE EXISTENCE OF MERMAIDS'. As it happens that there are mermaids does not correctly characterize the world. Note that our upper case convention does not extend to the form 'that s'. Thus we say that 'there are elks' expresses the proposition that there are elks. Equivalently we could say it expresses the proposition BEING {ELK} ISH, THERE BEING ELKS, THE EXISTENCE OF ELKS. To every true characterization there corresponds a characteristic ofthe world. Consider the statement, 'there are elks'. This statement expresses the true proposition that there are elks, a proposition that is true of the world because of the presence of elks, an existential characteristic of the world. In general, a totality that has a Q constituent has the characteristic of {Q}ishness. That is to say, the existence (or presence) of a Q thing ({Q}ishness) is a (constitutive or 'existential') characteristic ofthe totality. If the world is Q-ish, then BEING {Q}ISH (the proposition expressed by 'there are Q things') corresponds to being Q-ish or {Q}ishness (a world characteristic). Consider 'there are no K things'. If the totality is un {K} ish, then it is characterized by the nonexistence ofK things (by un{K}ishness, by

Reasoning

19

being un{K}ish). The proposition that there are no K things then corresponds to the un{K}ishness of the world. Un{elf}ishness (the nonexistence of elves) is a negative existential characteristic ofthe world; {elk}ishness (the existence of elks) is a positive existential characteristic of the world. A more familiar term for a positive or negative world characteristic is 'fact'. The existence of elks ( {elk}ishness) is a positive fact; the nonexistence of elves (un{elf}ishness) is a negative fact.

12. Truth and Correspondence to Facts The world's existential characteristics constitute the facts. Facts are what make true propositions true. A world characterization is true (or true of the world) if it corresponds to a characteristic of the world. Each of the above characterizations corresponds to an existential characteristic of the world. For example, the existence of women farmers ( {women farmer} ishness), is the fact that corresponds to and confers truth on such characterizations as SOME WOMEN BEING FARMERS, BEING {WOMAN FARMER}ISH, THE EXISTENCE OF WOMEN FARMERS, THERE BEING WOMEN FARMERS, {WOMEN FARMER}ISHNESS, and that there are women farmers, which are all different but equivalent expressions standing for [some women are farmers], the proposition expressed by 'some women are farmers'. A statement that expresses a true proposition is true. Since [some women are farmers] corresponds to <some women are farmers>, it is a true proposition and the statement expressing it is a true statement. A proposition (and the statement that expresses it) is false if the proposition does not correspond to any characteristic of the world. For example, SOME THING BEING AN ELF is a false characterization that does not correspond to any fact, there being no such fact as {elf}ishness. Among the world's existential characteristics (facts) are the following: the existence of horses; {horse} ishness, being {horse} ish the nonexistence of elves; un {elf} ishness, being un {elf} ish the existence of women farmers; {woman farmer}ishness the existence of elks; {elk} ishness the nonexistence of mermaids; un{mermaid}ishness Using the angle bracket notation for the facts that are characteristics of the world, we represent the fact signified by 'there are elks' by '' and the fact signified by 'there are no elves' by ''. This negative fact, corresponds to the proposition that there are no elves, making it true.

****************************************************************** Exercises: What fact makes the following true: 1. that there are no mermaids answer: the nonexistence of mermaids 2. THERE BEING RICH BACHELORS answer: ? 3. that some even number is prime answer: the existence of 4. [Bertrand Russell did not write Waverly] answer:< ? > 5. that no pope is female answer: the nonexistence of ?

****************************************************************** 13. Propositions Generally, any statement 's' expresses as its sense the proposition that s. (which we symbolically represent as '[s]'). The form 'that s' is so common a way oftalking about the sense of's' that we continue to use lower case letters for it. For example, in asserting 'some women are farmers' we claim that SOME WOMEN BEING FARMERS obtains, (is a true characterization of the world). Equivalently we are claiming truth for the proposition that some women are farmers. As it happens, the existence of women farmers is a fact; [some women are farmers] corresponds to the fact <some women are fanners>. So [some women are farmers] is a true proposition.

