This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] Acknowledgements We were unfortunately remiss in the first edition in failing to thank Harry Stanton for his encouragement to write this text. We gratefully acknowledge the comments we have received from colleagues who have taught from the first edition , particularly Jon K vanvig and Chris Menzel . A number of typo graphical errors were identified by an extremely meticulous self - study reader from the Midwest whose identity has become lost to us and whom we encourage to contact us again (and to accept our apologies) . Chris Menzel also receives credit for his extensive (and voluntary ) work on the web software. Finally , Amy Kind deserves special thanks for her help with the website and for her most useful comments on the manuscript for the second edition .
This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] Preface To the Student The most important thing for you to know about this book is that it is designed to be used with a teacher. You should not expect to learn logic from this book alone (although it will be possible if you have had experience with formal systems or can make use of the web site at http :// mitpress.mit .edu/ LogicPrimer /) . We have deliberately reduced to a minimum the amount of explanatory material , relying upon your instructor to expand on the ideas. Our goal has been to produce a text in which all of the material is important , thus saving you the expense of a yellow marker pen. Consequently, you should never turn a page of this book until you understand it thoroughly . The text consists of Definitions , Examples, Comments, and Exercises. (Exercises marked with asterisks are answered at the back of the book .) The comments are of two sorts. Those set in full - size type contain material we deem essential to the text. Those set in smaller type are relatively incidental - the ideas they contain are not essential to the flow of the book , but they provide perspective on the two logical systems you will learn .
In this age of large classesand diminished personalcontact between studentsand their teachers , we hope this book promotesa rewarding learningexperience. To the Teacher We wrote this book because we were dissatisfied with the logic texts now available . The authors of those texts talk too much. Students neither need nor want page after page of explanation that require them to turn back and forth among statements of rules , examples, and discussion. They prefer having their teachers explain things to them- after all , students take notes. Consequently, one of our goals has been to produce a text of minimal chattiness, leaving to the
Preface
in Sb"uctor the task of providing explanations . Only an insb"uctor in a given classroom can be expected to know how best to explain the material to the students in that class, and we choose not to force upon the insb"uctor any particular mode of explanation .
Another reason our for dissatisfaction was that most texts contain material that we are not interestedin teachingin an introductory logic class. Some logic texts, and indeed some very popular ones, contain chapterson informal fallacies, theoriesof definition, or inductive logic, and some contain more than one deductive apparatus . Consequently , we found ourselvesordering texts for a single-semestercourse and covering no more than half of the material in them. This book is intended for a one-semestercourse in which propositional logic and predicatelogic are introduced, but no metatheory. (Any studentwho has masteredthe material in this book will be well preparedto take a second course on metatheory, using Lemmon' s classic, Beginning ' Logic, or evenTennants Natural Logic.) We prefer systemsof natural deductionto other ways of representing ' , and we have adopted Lemmon s technique of explicitly arguments tracking assumptionson each line of a proof. We find that this techniqueilluminates the relation betweenconclusionsand premises better than other devices for managingassumptions . Besidesthat, it allows for shorter, more elegant proofs. A given assumptioncan be dischargedmore than once, so that it need not be assumedagain in order to be dischargedagain. Thus, the following is possible, and there is no needto assumeP twice:
1 2 1,2
(1) (2) (3)
P~ (Q & R) P Q& R
assume assume from 1,2
Preface
1,2 1,2 1 1 1
(4) (5) (6) (7) (8)
Q R P-+Q P-+R (P - + Q) & (P - + R)
from 3 from 3 from 4, discharge2 from 5, discharge2 from 6,7
---
Clearly, the notion of subderivation has no application in such a system. The alternative approachinvolving subderivationsallows a given assumptionto be dischargedonly once, so the following is needed:
P-+ (Q & R) P Q& Q p~ Q P Q& R
(8) (9)
R P-+R
(10) (P~ Q) & (P~ R)
assume assume from 1,2 from 3 from 4, discharge2 assume from 1,6 (sameinferenceas at 3!) from 7 from 8, discharge6 from 5,9
The redundancyof this proof is obvious. Nonetheless , an instructor who preferssubderivationstyle proofs can useour systemby changing the rules concerningassumptionsetsas follows: (i ) Every line has the assumptionset of the immediately preceding line, except when an . (ii ) The only assumption available for assumption is discharged at a line is the highest-numberedassumptionin the discharge given , that line assumptionset. (iii ) After an assumptionhasbeendischarged number can never again appearin a later assumptionset. (In other
Preface
words , the assumption - set device becomes a stack or a flfSt - in - last - out memory device .)
There are a number of other differences between our system and Lemmon' s, including a different set of primitive rules of proof. What follows is a listing of the more significant differences betweenour ' systemand Lemmon s, togetherwith reasonswe prefer our system. .
Lemmon disallows vacuousdischargeof assumptions . We allow it . Thus it is correct in our systemto dischargean assumptionby reductio ad absurdum when the contradictiondoesnot dependon that assumption. Whenevervacuousdischargeoccurs, one can obtain a Lemmon-acceptable deduction by means of trivial additionsto the proof. We prefer to avoid theseadditions. (Note that Lemmon' s preclusion of vacuous discharge means that accomplishingthe same effect requires redundantsteps of & introduction and & -elimination. For instance, Lemmon requires (a) to prove P ~ Q ~ P, while we allow (b) . (a) 1 2 1,2 1,2 1
( 1) (2) (3) (4) (5)
P Q Q& P P Q~ P
assume assume from 1,2 from 3 from 4, discharge2
(b) 1 2 1
(1) (2) (3)
P Q Q~ P
assume assume
from 1, discharge2
xiii
Preface
Lemmon ' s characterization of proof entails that an argument has been established as valid only when a proof has been given in which the conclusion depends on all of the argument' s premises. This is needlessly restrictive , since in some valid arguments the conclusion is in fact provable from a proper subset of the premises. We remove this restriction , allowing a proof for a given argument to rest its conclusion on some but not all of the ' argument s premises. We have replaced Lemmon ' s primitive v -Elimination rule by what is normally known as Disjunctive Syllogism (DS ) . We realize that Lemmon ' s rule is philosophically preferable, as it is a pure rule ; however , DS is so much easier to learn that pedagogical considerations outweigh philosophical ones in this case. Despite the preceding point , we have kept the 3 -elimination rule used by Lemmon . Although slightly more complicated than the more common rule of 3 - Instantiation , this rule frees the student from having to remember to instantiate existential quantifications before instantiating universal quantifications . It also frees the student from having to examine the not- yet-reached conclusion of the argument, to determine which instantial names are unavailable for a given application of 3 - Instantiation . Furthermore , at any point in a proof using 3 -elimination , some argument has been proven . If the proof has reached a line of the form
m, ...,n
(k)
z
tilen tile sentencez has been establishedas provable from the premise set { m, . . .,n } . (Here the right-hand ellipsis indicates
xiv
Preface
which role was applied to yield z, and which earlier sentencesit was applied to.) This is quite useful in helping the student understand what is going on in a proof . In a system using 3 instantiation , however , this feature is absent: there are correct proofs some of whose lines do not follow from previous lines , since the role of 3 -instantiation is not a valid role . For instance, the following is the beginning of a proof using 3 -instantiation . 1 1
( 1) ( 2)
3xFx Fa
assumption 1 3 - instantiation
Line 2 does not follow from line 1. This difference between 3 elimination and 3 -instantiation can be put as follows : in an 3 elimination proof , you can stop at any time and still have a correct proof of some argument or other , but in an 3 -instantiation proof , you cannot stop whenever you like . It seems to us that
these implications of 3-instantiations invalidity outweigh the additional complexity of 3-elimination. In an 3-elimination system, not only is the systemsoundas a whole, but every rule is individually valid; this is not true for an 3-instantiationsystem. .
Whereas Lemmon requires that existentialization (existential generalization) replace all tokens of the generalizedname by tokensof the bound variable, we allow existentializationto pick up only someof the tokensof the generalizedname. We have abandoned Lemmon' s distinction between proper namesand arbitrary names, which is not essentialin a natural deductionsystem. The conditionson quantifier rules ensurethat the instantial name is arbitrary in the appropriatesense. ( We commenton this motivation for the conditionsin the text.)
Preface
In many cases, we have deliberately not used quotation marks to indicate that an expression of the formal language is being mentioned. In general, we use single quotes to indicate mention only when confusion might result. ( We hope no one is antagonized by this flaunting of convention . Trained philosophers may at first find the absence of quotes disconcerting , but we believe that we are making things easier without leading the student astray significantly .)
We have tried to presentthe material in a way that revealsclearly the systematicorganizationof the text. This mannerof presentationmakes it especiallyeasy for studentsto review the material when studying, and to look up particular points when the need arises. Consequently , thereis little discursiveprosein the text, and what seemedunavoidable . We hope to have produced a has been relegatedto the Comments small text that is truly studentoriented but that still allows the instructora maximumof flexibility in presentingthe material. TheSecondEdition With one exception, the changesto the second edition have been minimal. We haveaddeda ueatmentof identity to chapters3 and 4. In chapter3 this requiredmerely a slight modification to the definition of wff , somecommentson uanslation, and the inclusion of inuoduction and elimination rules for identity. The changesmadeto chapter4 are more extensive. In the first edition we avoided overt referenceto the object language/ metalanguage distinction and had no need to introduce into the specification of interpretations the extensions (denotations, referents) of names, but the inclusion of identity in the language necessitatesthem. To keep matters simple , when giving interpretations for sentences that involve identity we use italicized names in the metalanguage, and we recommend that no member of the universe of an interpretation be given more than one metalinguistic name. This makes it easy to specify whether or not two names of the object
Preface
language have the same extension in an interpretation , for the same metalinguistic name will be used for names denoting the same object . Expansions now involve the use of italicized names, so that strictly speaking they are not wffs of the object language. This does not affect their use in determining truth values of quantified wffs in an interpretation , and facilitates their use in determining truth values of wffs involving identity . ( We realize that italicization is not available for hand- written exercises, so we recommend that instructors adopt a convention such as underlining for blackboard presentations.) The addition of the material on identity is supplemented with new exercises in chapters 3 and 4. We have tried to organize the new material in such a way that an instructor who wishes to omit it can do so easily .
In chapter 1, a set of exerciseshas beeninsertedwhoseproofs do not require ~ I and RAA . That is, these proofs do not involve the . These exercises are intended to allow discharge of assumptions studentsto become comfortable with the remaining rules of proof before they are forced to learn the more complicatedmechanicsof ~ I andRAA . In chapter 3 we have waited until after the section on translations to introduce the notions of a wff s universalization , existentialization , and instance. This change reduces the chance of the student' s confusing the rules for constructing universally quantified wffs , where at least one occurrence of a name must be replaced by a variable , and univers alization , where all occurrences of the name must be replaced.
Preface
xvii
Web Support A variety of interactive exercises and an automated proof checker for the proof systems introduced in this book can be accessed at http :// mitpress.mit .edu/ Logic Primer/ . Use of the software requires nothing more than a basic web browser running on any kind of computer .
This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] Chapterl Sentential Logic 1.1
notion. Basic logical
argument , premises , conclusion
Definition. An ARGUMENT is a pair of things: . a setof sentences , the PREMISES . a sentence , the CONCLUSION . Comment. All argumentshave conclusions, but not all argumentshave premises: the set of premisescan be the empty set! Later we shall examine this idea in somedetail. Comment . If the sentencesinvolved belongto English (or any other natural language), we need to specify that the premisesand the conclusion are sentencesthat can be true or false. That is, the premisesand the conclusionmust all be declarative(or indicative) sentencessuch as ' The cat is on the mat' or ' I am here' , and not sentencessuchas ' Is the cat on the mat?' (interrogative) or ' Come here!' (imperative) . We are going to construct some formal languagesin which every sentenceis either true or false. Thus this qualification is not presentin the definition above.
validity
Definition . An argument is VA Lm if and only if it is necessary that if all its premises are true, its conclusion is true.
Comment . The intuitive idea captured by this definition is this: If it is possiblefor the conclusionof an argument to be false when its premisesare all b1Je , then the argumentis not reliable (that is, it is invalid).
1 Chapter
If true premisesguaranteea true conclusion then the argumentis valid. Alternateformulation of the definition. An argumentis V A Lm if and only if it is impossible for all the premisesto be true while the conclusionis false. entaUment
Definition. When an argumentis valid we say that its premisesENTAll.., its conclusion.
soundness
Definition . An argument is SOUND I if valid and all its premises are b"ue.
andonly if it is
. It follows that all sound argumentshave Comment hue conclusions. Comment. An argument may be unsound in either of two ways : it is invalid , or it has one or more false premises. with validity rather Comment . The restof this bookis concerned than soundness.
Exercise1.1
Indicate whether each of the following sentencesis True or False.
i* ii * iii * iv* v*
Every premise of a valid argument is hUe. Every invalid argument has a false conclusion . Every valid argument has exactly two premises. Some valid arguments have false conclusions. Some valid arguments have a false conclusion despite having premises that are all hUe.
Chapterl
vi * vii * ... Vlll * ix * x*
A sound argument cannot have a false conclusion . Some sound arguments are invalid . Some unsound arguments have true premises. Premises of sound arguments entail their conclusions. If an argument has true premises and a true conclusion then it is sound.
1.2
A Formal Language
fonnal
Comment . To representsimilarities among arguments of a natural language, logicians introduce formal . The first formal languagewe will introduce languages is the language of sentential logic (also known as propositionallogic) . In chapter3 we introduce a more sophisticatedlanguage: that of predicatelogic.
language
vocabulary
for Sentential
Logic
. TheVOCABULARY OF SENTENTIAL Definition -
LOGICconsists of . SENTENCE LETTERS , . CONNECTIVES and , . PARENTHESES . sentence
Definition. A SENTENCE LEIf ER is anysymbol
letter
from the following list:
A, ... , Z, Ao'... , Zo'AI' ... ,Zit....
sentence variable
Comment . By the useof subscriptswe make available an infinite numberof sentenceletters. Thesesentence letters are also sometimescalled SENTENCE VARI ABLES , becausewe use them to stand for sentences of naturallanguages .
Chapter1
Definition. The SENTENTIAL CONNECTIVES (often just called CONNECTIVES ) are the members of the following list: - , & , v , - +, H .
connectives
Comment. The sentential connectives correspond to various words in natural languages that serve to connect declarative sentences.
tilde
-
The TILDE correspondsto the English ' It is not the casethat' . (In this casethe useof the term ' connective' is odd, since only one declarativesentenceis negated at a time.)
' ampersand & The AMPERSAND correspondsto the English Both ... and ... ' . The WEDGE correspondsto the English ' Either ... or . . . ' in its inclusivesense.
wedge
v
an - ow
~ The ARROW corresponds to the English
then
,
... .
double arrow
The DOUBLE - ARROW correspondsto the English ' ' if and only if .
Chapterl
Comment. Natural languagestypically provide more than one way to expressa given connectionbetweensentences . For instance, the sentence' John is dancing but Mary is sitting down' expresses the same logical relationship as ' John is dancing and Mary is sitting down' . The issue of translation from English to the formal languageis takenup in section 1.3.
) and ( . expression
The right and left parentheses are used as punctuation marks for the language .
Definition. An EXPRESSION of sentential logic is any sequenceof sentenceletters, sentential rives, or left andright parentheses . Examples. (P ~ Q) is an expressionof sententiallogic. )PQ~ - is alsoan expressionof sententiallogic. (3 ~ 4) is not an expressionof sententiallogic.
metavariable Definition. Greek letters such as , and 'I' are used as METAVARIABLES . They are not themselvesparts of the language of sentential logic, but they stand for expressionsof the language. Comment. (, ~ '1') is not an expressionof sentential : an expressionof logic, but it may be usedto represent sententiallogic.
ChapterI
Definition. A WELL - FORMED FORMULA ( WFF) of sententiallogic is any expressionthat accordswith the following sevenrules:
weD - formed
f OrDlu1a
( 1) A sentenceletter standing alone is a wff .
[Definition.The sentence lettersaretheATOMIC of thelanguage: of sententiallogic.] SENTENCES
atomic sentence
(2) If , is a wff , then the expression denoted by - , is also awIf . [Definition. A wff of this form is known as a NEGA TION , and - ~ is known asthe NEGATION OF ~.]
negation
(3) If ~ and 'I' are both wffs, then the expression denoted by (~ & '1') is a wff .
[Definition. A wff of this form is knownas a CONJUNCTION. ct>and'If areknownasthe left andright .] CONJUNCTS,respectively
conjunction
(4) If cj) and 'If are both wffs, then the expression denoted by (~ v '1/) is a wff .
disjunction
[Definition. A wff of this form is known as a DIS JUNCTION . ~ and '1/ are the left and right DISJUNCTS, respectively.]
(5) If (j) and 'I' are both wffs, then the expression denoted by (j) - + '1') is a wff .
Chapter1
conditional , antecedent , consequent
[Definition. A wffofthis Connis known as a O : NXnO NAL. Thewff , is knownas theANTECEDENT of the conditional . The wff 'If is known as the .] CONSEQUENTof theconditional
(6) If , and 'If are both wiTs, then the expression denoted by (cj) H '1') is a wff .
biconditional [Definition. A wff of this form is known as a BI CONDm ONAL. It is alsosometimes knownasan . EQUIVALENCE] (7) Nothingelseis awIf . binary and unary connectives
Definition. & , v , ~ , and H are BINARY CONNECTIVES , since they connect two wffs together. - is a UNARY CONNECTIVE , sinceit attachesto a single wff .
sentence
Definition. A SENTENCE of the fonnallanguage is a wff that is not part of a largerwff .
denial
Definition . The DENIAL of a wff (j) that is not a negation is - (j). A negation, - (j), has two DENIALS : (j) and - - (j).
Example. - (P - + Q) hasone negation: - - (P - + Q) It hastwo denials: (P - + Q) and - - (P - + Q) . (P - + Q) hasjust one denial: its negation, - (P - + Q).
Chapterl
Comment. The reason for introducing the ideas of a sentence and a denial will be apparent when the rules of proof are introduced in section 1.4.
Exercise 1.2.1 Which of die following expressions are wffs ? If an expression is a wff , say whedier it is an atomic sentence, a conditional , a conjunction , a disjunction , a negation, or a biconditional . For die binary connectives , identify die component wffs (antecedent, consequent , conjuncts , disjuncts , etc.) .
0 1* 00 n* 000 U1* iv * v* vi * vii * viii * ix * x* xi * xii * xiii * xiv * xv *
A (A (A ) (A - + B)
(A - + ( (A - + ( B - + C P & Q) - + R)
A & B) v (C -+ (0 H G ) - (A -+ B) - (P -+ Q) v - (Q & R) - (A) (- A) -+ B (- (P& P) & (PH (Q v - Q ) (- B v P) & C) H 0 v - G) -+ H (- (Q v - (B v (EH (0 V X )
1 Chapter parenthesisdropping conventions
Comment. For easeof reading , it is often convenient to drop parentheses from wffs , so long as no ambiguity results. If a sentence is surrounded by parenthesesthen these may be dropped. Example . p ~ Q will be read as shorthand for (P ~ Q) . Comment. Where parentheses are embedded within sentences we must be careful if we are to omit any parentheses. For example, the expression P & Q ~ R is potentially ambiguous between P & Q) ~ R) and (P & (Q ~ R . To resolve such ambiguities , we adopt the following convention: - binds more strongly than all the other connectives; & and v bind component expressions more strongly than ~ , which in turn binds its components more strongly than H .
. Examples - P & Q ~ R is readas - P & Q) ~ R). P ~ Q H R is readas P ~ Q) H R). P v Q & R is not allowed,asit is ambiguous between v P ( (Q & R and P v Q) & R). P ~ Q ~ R is notallowed,asit is ambiguous between (P ~ (Q ~ R and P ~ Q) ~ R). Comment. The expressions admitted by these parenthesis -dropping conventions are not themselves well fonned fonnulas of sentential logic .
1 Chapter
Exercise1.2.2 Rewrite all the sentencesin exercise1.2.1 above, using -dropping conventions. Omit any parentheses the parenthesis you can without introducingambiguity. Exercise 1.2.3 State whether each of the following is ambiguous or unambiguous, given the parenthesis-dropping conventions . In the unambiguous cases, write out the sentences and reinstate all omitted parentheses. 0 1* 00 n* 000 U1* iv *
vi * vii * viii * ix *
PH - QvR PvQ ~ R & S PVQ ~ RHS PvQ & R ~ - S P~ R & S~ T P~ Q ~ R ~ S P& QH - RvS - P& QVR ~ SHT P~ Q & - R H - Sv T ~ U P~ Q & - R ~ - Sv T H U
1.3
Translation of English to Sentential Wffs
translation
Definition . A TRANSLATION SCHEME for the lan guage of sentential logic is a pairing of sentence letters with sentencesof a natural language. The sentencesin a translation scheme should be logically simple . That
scheme
is , they should not contain any of the words corresponding to the sentential connectives.
ChapterI
logical form
Definition. The LOGICAL FORM of a sentenceof a natural language relative to a translation scheme is given by its translation into a wff of sententiallogic accordingto that translationscheme. Example. Under the translationscheme P: Johndoeswell at logic Q: Bill is happy The sentence If Johndoeswell at logic, then Bill is happy hasthe logical form (P ~ Q) . Comment. English provides many different ways of stating negations, conditionals, conjunctions, disjunctions , andbiconditionals. Thus, manydifferent sentences of English may havethe samelogical form.
sty Ustic varianu
Definition. If two sentencesof a naturallanguagehave the same logical form relative to a single translation scheme, they are said to be STYLISTIC VARIANTS of eachother. Comment. There are far too many stylistic variants of negations, conjunctions, disjunctions, conjunctions, and biconditionals to list here. The follow is a partial list of stylistic variantsin eachcategory.
1 Chapter
negations
conditionals
Let P b' anslate the sentence ' John is conscious.' Here are a few of the ways of expressing - P: John is not conscious. John is unconscious. It is not the case that John is conscious. It is false that John is conscious.