Reasoning

21

Just as the proposition, [s], expressed by a statement 's' can be spoken of in different ways, so the fact, <s>, that 's' signifies, can be spoken of in different ways. Here are some of the ways we may speak of the fact that some women are farmers. some women being farmers the existence of women farmers {woman farmer} ishness the state of affairs in which some women are farmers All these phrases are equivalent ways of talking about one and the same fact: the existence of women farmers. We have now shown how to answer the question: What is there about the world that makes a true statement true? Consider any statement of form 'some thing is a Q thing'. Ifthe world is characterized by {Q}ishness, the statement signifies a fact that corresponds to its sense. That fact --the existence of a Q thing-makes the statement true. If the world is un{Q}ish, the statement is false; in that case un {Q} ishness is a fact and the contradictory statement 'no thing is a Q thing' is true. Consider again the true statement 'there are no mermaids'. This statement expresses the true negative proposition that there are no mermaids. Equivalently, we may think of the proposition expressed as a STATE OF AFFAIRS: THE NONEXISTENCE OF MERMAIDS. The proposition is true (the STATE obtains) because un{mermaid}ishness, a negative existential characteristic of the world is a fact. This fact, which is signified by 'there are no mermaids', makes the proposition true. And that in tum means that the statement expressing it is true.

14. 'States of Affairs' We commonly speak of a statement as expressing a 'state of affairs'. Here one should distinguish between STATES that are expressed and the states that are signified. What a statement expresses is a proposition or STATE OF AFFAIRS, what it signifies, if anything, is a fact or state of affairs (lower case). False statements express STATES but they do not signify states. For example, THE EXISTENCE OF MERMAIDS is the STATE OF AFFAIRS expressed by 'there are mermaids'. But there is no such state of affairs as the existence of mermaids. So THE EXISTENCE OF MERMAIDS does not

22


obtain. (A STATE that corresponds to a state characterizes the world and is said to 'obtain'.)

15. The facts and the FACTS Here again we distinguish between an upper and lower case meaning of a key word. In its primary meaning the word 'fact' denotes an existential characteristic of the world but 'fact' is also often used as a synonym for 'true proposition' or STATE OF AFFAIRS that obtains. Taken as a synonym for 'true proposition' the word 'fact' should be written in upper case. For example, that some farmers are women is a true proposition; we may call this proposition a FACT. FACTS are true of the world. That there are mermaids (The EXISTENCE OF MERMAIDS) is not a FACT. On the other hand; the NONEXISTENCE of MERMAIDS, that there are no mermaids, is a FACT or true proposition. The nonexistence of mermaids is a fact (lower case). This fact is a negative existential characteristic of the world. FACTS correspond to, are made true by, facts.

16. What Statements Denote Suppose I am at the zoo and say 'that elk keeps staring at me'. The nonvacuous term 'elk' signifies the characteristic of being an elk and it denotes something that has the characteristic signified. Just as non-vacuous terms denote what they characterize so do true statements. A true statement such as 'there are elks' signifies a (positive, existential) characteristic of the world ( {elk} ishness) and it too denotes something that has the signified characteristic. Since it is the world that possesses the characteristic of {elk} ishness, the statement 'there are elks' denotes the world. A true negative statement such as 'there are no elves' signifies a (negative existential) characteristic of the world (its un{elf}ishness) and it too denotes what has the signified characteristic. Thus 'there are elks' and 'there are no elves' signify different facts but both denote one and the same world. The false statement 'there are elves' expresses a characterization ({ELF} ISHNESS) but, like the vacuous term, 'elf', the statement denotes nothing and signifies nothing. True statements signify positive or negative facts. Facts differ from one another. All true statements denote one and the same world. As for false statements, they have expressive meaning but apart from that they are vacuous.