Stylistic variants whose logical fonn is include these: . or 'l' . Either . or '1'. . unless'1'. ' Comment. , unless '1" is also commonly translated as ( - 'I' - + , ) . The proof techniques introduced in section 1.4 can be used to show that this is equivalent to (, v '1') . biconditionals
Variants having the logical form (, H '1') include the
following: . . , , neither. nor ...
if andonly if '1/. is equivalentto '1/. . is necessaryand sufficient for '11 . just in case'11
of theforill ' Neither~ nor'1/' have Englishsentences thelogicalforill - (~ v '1/), or, equivalently , (- ~ & - '1/).
Chapter1
tenses
Comment. In English , the sentences ' Mary is dancing ' and ' Mary will dance' have different meanings because of the tenses of their respective verbs. In some cases, when one is analyzing arguments it is important to preserve the distinction between tenses. In other cases, the distinction can be ignored . In general, a judgment call is required to decide whether or not tense can be safely ignored .
Example. Considerthe following two arguments: A If Mary is dancing, Johnwill dance. Mary is dancing. Therefore, Johnis dancing. B
If Mary dances, John will dance. If John dances, Bill will dance. Therefore , if Mary dances, Bill will dance.
In A , if the difference between ' John will dance' and ' John is dancing ' is ignored , then the argument will look valid in translation . But this seems unreasonable
on inspectionof the English. In B , ignoring the difference between ' John will dance' and ' John dances' also makes the argument valid in translation . In this case , however , this seems reason -
able.
Chapter1
In the translation exercisesthat follow, assumethat tensedistinctionsmay be ignored. Exercise 1.3
Translatethe following sentencesinto the languageof sententiallogic.
for 1- 20
1* 2* 3* 4* 5* 6*
10*
12* 13*
Johnis dancingbut Mary is not dancing. If Johndoesnot dance, then Mary will not be happy. John' s dancingis sufficient to makeMary happy. John' s dancingis necessaryto makeMary happy. John will not danceunlessMary is happy. If John' s dancing is necessaryfor Mary to be happy, Bill will be unhappy. If Mary dancesalthough John is not happy, Bill will dance. If neitherJohnnor Bill is dancing, Mary is not happy. Mary is not happy unless either John or Bill is dancing. Mary will be happyif both Johnand Bill dance. Although neither John nor Bill is dancing, Mary is happy. If Bill dances, then if Mary dancesJohnwill too. Mary will be happyonly if Bill is happy.
I Chapter 14* 15
*
16* 17
*
18* 19* 20*
Neither Johnnor Bill will danceif Mary is not happy. If Mary dancesonly if Bill dancesand John dances only if Mary dances, then John dancesonly if Bill dances. Mary will danceif Johnor Bill but not both dance. If John dancesand so does Mary, but Bill does not, then Mary will not be happybut Johnand Bill will . Mary will be happyif and only if Johnis happy. Provided that Bill is unhappy, John will not dance unlessMary is dancing. If John danceson the condition that if he dancesMary dances, then he dances. ira P: Q: R: S: T:
21*
22* 23
*
24*
DSlation schemefor 21- 25 A purposeof punishmentis deterrence . Capital punishmentis an effectivedeterrent. Capital punishmentshouldbe continued. Capital punishmentis usedin the United States. A purposeof punishmentis rebibution.
If a purpose of punishment is deterrence and capital punishment is an effective deterrent , then capital punishment should be continued . Capital punishment is not an effective deterrental though it is used in the United States . Capital punishment should not be continued if it is not an effective deterrent , unless deterrence is not a purpose of punishment . If retribution is a purpose of punishment but deterrence is not , then capital punishment should not be continued .
Chapter1
25
*
Capital punishment should be continued even though capital punishment is not an effective deterrent provided that a purpose of punishment is retribution in addition to deterrence.
Primitive Rules of Proof turnstile
Definition. The TU RN STa Eis the symbol ~ .
sequent
Definition. A SEQUENT consists of a number of sentencesseparatedby commas(correspondingto the premises of an argument), followed by a turnstile, followed by another sentence(correspondingto the conclusionof the argument). Example. (P & Q) - + R, - R & P ~ - Q more than a Comment. Sequents are nothing convenient way of displaying arguments in the formal notation . The turnstile symbol may be read as ' therefore ' .
proof
Definition. A PROOF is a sequence of lines containing . Eachsentenceis either an assumptionor sentences the result of applying a rule of proof to earlier . The primitive rules of proof sentencesin the sequence are statedbelow.
Chapter1
Comment. The purpose of presenting proofs is to demonsuate unequivocally that a given set of premises entails a particular conclusion . Thus, when presenting a proof we associate three things with each sentence in the proof sequence:
annotation
On the right of the sentencewe provide an ANNO TADON specifying which rule of proof was applied to which earlier sentencesto yield the given sentence .
assumptionset On the far left we associatewith eachsentencean ASSUMPTIONSET containingthe assumptions on which the given sentencedepends. line number
Also on the left , we write the current LINE NUM BER of the proof.
Hneof proof
Definition. A sentenceof a proof, togetherwith its annotation setandthe line number , its assumption , is calleda LINE OF TIlE PROOF. . Example 1,2 (7) P - + Q & R i Linenumber i set Sentence Assumption
proof for a given argument
6 ~ I (3) Annotation
Definition. A PROOF FOR A GIVEN ARGUMENT is a proof whose last sentence is the argument's conclusion depending on nothing other than the ' arguments premises.
Chapter1
primitive
rules
assumption
Definition. The ten PRI Mm V E RULES OF -inttoPROOF are the rules assumption , ampersand -elimination, wedge-inttoduction, duction, ampersand wedge-elimination, arrow-inttoduction, arrow-elimination , reductio ad absurdum, double-arrow-inttoduction, anddouble-arrow-elimination, as describedbelow. Assume any sentence .
Annotation : Assumption set: Comment:
A The currentline number. Anything may be assumed at any time . However , some assumptions are useful and some are not !
Example.
(1) ampersand . intro
Pv Q
Given two sentences ( at lines m and n ) , conclude conjunction of them .
Annotation : Assumption set: Comment:
Also known as:
a
m, n & 1 The union of the assumption sets at lines m and n. The order of lines m and n in the proof is irrelevant . The lines referred to by m and n may also be the same. Conjunction (CONI ) .
I Chapter
. Examples 1 ( 1) 2 (2) 1,2 (3) 1,2 (4) 1 (5) ampersand -
elim
p Q P& Q Q& P P& P
Given a sentence that is a conjunction clude either conjunct . Annotation : Assumption set : Also known as:
wedge-intro
A A
1,2 &1 1,2 &1 1,1&1 ( at line m ), con -
m &E The sameasat line m. Simplification (S).
. Examples (a) 1 (1) P& Q 1 (2) Q" 1 (3) P
l &E l &E
(b) 1 1
l &E
(1) (2)
P& (Q ~ R) Q~ R
Given a sentence (at line m), conclude any disjunction having it as a disjunct . Annotation : Assumption set: Also known as:
m vI The same as at line m. Addition (ADD ) .
Chapterl
Examples.
wedge-elim
(a) 1 1 1
(1) (2) (3)
P Pv Q (RH - T) v P
A 1vI 1vI
(b) 1 1
(1) (2)
Q -+ R (Q -+ R) v (P & - g)
A 1 vI
Given a sentence(at line m) that is a disjunction and anothersentence(at line n) that is a denial of one of its disjunctsconcludethe other disjunct. Annotation : Assumption set: Comment : Also known as :
m, n v E The union of the assumptionsetsat lines m andn. The order of m and n in the proof is irrelevant. Modus Tollendo Ponens (MTP), Disjunctive Syllogism (OS) .
Examples.
(a) 1 2 1,2
(1) (2) (3)
Pv Q -P Q
A A
(b) 1 2 1,2
(1) (2) (3)
P v (Q -+ R) - (Q -+ R) P
A A
1,2 vB
1,2 vE
Chapter1
(c) 1 2 1,2 arrow - in tro
(1) (2) (3)
Pv - R R P
A A
1,2 v E
Given a sentence (at line n), conclude a conditional having it as the consequent and whose antecedent appears in the proof as an assumption (at line m) . Annotation :
n - +1 (m )
Assumption set :
Everything in the assumption set at line n excepting m, the line number where the antecedent was assumed. The antecedent must be present in the proof as an assumption . We this assumption speak of DISCHARGING when applying this rule . the number m in parentheses Placing indicates it is the discharged assumption . The lines m and n may be the same. Conditional Proof (CP) .
Comment :
Also known as:
. Examples (a) 1 (1) 2 (2) 1,2 (3) 1 (4)
- PvQ P Q P~ Q
A A
1,2 vE 3 ~ I (2)
ChapterI
(b) 1 2 2 (c) 1
armw - elim
(1) (2)
R P
A A
(3)
P~ R
1-+1(2)
( 1) (2)
P P-+P
A 1 - +1 ( 1)
Given a conditional sentence(at line m) and another sentencethat is its antecedent(at line n), concludethe consequentof the conditional. Annotation : Assumption set:
m, n - +E The union of the assumption sets at lines m and n-
Comment: Also known as:
. Example 1 ( 1) 2 (2) 1,2 (3)
The order of m and n in the proof is irrelevant. Modus Ponendo Ponens (MPP), Modus Ponens (MP), Detachment , Affirming the Antecedent.
P~ Q P Q
A A
1,2 -.+E
Chapterl
reductio ad absurdum
Alsoknownas:
IndirectProof(IP), - Intra! - Elim.
Examples.
(a) 1 2 3 1,3 1,2
(1) (2) (3) (4) (5)
---
(b) 1 2 3 2,3 1,2,3 1,3
P-+ Q -Q P Q -P
A A A 1,3 ~ E
PvQ -P - P~ - Q -Q P P
A A A
2,4 RAA(3)
2,3 -+E 1,4 vE 2,5 RAA(2)
I Chapter
(c) 1 2 3 2,3
(1) (2) (3) (4)
p Q -Q -p
A A A
2,3 RAA(1)
double - armw . Given two conditional sentences having the forms intro cI> - + 'I' and 'I' - + cI> (at lines m and n ), conclude a
biconditional with cI>on one side and 'I' on the other. Annotation :
m , n HI set: The union of the assumption sets at Assumption lines m and n. Comment: The order of m and n in the proof is irrelevant .
. Examples 1 (1) 2 (2) 1,2 (3) 1,2 (4)
P~ Q Q~ P PHQ QH P
A A
1,2 HI 1,2 HI
double-arrow - Given a biconditionalsentence, H ' I' ( at line m ), con eUm clude either . ~ 'If or 'If ~ . . Annotation : Assumption set : Also known as :
mHE the sameas at m. Sometimesthe rules HI and HE aresubsumedasDefinition of Biconditional (df.H ) .
Chapter1
. Examples 1 (1) PHQ 1 (2) P- + Q 1 (3) Q - +P
A tHE tHE
Comment . These ten rules of proof are truth preserving . Given b' Ue premises, they will always yield b' Ueconclusions. This entails that if a proof can be consb' Uctedfor a given argument, then the argument is valid . aid in the discoveryof proofs, . A numberof strategies Comment . Wedonotprovideanyprooffor practice butthereis no substitute ' s . We in this book- that is the instructor job discoverystrategies lack of should be no so there do provideplenty of exercises , . opportunityto practice Exercise 1.4.1 Fill in the blanks in the following proofs .
P, - Q ~ P& - Q 1 (1) P (2) - Q (3) P& - Q PvQ , - QvR,
(1) Pv Q (2) - Qv R (3) (4) Q (5)
1,3 vE 2,4
Nil .>~ ~,
~ > l 0. t -~t ~.
Qt 0 . ~ > l at to -Qt ~-~( t ~
N
N~
~
. ~ t ~
--- I Q: 0
.*
3,5 RAA
--- Qt 0
~ t oi ~ > 0 . 0
: *
I -
1 Chapter
Exercise 1.4.2 Give proofs for the following sequents . All of these proofs may be completedwithout using the rules - +1 or RAA.
81* 82 83* 84* 85 86 87* 88 89* 810
P v - R, - R - + S, - P ~ S P v - R, - R - + S,- P ~ S & - R P - + - Q, - Q v R - + - S, P & T ~ - S P & (Q & R), P & R - + - S, S v T ~ T P - + Q, P - + R, P ~ Q & R P, Q v R, - R v S, - Q ~ P & S - P, Rv - PHPVQ ~ Q (P H Q) - + RiP- + Q,Q - + P ~ R - P- + Q & R, - Pv S - + - TU & - P ~ (U & R) & - T (Q v R) & - S - + T, Q & U, - S v - U ~ T & U
Sequents and Derived Rules double turnstile
Comment. If a sequent has just one sentence on each side of a turnstile , a reversed turnstile may be inserted ( -i ) to represent the argument from the sentence on the right to the sentenceon the left .
Example. P ~ ~ P v P Comment. This examplecorrespondsto two sequents : p ~ p v P and P v P ~ P. You may read the example as saying ' P thereforeP or P, and P or P thereforeP' . When proving cI ~~ 'If, one must give two proofs: one for cI ~ 'If and one for 'If ~ cI.
ChapterI
. Example ProveP -i .- P v P. (a) ProveP .- P v P. 1 ( 1) P 1 (2) P v P
A 1v I
(b) ProveP v 1 ( 1) 2 (2) 1,2 (3) 1 (4)
A A 1,2 v E 2,3 RAA (2)
P.- P. Pv P -P P P
Exercise1.5.1 Give proofs for the following sequents , using the primitiverulesof proof.
S11* S12* S13 S14* S15 S16* S17* S18* S19 S20 S21* S22 S23 S24 S25
P -i .- - - P P - + Q, - Q .- - P P - + - Q, Q .- - P - P --+Q, - Q I- P - P --+- Q, Q I- P P --+Q, Q --+RIP --+R PI - Q --+P - PIP --+Q PIP - -+Q P --+Q, P --+- Q I- - P - P v Q ~I- P -:0+Q P v Q ~I- - P --+Q P v Q ~I- - Q --+P P v - Q ~I- Q --+P P v Q, P --+R, Q --+R I- R
DoubleNegation ModusTollendoTollens MTf MTf MTf HypotheticalSyllogism TrueConsequent FalseAntecedent FA Antecedent Impossible WedgeArrow (v~ ) v - + v - + v - +
Simple DUemma
Chapterl
S26* S27 S28* S29 S30 S31 S32* S33 S34 S35 S36 S37* S38* S39 S40 S41* S42* S43 S44 S45 S46 S47 S48 S49 S50 S51 S52
P v Q, P - + R, Q - + S ... R v S Complex Dilemma P - + Q, P - + Q ... Q SpecialDilemma - (P v Q) ~ ... - P & - Q DeMorgan' s Law - (P & Q) ~... - P v - Q OM P & Q ~ ' " ( P v Q) OM P v Q ~ ' " ( P & Q) OM - (P - + Q) ~ ... P & - Q NegatedArrow (Neg- +) - (P - + - Q) ~... P & Q Neg- + P - + Q ~ ... (P & Q) Neg- + P - + Q ~... (P & Q) Neg- + P & Q ~... Q & P & Commutativity P v Q ~ ... Q v P v Commutativity P H Q ~ ... Q H P H Commutativity P - + Q ~ ... Q - + P trausposition P & (Q & R) ~ ... (P & Q) & R & Associativity P v (Q v R) ~ ... (P v Q) v R v Associativity P & (Q v R) ~ ... (P & Q) v (P & R) &Iv Distribution P v (Q & R) ~... (P v Q) & (P v R) v/& Distribution P - + (Q - + R) ~ ... P & Q - + R Imp/ Exportation P H Q, P ... Q Biconditional Ponens P H Q, Q ... P BP P H Q, P ... Q Biconditional Tollens P H Q, - Q ... - P BT P H Q ~ ... - Q H - P Bi' Ira DSposition P H - Q ~... - P H Q Bi Trans - ( PH Q) ~ ... P H - Q NegatedH - ( P H Q) ~... - P H Q NegH
Chapter!
Exercise 1.5.2 Give proofs for the following sequents using the primitive rules of proof.
S53* S54 S55* S56* S57 S58 S59 S60 S61
S62 S63 substitution instance
PH Q -It- (P & Q) V (- P& - Q) P - + Q & R, R V - Q - + S & T, T HUt - P - + U (- P V Q) & R, Q - + St- P - + (R - + S) Q & R, Q - + P V S, - (S & R) t- P P - + R & Q, S - + - R V - Q t- S & P - + T R & P, R - + (S V Q), - (Q & P) t- S P & Q, R & - S, Q - + (P - + T) , T- +(R- +SvW) t- w R - + - P, Q, Q - + (P V - g) t- S - + - R P - + Q, P - + R, P - + S, T - + ( U- + (- V - + - S , Q - + T, R - + ( W- + V), V - + - W, W t- - P P H - Q & S, P & (- T - + - g) t- - Q & T PVQHP & Qt - PHQ Definition . A SUBSTITUTION INSTANCE of a sequent is die result of uniformly replacing its sentence letters with wffs .
Comment. This definition states that each occurrence of a given sentence letter must be replaced with the
samewff throughoutthe sequent. Example. The sequent P v Q J- - P - + Q hasas a substitutioninstancethe sequent (R & S) v Q J- - (R & S) - + Q accordingto the substitutionpattern P/ (R & g); Q/Q.
Chapter}
Comment. The given substitution pattern shows that the sentence letter P was replaced throughout the original sequent by the wff (R & S) , and the sentence letter Q was replaced throughout by itself .
Exercise1.5.3 Identify each of the following with a sequentin . exercise1.5.1 andidentifythesubstitution pattern i* ii * iii * iv* v* vi * vii * viii * ix* x* derived
R - + S -1.- - S - + - R - P - + Q v R, Q v R - + S.- - P - + S (P & Q) v R -1.- R v (P & Q) (PvQ) & (- Rv - S) -I'- PvQ) & - R) v PvQ) & - S) R v S -1.- - - (R v S) (P v R) & S -1.- - (P v R - + - S) P v (Q v R) -1.- - P - + Q v R - (P & Q) .- R - + - (P & Q) - P & Q) v (R & S -1.- - (P & Q) & - (R & S) P v (R v S), P - + Q & R, R v S - + Q & R .- Q & R rule
thatonehasprovedusingonly . Anysequent Comment be usedas a the primitiverulesmay subsequently DERIVEDRULEof proofif in the proof are the (i ) somesentences appearing premisesof the sequent,or (ii ) some sentencesappearing in the proof are the premisesof a substitutioninstanceof the sequent. In case (i ) the conclusion of the sequent may be assertedon the current line; in case(ii ) the conclusion . of the substitutioninstancemay be asserted
Chapterl
Annotation :
Assumptionset:
The line numbers of the premises followed by Sf , where S# is the number from the book, or the name of the sequent (see comment below ) . The union of the assumption sets of the premises.
Comment. All of the sequentsin exercise1.5.1 (SII 552) are used so frequently as rulesof proof thatthey have the names we have indicated. (Indeed, in some systems of logic some of our derived rules are given as
primitive rules.)
. Examples (a) ProveR v 5 - + T, - T ~ - R. 1 ( 1) R v 5 - + T 2 (2) - T 1,2 (3) - (R v 5) 1,2 (4) - R & - 5 1,2 (5) - R
A A
1,2 M1T 30M 4 &E
(b) ProveP v R ~ ST ~ - S .- T ~ - (P v R). 1 A ( 1) P v R ~ S 2 A (2) T ~ - S 1 S ~ (P v R) 1Trans (3) 1,2 2,3 HS (4) T ~ (P v R)
Chapter1
Comment . Requiring that the sequentto be used as a derived rule hasbeenproved using only primitive rules is unnecessarilyresbictive. If the sequentsare provedin a sbict order and no later sequent in the series is used in the proof of an earlier sequent , then no logical errors can result. We suggestthe strongerresbiction only becauseit is good practice to construct proofs using only the primitive rules. Exercise 1.5.4
Prove the following using either primitive or derived from the previous exercises . If you like a challenge , prove them again using primitive rules only . rules
S64 S65 S66* S67* S68 S69 S70 S71 S72 S73* S74 S75 S76* S77 S78* S79* S80* S81*
- p - + PiP PH Q -i .- - P - + Q) - + - (Q - + P P H Q -i .- P v Q - + P & Q PH Q -i .- - (P v Q) v - (- P v - Q) P H Q -i .- - (P & Q) - + - (P v Q) PH Q -i .- - (- (P & Q) & - (- P & - Q P v Q - + R & - P , Q v R, - R .- C - PHQ , P- + R, - R .- - QHR - P H - Q) H R), S - + P & (Q & T), R v (P & S) .- S & K - + R & Q (P & Q) v (R v S) .- P & Q) v R) v S P & (~ & - R), P- +(- S - + T), - S - + (T H R v Q) .- S P & - Q - +-R, (- S - + - P) H - R .- R H Q & (P & - S) P v Q, (Q - + R) & (- P v S), Q & R - + T .- T v S P - + QVR, (~ & S) v (T- +-P), - (- R - + - P) '- - T & Q P v Q, P - + (R - + - S), (- R H T) - + - P .- S & T - + Q (PH ~ - + - R, (- P& S) v (Q& 1) , SvT - + R .- Q - + P - S v (S & R), (S - + R) - + P .- P Pv ( RVQ ) , (R- +S) & (Q- +1) , S v T - + P v Q, - P .- Q
ChapterI
S82* S83* S84* S85* S86
(P - + Q) - + R, S - + (- Q - + T) t- R v - T - + (S - + R) P & Q - + R v St - (P - + R) v (Q - + S) (P- +Q) & (R - + f ), (PvR) & - (Q&R) t- (P& Q) & - R P & Q - + (R v S) & - (R & S), R & Q - + S, S - + R & Q) v (- R & - Q v - P t- P - + - Q - (P& --Q)v- (- R& -s), - S& --Q, T- +(- S- +- R& p) t- - T
1.6
Theorem.
theorem
Definition. A THEOREM is a sentence that can be proved from the empty set of premises. Comment. We can assert that a given sentence is a theorem by presenting it as the conclusion of a sequent with nQthing to the left of the turnstile .