Reasoning

23

17. Summary and Discussion on the Meaning of Statements If 's' is a statement then 1. 's' expresses [s] (the proposition that s) If [s] is true of the world, then <s>, a fact, is an existential characteristic of the world and 2. 's' signifies <s>, 3. [s] corresponds to <s>, 4. [s] 'obtains', is true, is a FACT, 5. 's' denotes the world, 6. 's' is true. If there is no such fact as <s> then 1* [s] does not correspond to any fact, 2* [s] does not obtain, is false, is not a FACT, 3* 's' does not denote the world, 4 * 's' does not signify a fact, 5* 's' is false.

******************************************************************* Exercises: I. What proposition or STATE OF AFFAIRS is expressed by each of the following statements? What fact or state (if any) does it signify? What, if anything does it denote? (The answers to (1) and (2) are given by way of illustration.) 1. no senator is a citizen: expresses TilE NONEXISTENCE OF SENATORS WHO ARE CITIZENS claiming that it obtains. But it fails to signify a fact. (An equally good answer is: ( 1) expresses the proposition that there are no senators who are citizens but this proposition does not correspond to any state of affairs and so ( 1) is false and it does not denote the world. 2. some waiters are not friendly: expresses the proposition that some waiters are not friendly. (Equivalently it expresses SOME WAITERS NOT BEING FRIENDLY (a FACT) or THE EXISTENCE OF UNFRIENDLY WAlTERS.) It signifies a well known fact, the existence of unfriendly waiters. (Another technically correct answer is: (2)

24


expresses [some waiters are not friendly], a true proposition that corresponds to <some waiters are not friendly>, the fact signified by (2). THE PRESENCE OF UNFRIENDLY WAlTERS correctly characterizes (is true of) the world so the statement expressing this FACT denotes the world.) 3. 4. 5. 6.

not a creature was stirring some mammals lay eggs. some birds do not fly. no bird is immortal

II. Which of the following statements denote the world? 1. There are no elves. 2. Some citizens are not farmers. 3. All women are citizens. 4. Elvis lives. 5. The France is a republic. III. Which ofthe following is incorrect? 1. Any statement is a truth claim made with respect to a specific domain, called the domain of the claim (DC). 2. A statement's' signifies <s>, only if <s> is an existential characteristic of the DC of 's'. 3. If's' is a true statement, then [s] corresponds to the world. 4. A statement, 's' denotes its DC, if and only if <s> is a fact. 5. A statement, 's,' denotes its DC if and only if[s] is true of its DC. 6. A fact is a property of the world. 7. FACTS are true propositions. 8. FACTS correspond to facts. 9. Vacuous statements are meaningless. 10. That some dogs are not friendly is a negative FACT. 11. That no dogs are friendly is a negative FACT. 12. False statements are not vacuous.

*******************************************************************

2 Picturing Propositions

1. State Diagrams The propositions expressed by statements are STATES OF AFFAIRS. In what follows we sometimes use 'STOA' or 'STATE' as an abbreviated way ofwriting 'STATE OF AFFAIRS'. A STOAmay or may not be a FACT. A STOA or proposition is true (obtains, is a FACT) if it corresponds to a fact and false if it does not. Calling a statement true (or false) is just a convenient way of saying that the proposition it expresses is true (or false). John Venn, a nineteenth century logician, introduced a way of depicting simple propositions or STOAs by means of diagrams. The two Venn Diagrams below show how to represent two STATES the world could be in with respect to the presence or absence of mermaids. Figure 1

Figure 2 mermaids

25

26


A square figure represents all the things in the domain. Any things inside the circle are mermaids. Any things outside are nonmermaids. By placing a cross inside the circle labeled 'mermaids' we indicate that the circle is not empty. Thus the first diagram represents the positive STATE OF AFFAIRS: SOME THINGS BEING MERMAIDS or THE EXISTENCE OF MERMAIDS. In the second diagram the mermaid circle is shaded. By shading the circle we signify that it is empty. Figure 2 represents the negative STATE OF AFFAIRS: the NONEXISTENCE OF MERMAIDS. Figure 2 represents a STOA that is a FACT. Figure 1 represents a STOA that is not a FACT. The two statements that express the depicted STATES are: 1. 2.