. Example Provet- P& Q -+ Q & P. 1 (1) P& Q 1 (2) Q 1 (3) P 1 (4) Q & P (5) P& Q -+ Q & P
A 1 &E 1 &E 2,3 &1 4 - +1( 1)
Comment . Note that in step 5 we discharge assumption 1. Hence, the final conclusion rests on no . assumptions
ChapterI
Exercise 1.6.1 Provethe following theorems, (i ) using primitive rules only and (ii ) using primitive rules together with derivedrules establishedin a previousexercise.
Tl * T2* T3 T4* T5* T6 17 T8* T9* TIO* Til * T12
*
T13* T14* T15 T16 T17* TIS T19* T20 T21* T22 T23 T24 T25 T26 T27*
t- P ~ P Identity ExcludedMiddle t- P v - P Non-Contradiction t- - (P & - P) t- P ~ (Q ~ P) Weakening t- (P ~ Q) v (Q ~ P) Paradoxof Material Implication Double Negation t- P H - - P t- (PHQ ) H (QHP ) t- - (PHQ ) H (- PHQ ) Peirce's Law t- P ~ Q) ~ P) ~ P t- (P ~ Q) v (Q ~ R) t- (PHQ ) H (- PH - Q) ~ (- P ~ Q) & (R ~ Q) H (P ~ R) ~ Q ~ ~ ~ ~ ~ ~ ~
& Idempotence PH P& P v Idempotence V PH P P (PHQ ) & (RHS ) ~ P ~ R) H (Q ~ S (PHQ ) & (RHS ) ~ (P & RHQ & S) (PHQ ) & (RHS ) ~ (Pv R H Q V S) (PHQ ) & (RH S) ~ PHR ) H (QH S (PHQ) ~ (R~ P) H (R~ Q & P~ R) H (Q ~ R
.- (PHQ ) ~ (R & PHR & Q) .- (PHQ ) ~ (R V P H Rv Q) .- (PHQ ) ~ RHP ) H (RHQ .- P & (Q H R) ~ (P & Q H R) .- P~ (Q ~ R) H P~ Q) ~ (P~ R .- P ~ (Q ~ R) H Q ~ (P ~ R) '- P~ (P~ Q) HP ~ Q .- (P~ Q) ~ QH (Q ~ P) ~ P
1 Chapter
T28 T29 T30* T31* T32
*
T33 T34 T35 T36 T37 T38 T39
theorems as derived rules
~ P~ - QHQ ~ - P ~ - P~ PHP ~ (P& Q) v (R & S) H P v R) & (P v S & Q v R) & (Q v S ~ (P v Q) & (R v S) H P & R) v (P & S v Q & R) v (Q & S ~ (P~ Q) & (R ~ S) H - P & - R) v (- P & S v Q & - R) v (Q & S ~ (PV - P) & QHQ ~ (P& - P) VQHQ ~ P v (- P & Q) H P v Q ~ P& (- PVQ) HP & Q ~ PHPV (P& Q) ~ PHP & (PVQ ) ~ (P ~ Q & R) ~ (P & Q H P & R)
Comment. We now consider a special case of the use of sequents as derived rules. Since it is the conclusion of a sequent without premises, a theorem or a substitution instance of a theorem can be written as a line of a proof with an empty assumption set. For a theorem to be used this way , it must have been proved already by means of primitive rules alone. The annotation should be the name of the theorem or T# (the theorem' s number) .
Chapterl
. Example ProveP ~ Q, - p ~ Q .- Q. 1 ( 1) P ~ Q 2 (2) - p ~ Q (3) P v - p 1,2 (4) Q
A A
T2 1,2,3 SimDil
. In the precedingexample, the annotationfor Comment line 3 gives the number of the theorem introduced. Since this theorem has a name, the annotation ' ExcludedMiddle' would alsohavebeen . acceptable Comment. As with sequentintroductions. requiring that dleorems first
be proved using only
primitive
rules is unnecessarily
restrictive .
Exercise1.6.2 Using theoremsas derivedrules, attemptto construct alternativeproofs of sequentsappearingin exercise 1.5.4.
This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] Chapter2 Truth Tables 2.1 truth value
Definition. Truth and Falsity (abbreviatedT and F) are TRUm VALUES.
truth table
Comment . When an argumentis valid, its conclusion cannotbe false when its premisesare all true. One way to discoverwhetheran argumentis valid is to consider explicitly all the possible combinationsof truth values amongthe premisesand the conclusion. In this chapter we show how to do this. The idea is to assign truth valuesvariously to the sentencelettersof the argument and seehow the premisesand the conclusionturn out. The following rules, codified in TRUTH TABLES (17s ), enableus to do this. Comment. For this method to work , it has to be the case that the truth values of compound sentences are determined by the truth values of the sentence letters that appear in them.
truth functional connectives
Comment. All the sententialconnectives introducedin chapter I have the property described in the previous comment. Sincethe truth values of compoundsentencescontaining theseconnectives arefunctions of the truth values of the componentwffs, they are known as TRUTH -FUNCTIONAL CONNE Cfl VES. ( Not all English connectivesaretruth-functional.)
2 Chapter TT for negation
In order for a negation - , to be true , , must be false .
>
I
>
51
~. c~: 5I1Q j ~ >Q: ~ >51 -. ~ . : > I (
Answersto Chapter1 Exercises
, 1 Exercises Answersto Chapter
144 -R -+P -R -+Q
~~~ ~IU 0~tu5It 0: g t c t ~ .:
4v ~ 2,5 as 6v ~ 7 ~ I (4) A 9v ~ 3, 10 as 11 v ~ 12 ~ I (9) 8, 13 HI 14 ~ I ( 1)
RvQ RvP - + RvQ RvQ - R -+ Q - R -+ P RvP RvQ -+RvP R V P H Rv Q (PH Q) -+ ( RV PH R V Q)
c c
(5) (6) (7) (8) (9) (10) (11) (12) (13) ( 14) ( 15)
.
4 1,4 1,4 1 9 9 1,9 1,9 1 1
A [for -+1] A [for - +1] A [for RAA] A 2,4 - +E 3,5 RAA (4) A 1,7 - +E 6,8 RAA (7)
A tOyI 3.11vE 12-+1( 10) 9.13RAA(3) 14-+1(2) IS-+1( 1) A [for-+1] A [for-+1] A [forRAA] A 18.20-+E 19.21RAA(20) A 17.23-+E
Answersto ChapterI
Exercises
17,18,19(25) - (Q ~ P) 26 (26) Q 26 (27) Qv P 19,26 (28) P 19 29 ( ) Q~ P 17,18 (30) Q 17 31 ( ) (P~ Q) ~ Q 32 ( ) Q ~ P) ~ P) ~ P ~ Q) ~ Q) (33) (P~ Q) ~ QH (Q ~ P) ~ P
145
22.24RAA(23)
A 26vI 19,27vE 28-+1(26) 25,29RAA( 19) 30-+1(18) 31-+1(17) 16,32HI A 1Neg- + 2 &E 2 &E 4 Neg-+ 5 &E 5 &E 7FA
3,8 -+E 6,9 -+E 7,10RAA( 1) A 12Neg-+ 13&E 13&E 15Neg-+ 16&E 16&E 18FA 14,19-+E 17,20-+E 18,21RAA(12) 11,22HI
-~
Answersto Chapter 1 Exercises
J- (P& 0) v (R& S) H (Pv R) & (Pv S & Q v R) & (Q v S (i) primitiverulesonly A [for -+1] I (p& Q) v (R& S) (I ) A [for RAA] v R 2 2 (P ) () A [for RAA] v 3 (P S) (3) A 4 P 4 () A P& Q 5 (5) 5 &E P 5 (6) - (P& Q) 4,6 RAA (5) 4 (7) 1,7 vE R& S 1,4 (8) 8&E R 1,4 (9) 8&E S 1,4 (10) vI 9 Pv R 1,4 (11) 2 P 1,2 , 11RAA (4) ( 12) vI 12 Pv R 1,2 ( 13) 2, 13RAA (2) Pv R I (14) 10vI Pv S 1,4 ( 15) 3,15RAA (4) P 1,3 ( 16) 16v I Pv S 1,3 (17) 3 PvS I 18 ,17RAA (3) ( ) 14 v v R & S I 19 ,18&1 (P ) (P ) ( ) A 20 [for RAA] (Q v R) (20) A [for RAA] 21 (Q v S) (21) -Q A 22 (22) 5 &E 23 5 ( ) Q - (P& Q) 22,23RAA (5) 22 (24) 1,24 v E R & S 25 1,22 ( ) 25&E R 1,22 (26) 25&E S 1,22 (27) 26vI v R 1,22 28 ( ) Q 20,28RAA(22) 1,20 (29) Q 28vI v R 30 1,20 ( ) Q 20,30RAA(20) v R I 31 ( ) Q 27vI v 1,22 (32) Q S 21,32RAA(22) 1,21 33 ( ) Q 33vI 1,21 (34) Qv S 21,34RAA(21) I (35) Qv S 30,35&I I (Q v R) & (Q v S) (36)
V
(
(
' SA
~
mleg
1S~ ~ 1S1S
( (
---
I I I p I ~
If
H +
[
(
' S
S (
~ 3 ~ 3 3 3
( (
[
d
V V
' S
I
S
I
( 3A 3AI
d l
l
(
l
I
( 3A 3AI
+
IA
(
( (
( (
(
I (
' s ' SAd
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
Ot
Ltl
ChapterI Exercises
-
148
15 15 15 15 15 15 15 15 15 15 15 15 15
Answersto Chapter I Exercises
(7) (8) (9) (10) ( 11) ( 12) ( 13) ( 14)
(IS) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29)