there are mermaids (Figure 1) there are no mermaids (Figure 2)

(1) claims that the EXISTENCE OF MERMAIDS corresponds to a fact. (2) claims that the NONEXISTENCE OF MERMAIDS corresponds to a fact. What (1) claims is false; there is no such fact as the existence of mermaids. (2) is true; the nonexistence of mermaids is a fact. Logicians are particularly interested in the STATES expressed by certain basic statements. Let 'S' and 'P' be two terms. Then the STATE, expressed by 'someS is P' is THE EXISTENCE OF AN SP THING:

Figure 3

[someS is P)

The STATE expressed by 'noS is P' is THE NONEXISTENCE OF AN SP THING:

Picturing Propositions

Figure 4

27

[noS is P]

The STATE expressed by 'every Sis P' is THE NONEXISTENCE OF AN

S(-P) TIUNG: Figure 5

[every S is P]

The STATE expressed by 'someS is not P' is THE EXISTENCE OF AN S(P)TIDNG:

Figure 6

[someS is not P]

Because 'no S is P' contradicts 'some S is P' Figure 4 is shaded (indicating absence) where Figure 3 shows occupancy or presence. Similarly the Figure


28

for 'every S is P' shows absence where that for 'some S is not P' shows presence. Note that 'noS is notP' expresses the same STATE as 'every Sis P'. For example, 'no senator is not pragmatic' claims that the world is characterizedbyTHEREBEINGNONONPRAGMATICSENATORS. The same claim is made by 'every senator is pragmatic'; both statements express the STOA depicted by shading the S(-P) segment (figure 5), depicting the absence of anything that is S and non-P.

2. Representing Singular Propositions A statement that has a proper name or other uniquely denoting term in subject position is called singular. Examples of singular statements are: Socrates is wise. Garbo is beautiful. The President is tired. Bigfoot is hairy.

S* is W G* is B P* is T B* isH

As we saw in the first chapter, a singular term letter is affixed with a star to indicate that the term it stands for is a uniquely denoting term (UDT), i.e., a term that applies to no more than one individual. Note that 'president' in the phrase 'The president' is represented by a starred letter. In the context of a phrase ofthe form 'the S', the term'S' is a UDT and so we star it. The singular statement 'Bigfoot exists' expresses the singular proposition: THE EXISTENCE OF BIGFOOT:

Figure 7

B*()


29

All things other than Bigfoot are outside the circle. The circle represents a set of things which, if it is occupied, has only one thing in it, Bigfoot. The cross indicates that the set of things that are Bigfoot is not empty, thus the diagram represents the claim that some thing is Bigfoot, that Bigfoot exists. The corresponding negative proposition expressed by 'Bigfoot does not exist' would then be represented thus:

Figure 8

The shading indicates that nothing is Bigfoot. From a logical point of view, a singular statement, 'N* is P' has the form 'some X* is Y' since it claims existence. For example, 'Bigfoot is hairy' claims that the world is characterized by the presence of a hairy creature known as Bigfoot. However, since there is no more than one Bigfoot, 'some Bigfoot* is hairy' entails 'every Bigfoot* is hairy'. The STATE expressed by 'Bigfoot is hairy' is a STATE of absence as well as presence: Figure 9

[(some/every Bigfoot* is hairy]

H

30


******************************************************************* Exercises: Represent the propositions expressed by the following statements by means of Venn diagrams: 1. There are Eskimo senators. 2. There are no women moonwalkers.

3. No one is perfect. 4. Socrates is wise. 5. All humans are mortal. 6. There is no such person as the present King of France. (hint use 'K*' for 'present King of France' and treat the statement as 'The K* does not exist' or as 'Nothing is K*'. · 7. Russell was a genius. 8. Some actors are not rich. 9. Some who are rich are not actors. 10. Every fool is unwise.