PvR Pv S (Pv R) & (Pv S) Qv R Qv S (Q v R) & (Q v S) P v R) & (Pv S & Q v R) & (Q v S (P& Q) v (R& S) -+ P v R) & (Pv S & Q v R) & (Q v S P v R) & (Pv S & Q v R) & (Q v S P ( v R) & (Pv S) (Q v R) & (Q v S) PvR PvS QvR QvS (Pv R) & (Q v R) (Pv S) & (Q v S) (P& Q) v R (P& Q) v S P & Q) v R) & P & Q) v S) (P& Q) v (R& S) P v R) & (Pv S & Q v R) & (Q v S -+
(P& Q) v (R& S) (P& Q) v (R& S) H
5&E 6&E 7,8 &1 5&E 6&E 10,11&1 9,12&1 13-+1(1) A 15&E 15&E 16&E 16&E 17&E 17&E 18,20&1 19,21&1 22Dist 23Dist 24,25&1 26Dist 27-+1( 15) 14,28HI
P v R) & (Pv S & Q v R) & (Q v S T31
J- (Pv Q) & (Rv S) H P & R) v (P& S v Q & R) v (Q & S
(i) primitiverulesonly 1 A [for -+1] (I ) (Pv Q) & (R v S) - (P& R) v (P& S v Q& R) v (Q& S ) A [for RAA] 2 (2) 3 P& R A (3) 3 3 vI (4) (P& R) v (P& S) 3 4vI (5) P & R) v (P& S v Q& R) v (Q & S - (P& R) 2 2,5 RAA (3) (6) 7 P& S A (7)
149
Answersto Chapter1 Exercises
tf' tfr
M M
(35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45)
Q& R) v (Q& S
Q & R) v (Q & S
7 vI 8 vI 2,9 RAA (7) A
'--
(P& R) v (P& S) P & R) v (P& S v - (P& S) Q& R (Q& R) v (Q & S) P & R) v (P& S v - (Q& R) Q& S (Q & R) v (Q & S) P & R) v (P& S v - (Q & S) PvQ RvS P R P& R -R S P& S -P Q Q& R -R S Q& S P & R) v (P& S v (Pv Q) & (R v S) -+ P & R) v (P& S v P & R) v (P& S v - (pvQ) (P& R) v (P& S) P& R P PvQ - (P& R) P& S P PvQ - P& R) v (P& S
-->-
(8) (9) (10) (II ) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34)
Q & R) v (Q & S I &E l &E A
Q& R) v (Q & S Q& R) v (Q& S Q & R) v (Q& S
A 21,22&1 6,23RAA(22) 20,24vE 21,25&1 10,26RAA(21) 19,27vE 22,28&1 14,29RAA(22) 20,30vE 28,31&1 18,32RAA(2) 33-+1(1)
cS ~~ .~ ~~-
35 36 37 38 38 38 36
-
-
7 7 2 II II II 2 15 15 15 2
A for - +1
150
f. ~ ~
Q & R) v (Q & S Q& R Q PvQ - (Q & R) Q& S Q PvQ PvQ - (Rv S) (Q & R) v (Q & S) Q& R R
~-
>~ ~r--
36,53RAA (36) A [for RAA] A A 57&E 58 vI 55,58RAA (57) 56,00 v E 61&E 62 v I 55,63RAA (56) 35,64 v E A 66&E 67 vI 55,68 RAA (66) 65,69 v E 70&E RvS 71 vI RvS 55,72 RAA (55) v v & S 54,73&1 (P Q) (R ) P & R) v (P& S v Q & R) v (Q & S -+ 74--+1(35)
~>r
(46) (47) (48) (49) (50)
-~--
35,36 47 47 47 36 35,36 35,36 35,36 35 55 56 57 57 57 55 55,56 55,56 55,56 55 35,55 66 66 66 55 35,55 35,55 35,55 35 35
Exercises
~ -
Answers to Chapter
(Pv Q) & (Rv S) (Pv Q) & (Rv S) H P & R) v (P& S v Q& R) v (Q & S (ii ) derivedrulesallowed 1 ( 1) (Pv Q) & (R v S) 1 (2) (P& ( Rv S v (Q & ( Rv S 3 P & ( Rv S) (3)
34,7SHI
A 1 Dist A
151
19 19
t~>0-~~>(-
PvQ (Pv Q) & (Rv S) (Q& R) v (Q & S) -+ (Pv Q) & (Rv S) (Pv Q) & (Rv S) P & R) v (P& S v Q & R) v (Q& S -+
11
(28) T32
(P& R) v (P& S) P& (Rv S) -+ (P& R) v (P& S) Q& (Rv S) (Q& R) v (Q & S) Q & (Rv S) -+ (Q& R) v (Q & S) P & R) v (P& S v Q & R) v (Q& S (Pv Q) & (Rv S) -+ P & R) v (P& S v Q & R) v (Q & S P & R) v (P& S v Q & R) v (Q & S
~~~-~~(-
-0
11 12 12 12 12 12 12
(4) (5) (6) (7) (8) (9) ( 10)
----
\
Answersto Chapter1 Exercises
(Pv Q) & (Rv S) (Pv Q) & (Rv S) H P & R) v (P& S v Q& R) v (Q& S
3 Dist 4 -+1(3) A 6 Dist 7 -+1(6)
A A 12Dist 13&E 13&E 14vl 15.16&I 17-+1( 12) A 19Dist 20&E 20&E
21vI 22.23&I 24-+1(19) 11.18.25SimDil 26-+1(11) 10.27HI
.- (P-+Q) & (R-+ S) H - P& - R) v (- P& S v Q & - R) v (Q & S
(i) primitiverulesonly 1 (1) (p-+Q) & (R-+ S) - (- P& - R) v(- P& S v Q& - R) v(Q & S 2 (2) 1 P-+Q (3) 1 4 R -+ S () 5 5 () (- P& - R) v (- P& S)
A [for-+1] ) A [forRAA] 1&E 1&E A
-
Answersto Chapter I Exercises
- P& - R) v (- P& S v Q & - R) v (Q& S - - P& - R) v (- P& S (Q& - R) v (Q& S) - P& - R) v (- P& S v Q & - R) v (Q & S - Q& - R) v (Q & S (- P& - R) (- P& - R) v (- P& S) - (- P& - R) - P& S (- P& - R) v (- P& S) - (- P& S) Q& - R (Q& - R) v (Q& S) - (Q & - R) Q& S (Q & - R) v (Q & S) - (Q& S) -P
-r
~
5 2 8 8 2 11 11 2 14 14 2 17
-~-
152
& -R
37 38 39 40 40 39 43
R S - P& S P Q Q& - R R S Q& S - P& - R) v (- P& S v Q& - R) v (Q & S (P-+Q) & (R-+ S) -+ - P& - R) v (- P& S v Q & - R) v (Q& S - P& - R) v (- P& S v Q & - R) v (Q & S P -Q Q& - R Q - (Q & - R) Q& S
5 vI 2,6 RAA(5) A 8 vI 2,9 RAA(8) A 11vI 7,12RAA(11) A 14vI 7,15RAA(14) A 17vI 10,18RAA(17) A 20vI 10,21RAA(20) ARAP 23,24&1 13,25RAA(24) 4,26-+E 23,27&1 16,28RAA(23) 3,29-+E 24,30&1 19,31RAA(24) 4,32-+E 30,33&1 22,34RAA(2) 35-+1( 1) A [for-+1] A [for-+1]
A [for RAA] A 4O&E 39,41 RAA (40) A
153
Answersto Chapter1 Exercises
43 (44) 39 (45) 46 (46) 39,46 (47) 39 (48) 37,39 (49) 50 (50) 50 (51) 38 (52) 37,38,39(53) 37,38,39(54) 37,38 (55) 37 (56) 57 (57) 58 (58) 59 (59) 59 (60) 58 (61) 62 (62) 58,62 (63) 58,62 (64) 57,58 (65) 37,57,58(66) 67 (67) 67 (68) 57 (69) 37,57,58(70) 37,57,58(71) 37,57 (72) 37 (73) 37 (74) (75) (76)
43 &E
39,44RAA(43) A 42,46 v B
45,47 RAA (46) 37,48 v E A 50&E 38,51 RAA (50) 49,52 v E 53&E 38,54RAA (39) 55 ~ I (38) A [for ~ I] A [for RAA] A 59&E 58,60RAA (59) A 61,62 v E 63&E 57,64RAA (62) (- P & - R) v (- P & S) - P& - R -R - (- P & - R) - P& S S S R -+ S (P-+ Q) & ( R-+ S)
37,65vE A 67&E
57,68RAA(67) 66,69vE 70&E 58,71RAA(58) 72-+1(57) 56,73&1 - P& - R) v (- P& S v Q& - R) v (Q & S -+ 74-+1(37) (P-+ Q) & (R-+ S) 36,7SHI (P-+Q) & (R-+ S) H - P& - R) V (- P& S V Q& - R) V (Q& S
(ii ) derivedrulesallowed 1 (p - + Q) & ( R-+ S) ( 1) P -+ Q 1 (2)
A l &E
154
~ 16 16 16 16 16 16 16
(31)
0 -_ ~-
~
-.~Ai> f -
~- - 1~ > f ~
~ f-
~ f --
i
- ~~
0~
( 16) ( 17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30)
f
16 17 17
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
~i
- -
1 1 1 1 1 8 8
Answersto Chapter Exercises
15,30HI (P-+ Q) & (R-+ S) +-+ - P& - R) v (- P& S v Q & - R) v (Q & S
155
Answers to Chapter 2 Exercises
Chapter 2
~O
Exerdse2.1
PQ
PQ TT TF FT FF
""
QR TT TF FT FF TT TF FT FF
AO ~
>
iii PQ TT TF FT FF
Iu! t Q-t -A
Pv (- PvQ ) TF T TF F TT T TT T
(P& Q H Q) -+ (Q -+P) T T T T F T T T F F T F F T T T
ii - {P& Q) v P TTFT T TFTFT FTTFT FFTFT
iv (pv Q) v (- P& Q) T TF F T TF F T TT T F FT F vi RH - Pv (R& Q) TFT T TF F F FFF F TF F F TTT T FTT F TTT F FTT F viii PQ TT TF FT FF
(PH - Q) H (- P H - Q) FF F F TF TT FF FT TF FT FF FT FT TT
Answersto Chapter2 Exercises
156
ix PQR TTT TTF TFT TFF FTT FTF FFT FFF
(p +-+Q) +-+(pv R-+ (- Q-+ R T T T T F T T T T F T T F T T T T F T T FTF F F F T TFT F F F T F T T T T T T T T T F T T F
x P Q R S (P& Q) v ( R& S) -+ (P& R) v (Q & S) TT T T T T T T T T T TT T F T T F T T T F TTFT T T F T F T T TTFF T T F F F F F TF T T F T T T T T F TF T F F F F T T T F TF F T F F F T F F F TF F F F F F T F F F FT T T F T T T F T T FT T F F F F T F F F FTFT F F F T F T T FTFF F F F T F F F FF T T F T T F F F F FF T F F F F T F F F FF F T F F F T F F F FF F F F F F T F F F Exerdse2.2
A [forRAA] l &E 2,3 BP l &E 4,5 RAA(2)
-
' c>0
..
(~
~ tI
0~EtI a01 {~ 0I~~an. ~>E~ ac~ ~~
NN
~-
d d (
d
~ ~~tI ----fMMOI -01
.>
-
.cS. t ~
~EQ>0~0 tI~ OIWA 6~~tI 0I~E0I iE
M ~~ .(la
Q
~f
-
-Ir
S
01
Chapter
2 Exercises Answersto Chapter
o0
1
~:~~ t~>Ia -
5 5
P-+ (- P-+ R) Q - Q -+R Q -+ (- Q -+R) (- P-+ R) v (- Q -+R)
3 -+1(2) A [for-+1] SPA 6 -+1(5) 1,4,7 CornOil
~cS .~ -
-~-
158
VALID (1) (2) (3) R& - P (4) 5 R () 6 () Q (7) QvR - (Q v R) (8) (9) (QH R) xiv 1 2 3 4 1,2 1,2,4 1,2,3
VALID (1) (2) (3) (4) (5) (6) (7)
xv
INVALID P:F Q:F
P-+ (Q & R -+ S) P -S Q& R Q & R -+ S S - (Q& R)
R:T
A A A A [for RAA] 1,2 -+E 4,5 - +E 2,6 RAA (4)
159
,
~'
Answersto Chapter2 Exercises
INVALffi
P:T
Q:F
R:F
S:F
T:F
VALID
(1) (2) (3) (4) (5) (6) (7) (8)
4 4 3,4 8 8 (9) 2,8 (10) 1,2,8 (II ) 1,2,3,4,8 (12) 1,2,3,4,8 (13) 1,2,3,4 (14) 1,2,3 ( 15) 1,2,3 (16) xviii
Q -+ (P- +R & - Q) - Q -+ - (TvV ) U & SHP - (S -+ - V) S& U U& S P T TvV Q P-+R & - Q
A A A
A [for -+1] 4 Neg- + 5 Comm 3.6 BP A [for RAA] 8 vI 2.9 M1T 1.10-+E 7.11-+E 12&E 10.13RAA (8) 14-+1(4) 15v- +
~~I0
1 2 3 4
- (5 -+ - U) -+ - T (5 -+ - U) v - T
INVALID Q:F
R:F
S:F
T:F
i ii iii iv v vi vii viii ix x xi xii xiii 1
P:F PiT P:T P:F PiT P:F PiT P:F PiT P:T PiT P:F VALID ( 1)
Q:T Q:F Q:T Q:F Q:F Q:F Q:F Q:F Q:T Q:F Q:F Q:T
R:F RiT R:T R:F RiT R:T R:F RiT R:F R:F
ri5ri5 f
Exerdse 2.4.2
P - + (- Q - + - R & - 8 )
T:T T:F T:T TiT
U:F
V:T
W:F
Answers to Chapter2 Exercises
xiv xv xvi xvii I
2 1 1 1 1 1,2 1,2 1,2 1,2 -
xviii
1,2 1,2 1,2 1,2 1
-
I~ : CI
2 3 4 1,4 1,3,4 1,3,4 1,3,4 1,3,4 1,3,4 1,3,4 1,2,3
-Ntf -
160
A A A 1,4 -+E 3,5 -+E 6 &E 6 &E 7FA 8FA 9, 10HI
- Q -+ - R & - S -R& -S -R -S R -+S S-+R RHS -p
P:F PiT PiT VALID (1) (2) (3) (4) (5) (6) (7) (8) (9) ( 10) VALW (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Q:T Q:F Q:F
RiT RiT RiT
2,11RAA(4)
S:F S:F S:F
T:F T:F
U:F
- (p-+ - Q & R) - RH - P P& - (- Q& R) P - (- Q & R) - - Qv - R R --Q Q P& Q
1Neg-+ 3&E 3&E 50M 2,4 BT 6,7 vE SON 4,9 &1
(p -+ Q) & (- Q - + P & R) -+ (5 v T - + - Q) Q P- + Q - Q -+ P& R (p -+ Q) & (- Q - + P & R) 5vT -+ - Q - (S v T) - S& - T - (- S -+ T) Q -+ - (- S -+ T)
A A 2TC 2FA 3,4 &1 1,5 -+E 2,6 MTr 7DM 8Neg-+ 9 -+1(2)
A A
Answersto Chapter3 Exercises
8 9 8,9 8,9 8
xx 1 2 1,2 1 1 1 1
P v - Q) -+ (- P & - Q -+ - P -P Pv - Q -Q - P& - Q (Pv - Q) -+ (- P & - Q) - P -+ P v - Q) -+ (- P & - Q P v - Q) -+ (- P & - Q +-+- P
VALill (I ) (2) (3) (4) (5) (6) (7)
QH - Q Q -Q -Q Q QV (PH - P) PH - P
Pv - Q -+-P& -Q P Pv - Q -P& -Q -P -P
A [for-+1] A [forRAA] 2vl 2,3 -+E 4&E 2,5 RAA(2) 6 -+1(1)
4,6 v B
Chapter 3
.
. .
.
Exerdse 3.1.1 Not a wff Not a wff Universal Existential Not a wff Not a wff Universal Not a wff ( but acceptable biconditional abbreviation given the dropping convention )
-
I 2 2 1,2 1,2 I
VALill (I ) (2) (3) (4) (5) (6) (7) (8) (9) (10) (II ) (12) ( 13) ( 14)
-c2c2 .tt~ ~
xix
161
162 ix x Xl xii xiii xiv xv xvi xvii xviii xix xx xxi xxii xxiii xxiv xxx
Answersto Chapter3 Exercises Negation Not a wff (but acceptableconditional abbreviationgiven the parenthesis droppingconvention) Not a wff (but acceptableconditional abbreviation given the parenthesis convention ) dropping Negation Not a wff Existential Not a wff Atomic sentence Not a wff Not a wff Not a wff Universal Not a wff Not a wff Negation Biconditional Not a wff
Exercise 3.1.2 Fonnula Open i Fz ii None Gcax ill iv
v vi vii
Gxy Gyx (Gxy& Gyx) 'v'y(Gxy& Gyx) Gxy Hy Ax Fxx Fxy Hxyz Jz (Hxyz& Jz) 'v'z(Hxyz& Jz) ( Fxy- + 'v'z( Hxyz& Jz 'v'y( Fxy-+ 'v'z( Hxyz& Jz
ExampleWFF 3zFz
3y- VxGxy - Vy- Hy VxAx 3xFxx
Answers to Chapter,3 Exercises
viii
xi xii xiii xiv xv
163
Fxx Fxy
'v'yFxy Hz Ix ( Hzv Ix ) 3z( Hzv Ix ) - 3z( Hz v Ix ) Fxx (Hav Fxx) - (Hav Fxx) None Fx Fx Fyyy (Fyyy& P) Fzx Hxyz (FzyH Hxyz)
'v'xFx
Exercise 3.1. Translationschemeis providedonly whereit is not obvious. alt: indicatesan alternative, logically equivalentttanslation. . amb: indicatesnon- equivalentrenderingof an ambiguoussentence inc ! indicatesa common, but incorrectanswer.
I inc! 2 inc! 3 all: inc!
alt: inc!
Vx( Dx-+ Mx) Vx( Dx& Mx) 3x(Sx& Ox) 3x(Sx- + Ox) Vx(Fx-+ - Ex) - 3x(Fx& Ex) - Vx(Fx-+ Ex) - Vx(Fx-+ Px) 3x(Fx& - Px) 'v'x( Rx - + - Ex) & 'v'x(Ax - + - Ex) 'v'x( Rx v Ax - + - Ex) 'v'x( Rx & Ax - + - Ex)
164
inc! inc!
7 inc! 8 alt: 9 alt: 10 alt: 11 12 alt: inc! 13 14 15 alt: 16 17 18 alt: 19 arnb: 20 21 22 23-29
Answersto Chapter3 Exercises
3x(Px& Ax) & 3x(Rx & Ax) 3x Px& Rx) & Ax) 3x Pxv Rx) & Ax) 'v'x(Ox -+ Lx) 'v'x(Lx -+ Ox) 'v'x(Sx- + (Px -+ Tx v Bx 'v'x(Sx& Px -+ Tx v Bx) 'v'x(Mx H Px) 'v'x Mx -+ Px) & (Px -+ Mx 'v'x(Fx- + - Wx vEx ) 'v'x(Fx& Wx -+ Ex) (3x(Ox & Cx) & 3x(Ox & Mx & - 3x(Cx & Mx ) 'V'x(lx - + Px) 'V'x(- Px - + - Ix ) 'v'x(Px - + Ix ) 'v'x(Ax - + (- Wx - + Nx 'v'x(Ax - + (Nx - + - Wx
3x(Sx& (Px& Fx & - Vx(Px& Fx-+ Sx) 3x(Sx& (Px& Fx & 3x Px& Fx) & - Sx)
Wa& 'v'x( Wx-+ Mx) -+ Ma 'v'x(Sx-+ (- Nx v Mx 'v'x(Ox & Ex -+ - Px) - 3x Ox & Ex) & Px) 'v'x(Px -+ - Hx) - 'v'x(Px - + Hx) [possiblereadingin someregional dialects ofEnglish ] - 'v'x(Px - + Cx) 'v'x(Mx & Wx -+ Bx) 3x Mx & Wx) & Fx) TranslationschemeI using single-placepredicatesonly : Ta : a is a trick Wa : a is a whale Sa: Shamucan do a Ca: a can do a trick s: Shamu Note: Strictly we shouldusea letter from a-d for a name, but the useof s for . Shamuis more perspicuous
165
Answersto Chapter3 Exercises
'v'x(Tx-+ Sx) 23 'v'x(Tx-+ Sx) 24 - 'v'x(Tx -+ Sx) 25 'v'x(Tx-+ - Sx) 26 : - Cs thatis notlogicallyequivalent alternative translation 3x(Wx & Cx) -+ Cs 27 . Notedifferencein scope 'v'x( Wx& Cx -+ Cs) . Lessnaturalreading ambi: 'v'x( Wx-+ Cx) -+ Cs ambii: If anywhalecandoa trick, Shamucandothatsametrick. Thisreadingis notexpressible only. usingsingle-placepredicates Notescopeagain. 'v'x( Wx-+ Cx) -+ Cs 28 3x( Wx& Cx) -+ 'v'x( Wx-+ Cx) 29 . Thisreadingis lessnatural amb 'v'x( Wx-+ Cx) -+ 'v'x( Wx-+ Cx) 23-29
TranslationSchemen lI~ing many a is a hick Ta : : a can do p Cap a is a whale Wa : s: Shamu
23 24 25
-
place predicates
'v'x( Tx-+ Csx) 'v'x( Tx-+ Csx) - 'v'x(Tx-+ Csx) 3x(Tx & - Csx) alt: 'v'x( Tx-+ - Csx) 26 27 3xy Wx & Ty) & Cxy) - + 3z(Tz& Csz) ! 'v'xy ( Wx& Ty) & Cxy) - + 3z(Tz& Csz alt: Scope amb-i: 'v'x3y(Wx -+ Ty & Cxy) -+ 3z(Tz & Csz) amb-ii: 'v'xy ( Wx&Ty) & Cxy) -+ Csy) . scheme with thepreviousb' anslation Thisis theambiguous readingnotexpressible 'v'x3y( Wx-+ (Ty & Cxy -+ 3z(Tz& Csz) 28 'v'x( Wx-+ 3y(Ty & Cxy -+ 3z(Tz & Csz) alt: 29 3xy Wx & Ty) & Cxy) - + 'v'x( Wx-+ 3y(Ty & Cxy amb: 'v'x3y( Wx-+ ( Ty& Cxy -+ 'v'x( Wx-+ 3y(Ty& Cxy