****************************************************************** 3. Entailments We noted earlier that the truth of any statement claiming THE EXISTENCE OF SOMETHING THAT IS BOTH X AND Y entails the truth of its converse. In other words if the proposition expressed by 'some X is Y' is true, then the proposition expressed by 'some Y is X' must also be true. And again, to say that one statement entails another is a convenient way of saying that the proposition expressed by the first statement entails the proposition expressed by the second. We now turn to the task of explaining how one true proposition can entail the truth of another proposition. Consider again the statement s3

some citizen is a farmer

We represent the STATE expressed by s3 (that is, the STATE [s3]) by:


Figure 10

c

31

F

Figure 10 depicts the EXISTENCE OF A CITIZEN FARMER and s3 is true if [s3], the STATE depicted, characterizes the world. (As it happens, the existence of citizens who are also farmers is a fact. Thus [s3] is a true proposition and so the statement, s3, that expresses this proposition, is true.) Now consider s4

some farmer is a citizen

The STOA expressed by s4 is [s4], depicted in Figure 11

Figure 11

c

i.e., the EXISTENCE OF A FARMER WHO IS ALSO A CITIZEN. It is clear that the two Venn Diagrams are like two photos of the same state taken from different angles. It is clear, in other words, that the state that makes [s4] true is the very same state that makes [s3] true. Putting 'X' for 'citizen' and 'Y' for 'farmer' we see generally that any two statements of the form 'some X is a Y' and 'some Y is an X' will express one and the same STATE OF AFFAIRS. This being so, it will never be possible for one ofthese statements to be true and the other false. For if the STATE in question obtains, both will be true and ifthe STATE does not obtain both will be false. In this way the mutual entailment that holds between s3 and s4 is represented by the Venn diagrams that picture [s3] and [s4] as a single STATE.

32

An Invitation to Formal Reasoning Consider also the two statements:

s5 s6

no senator is an albino no albino is a senator

The following Venn Diagrams depict in different ways the single negative STATE expressed by s5 and s6.

Figure 12

In general, any two statements of the form 'no X is Y' and 'no Y is X' express one and the same (negative) STATE OF AFFAIRS, which makes it impossible for one to be true and the other false. When two statements express one and the same STATE OF AFFAIRS they mutually entail each other and we call them logically equivalent. Sometimes we have entailment one way but not the other way. Consider for example the following little argument: s7 s8

some farmer is a citizen and a poet /some farmer is a citizen

The two statements are not logically equivalent: [s7] entails [s8] but [s8] does not entail [s7]. We can see why by looking at the diagrams of the STATES each statement expresses.


Figure 13 F

33

Figure 14

c

The diagram on the left represents [s7], the STOA expressed by s7: The EXISTENCE OF A FARMER-CITIZEN-POET. The diagram on the right represents [s8], the STOA expressed by s8: the EXISTENCE OF A FARMER-CITIZEN. Note that the EXISTENCE OF A FARMER-CITIZEN, represented in diagram 14 by an 'x' in the overlap of the two circles is already shown in the diagram for 13 as part of the STATE which is the EXISTENCE OF A FARMER-CITIZEN-POET. For in diagram 13 we already have an x in the overlap of the 'farmer' and 'citizen' circles. Thus, if the EXISTENCE OF A FARMER CITIZEN-POET is a FACT, so is the EXISTENCE OF A FARMER-CITIZEN. But the converse does not hold: diagram 14 could represent a FACT even if Diagram 13 did not. For it could be the case that there is no farmer-citizen who is also a poet. The STATES depicted in the two diagrams are positive; the diagrams show that one of the STATES of EXISTENCE is included in the other, thereby grounding the entailment of one proposition by the other. More generally, if S 1 and S2 are any two statements and the STATE, [S 1] includes the STATE [S2], then Sl entails S2.

4. Negative Entailments A relation of entailment may hold between statements expressing negative STATES, STATES OF NONEXISTENCE. Ifthere are no mermaids, there are no flute-playing mermaids: the NONEXISTENCE OF MERMAIDS excludes the EXISTENCE OF FLUTE-PLAYING MERMAIDS ('includes' the NONEXISTENCE OF FLUTE-PLAYING MERMAIDS). And that is why 'nothing is a mermaid' will entail 'nothing is a flute-playing mermaid'.

34


Note that the STATE diagram for the premise in which the mermaid area is empty already shows the smaller area for flute-playing mermaids to be empty. Figure 15 [slO]

[s9]

Here, as in the case of s7 and s8, we have an entailment in one directin. More often than not, we have no entailment either way. We noted above that statements of form 'some X isn't Y' and 'some Y isn't X' are not equivalent and that neither entails the other. For example, an argument using either sl1, 'some farmer isn't a citizen', or s12, 'some citizen isn't a farmer', as premise with the other as conclusion is invalid. These statements express different STATES OF AFFAIRS neither of which includes the other. Here are the respective diagrams:

figure 16

figure 17

[sll] F

[s12]

F

As the diagrams show, the STOAS expressed by s11 and s12 are distinct; neither is included in the other. So there is no entailment in either direction.

35


******************************************************************** Exercises: Examine the following Venn diagrams. What proposition does each graphically represent? What proposition does it entail? (Explain how the entailed proposition, the conclusion, is depicted in the diagram that depicts the premise.)

c

F

c

F

****************************************************************** 5. STATES and states STATES that obtain correspond to states of affairs. For example the EXISTENCE OF A MILLIONAIRE WHO IS A FARMER AND A PHILOSOPHER obtains (is a FACT) because it corresponds to a fact: the existence of a millionaire who is a fanner and a philosopher. This fact includes another fact: the existence of a philosopher who is a millionaire. Thus, just as STATES include or exclude other STATES, so the corresponding states include or exclude other states. The nonexistence ofmennaids excludes the existence of unhappy mennaids so that 'there are no mermaids' entails the falsity of 'there are unhappy mennaids'. Equivalently we may say that the nonexistence ofmennaids includes the nonexistence of unhappy mennaids. so that 'there are no mennaids' entails 'there are no unhappy mennaids'. The existence of rich bridesmaids includes the existence of bridesmaids.

36


6. Positive and Negative 'Valence' In a Venn diagram the STOA expressed by a statement claiming absence (nonexistence) is represented by shading; the STOA expressed by a statement that claims presence is depicted as unshaded and marked by a cross. Let us call any statement that claims presence 'positive in valence' and any statement that claims absence 'negative in valence'. Venn diagrams graphically show that two logically equivalent statements express one and the same STATE OF AFFAIRS. Now ifthat STATE is a STATE of presence, the two statements that express it will be positive in valence. If it is a STATE of absence the two statements expressing it will be negative in valence. In effect when two statements are logically equivalent, both statements must be positive in valence or else both must be negative; thus no statement claiming presence can possibly be equivalent to a statement that claims absence. To put this P

An Invitation to Formal Reasoning: The Logic of Terms

An Invitation to Formal Reasoning

An Invitation to Formal Reasoning

forall x: An Introduction to Formal Logic

An Invitation to Astrophysics

An Invitation to Sin

An Invitation to Astrophysics

An invitation to astrophysics

An Invitation To Sin

An Invitation to Biomathematics

An Invitation to Sin

An Invitation To Seduction

An Invitation to Faith

An Invitation to Sin

An Invitation To Sin

An Invitation to Sin

An invitation to biomathematics

Formal Logic

An Invitation to Seduction

An Invitation to Pleasure

An Invitation to Pleasure

An Invitation to Seduction

An Invitation To Sin

An Invitation To Sin

An Invitation to Astrophysics

Outlines of formal logic

An Introduction to Reasoning

An invitation to Morse theory

An Invitation to Reflexive Sociology

An invitation to C-algebras

An Invitation to Morse Theory

An Invitation to Formal Reasoning: The Logic of Terms