30 31
Agb 3xAxb
166 32 33 34 35 36 amb: 37 amb: 38 39 40 41 alt: 42 alt:
Answersto Chapter3 Exercises
3xAgx VxAbx VxAxb 3xyAxy 3xVyAxy Vy3xAxy Vx3yAxy 3yVxAxy VxyAxy VxAxx 3xAxx Vx- Axx - 3xAxx 3xVy- Axy 3x- 3yAxy
Translation Scheme for 43-46. SaJiy; a saidIi to 'Y fa : a is a person 43 amb: 44 45 46
Vx(Px-+ 3yVz(Pz-+ Sxyz Vxy(Px& Py -+ 3zSxzy Vx(Px-+ 3yz(Pz& Sxyz Vx(Px-+ 3y(Py& - 3zSxzy Vxy(Px& Py-+ - 3zSxzy)
47 48 alt: amb: 49 50 amb: 51 52 amb: 53
3xyz Rx & (Cy & Sxy & ( Dz& Lxz 3x(Fx& Vy(Hy -+ Sxy 3xVy(Fx& (Hy -+ Sxy 3x(Fx& 3y( Hy& Sxy 3x(Fx& Vy( My-+ Sxy 3x( Wx& Vy(Fy& Exy -+ My 3x( Wx& Vy(Exy-+ Fy & My 3x( Wx& Vy(Fy& My -+ - Exy 3xy My & Fy) & Exy) -+ Vx(Sx- + 3y My & Fy) & Exy 3xy My & Fy) & Exy) -+ 3x(Sx& 3y My & Fy) & Exy ' Vwxyz Jw & Txw) & (Oy & Tzy) - + Lxz) [ Tali: a is Ii s tail]
Answers to Chapter 3 Exercises -- alt :
54 alt: 55 56
167
'v'wxyz (Jw & Tx) & Bxw) & Oy & Tz) & Bzy) -+ Lxz) (Bali: a belongs to Ii] 'v'x(3y(Cy & Sxy) -+ Ax) 'v'xy(Cy & Sxy-+ Ax) 'v'uvxy( Bu & Pvu - + (Hx & Pyx - + Mvy & 'v'uvwx(Ou & Evu - + Mw v Bw) & Exw - + Avx & - Mvx Ambiguous. i. The amounteatenby somewhalesis more than the amounteatenby any fish. Translationscheme: a is an amount(of food) Au : a is a fish Fa: : a eatsfi amount(of food) Eafi : a is greaterthan fi Gafi 3x( Wx& 3y Ay & Exy) & 'v'zw( Fz& Aw & Ezw - + Gyw ) ii . The amounteatenby somewhales is more than the amounteatenby all the fishescombined. Addition to translationscheme: the amounteatenby all the fishescombined a:
3xy Wx & Ay) & ( Exy& Gya 3x( Mx& 'v'y( My-+ (GxyH - Gyy ) ( Usingidentity) 3x( Cx & 'v'y(Cy -+ y =x 58 3x x=p 59 alt : 3x( x=p & 'v'y ( y=p - + y=x 60 3xy (Tx& Ty) & x * y) & (Ebx& Eby 'v'x( x~ -+ Exb) & - Ebb 61 'v'x( Dx -+ 3y Ty & Byx) & 'v'z(Tz& Bzx-+ y=z 62 Exerdse 3.3.1 i is an instanceof v ii is an instanceof vi ii is an instanceix
Answers to Chapter3 Exercises
168 iv is an instanceof i iv is an instanceof ill x is an instanceof ii Exerdse 3.3. 2
3x(Gx & - Fx), 'v'x(Gx -+ Hx) ~ 3x(Hx & - Fx) 3x(Gx & - Fx) ( 1) 'v'x(Gx -+ Hx) (2) 3 - + Ha Ga ( ) 4 Ga & - Fa ( ) Ga (5) Ha (6) - Fa (7) Ha& - Fa (8) 3x(Hx & - Fx) (9) 3x(Hx & - Fx) ( 10)
S88 1 2 2
3x(Gx& Fx). '9'x(Fx-+ - Hx) .- 3x- Hx 3x(Gx & Fx) (I ) 'v'x(Fx-+ - Hx) (2) Fa-+ - Ha (3) Ga& Fa (4) Ga (5) Fa (6) - Ha (7) 3x- Hx (8) 3x- Hx (9)
A A 2VE A [for 3E on I ] 4 &E 4 &E 3.6 ~ E 731 1.8 3E (4)
'v'x(Gx-+ - Fx), 'v'x(- Fx-+ - Hx) J- 'v'x(Gx-+ - Hx) 'v'x(Gx-+ - Fx) (1) 2 'v'x(- Fx-+ - Hx) () Ga-+ - Fa (3) - Fa-+ - Ha (4) Ga-+ - Ha (5) 'v'x(Gx-+ - Hx) (6)
I ' VE 2 ' VE 3,4 US S' v1
NN ~~
4 4 4
1 2 1 2 1,2 1,2
i :' ~~~i m ca ~.m ~g~.~-i
-N- mm ~~t ~~ ~
Answers to Chapter
'v'xGx - + 'v'xFx
'v'xFXH
Answersto Chapter. 3 Exercises
172 5
(
5
(
4
(
4
(
1
, 4
1
,
1
,
1
,
(
4
(
3
(
4
(
4
(
3
(
3
(
2
(
9
(
1
(
, 9
,
2
13
1
(
,
14
,
13
,
2
19
1
(
(
(
(
19
1
(
(
14
1
(
(
,
,
2
2
1
1
SI09 1 2
(
(
(
(
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
&
Ka
Ka
&
Ga
' V ' x
)
12
(
2
1
11
1
4,6 RAA (5) 7 vI 3,8 ~ E 9 Comm 10VI 2,11RAA (4)
(
(
Fx
H
&
' V'
-
t
)
yGy
)
Gx
)
Hx
' V ' x
(
-
H
Fx
v
' V ' xFx
' V '
' V' x
-
Fx
)
yGy
' V ' xFx
)
-
-
3x
)
-
Fx
Fa
)
-
3x
)
Kx
-
3x
)
Da
Ga
)
' V' x
1
1
)
10
(
v
Fa
)
9
Hx
Da
)
8
(
2
8107
1
7
-
3x
)
A 531
Da
)
6
(
4
-
5
Fx
Fa
)
' V ' xFx
)
-
3x
)
-
)
Fa
H
' V '
)
yGy
-
' V '
)
)
)
)
yGy
3xFx
Fa
' V'
yGy
-
)
3xFx
-
)
Fa
' V ' x
)
3xFx
Fa
)
)
yGy
' V'
)
)
' V ' yGy
-
)
Fx
Fa
)
-
Fx
' V ' xFx
' V ' xFx
-
v
+
' V ' x
' V' x
-
-
Fx
Fx
'v'x(Dx-+Fx) r 'v'z(Dz-+ ('v'y(Fy-+ Gy) -+Gz 'v'x(Dx-+Fx) (I ) Da (2)
A A [for-+1] A [forRAA] A 431 3,5 RAA(4) 6VI 2,7 RAA(3) A [for3Eon8] I ' VE 9,10BT 8,113E(9) A [forRAA] A [for3Eon13] 10,14BP 13,153E( 14) 12,16RAA(13) A [forRAA] 1831 17,19RAA(18) 20VI 21-+1(2) 22v-+
A A
173
Answersto Chapter3 Exercises
(3) (4) (5) (6) (7) (8) (9) ( 10)
SIll 1 2 3 3 5 5 5 3 2
11 12 1 1 1 1,12 1,12 1,12 1,11 1 1 1
'v'xFx I- - 3xOx+-+- (3x(Fx& Ox) & 'v'y(Oy - + Fy 'v'xFx - 3xOx 3x(Fx& Ox) & 'v'y(Oy -+ Fy) 3x(Fx& Ox) Fa& Oa Oa 3xOx 3xOx - (3x(Fx& Ox) & 'v'y(Oy -+ Fy - 3xOx-+ - (3x(Fx& Ox) & 'v'y(Oy -+ Fy 3xOx Oa Fa Oa- + Fa 'v'x(Ox -+ Fx) Fa& Oa 3x(Fx& Ox) 3x(Fx& Ox) & 'v'x(Ox -+ Fx) 3x(Fx & Ox) & 'v'x(Ox -+ Fx) 3xOx-+ 3x(Fx& Ox) & 'v'x(Ox -+ Fx) - (3x(Fx& Ox) & 'v'x(Ox - + Fx -+ - 3xOx - 3xOx+-+- (3x(Fx& Ox) & 'v'y(Oy - + Fy
A A [for-+1] A [forRAA] 3&E A [for3Eon4] 5&E 631 4,7 3E(5) 2,8 RAA(3) 9 -+1(2) A [for-+1] A [for3Eon11] 1'v'E 13TC 14VI 12,13&1 1631 15,17&1 11,183E(12) 19-+1(11) 20Trans 10,21HI
Sl 12 1 2 2
'v'x(3yFyx-+ 'v'zFxz) ~ 'v'yx( Fyx- + Fxy ) 'v'x( 3yFyx-+ 'v'zFxz) ( 1) Fab (2) (3) 3yFyb
A A [for-+1] 231
'v'y( Fy - + Gy) Da - + Fa Fa - + Ga Da - + Ga Ga 'v'y( Fy - + Gy) - + Ga Da - + ('v'y( Fy - + Gy) - + Ga) 'v'z(Dz - + ('v'y( Fy - + Gy) - + Gz
-
--
3 1 3 1,3 1,2,3 1,2 1 1
A IVE 3 ' v' E 4,5 HS 2,6 -+E 7 -+1(3) 8 -+1(2) 9 VI
(
I 9 '
-N-
-
(
~J~~
-
-
---
1j 1rI1rI
rn- -\
J 3AI
--- ~~g>j Ii -
~ -
C
-~-
-
r ~
N
HAl
tIi( i( f ~f tUZ Zq
-
~ma frill !~ -cS .~~ N ~ ~ a~ -(~~
fI-~*fI-N~
}~~"~tm ~~
9 9 9 '
S
01
-~
Chapter
175
Answersto Chapter3 Exercises
S127*
I 2 3 3 3 6 3 3,6 3 3,6 2,3 1,2 * S130 1 2 3 4 1 6 2 2 2,6 4 2,4,6 2,4,6 2,4,6 2,4,6 1,2,4 1,2,3 1,2
3x Fx & 'v'y( Fy-+ y=x & Ox), - Oa'- - Fa 3x Fx & 'v'y(Fy-+ y=x & Ox) ( 1) - Oa 2 ( ) 3 ( ) ( fb & 'v'y(Fy-+ y=b & Ob Fb& 'v'y(Fy-+ y=b) (4) 'v'y(Fy-+ y=b) (5) Fa (6) Fa-+ a=b (7) a=b (8) Ob ,(9) Oa ( 10) - Fa ( 11) - Fa 12 ( )
(I)
' v' x3y Gyx
(2 )
' v' xy ( Gxy
(3 )
3y ' v' x ( x : #y - + Gyx )
(4 )
' v' x ( x ~
(5 )
3yGya
(6 )
Gba
(7 )
' v' y ( Gay
- + - Gyx )
- + Gax )
- + - Gya )
- + - Gba
Gab
SVE 6,7 -+E 3&E 8,9 =E 2,10RAA(6) I ,ll 3E( 3)
- + - Gyx ) r - 3y ' v' x ( x : #y - + Gyx )
' v' x3y Gyx , ' v' xy ( Gxy
(8)
A A A 3 &E 4 &E A
(9 )
- Gab
( 10 )
~
( II )
b=a
( 12 )
Gaa
( 13 )
- Gaa
( 14 )
-
3y ' v' x ( x : #y - + Gyx )
( IS )
-
3y ' v' x ( x : #y - + Gyx )
( 16 )
-
3y ' v' x ( x : #y - + Gyx )
( 17 )
-
3y ' v' x ( x : #y - + Gyx )
- + Gab
A A
A [for RAA] A [for 3E] I' VE A [for 3E] 2 ' VE 7 ' VE 6,8 ~ E 4 ' VE 9,10MTr 6,11=E 9,11=E 12,13RAA (3) 5,143E (6) 3,153E (4) 3,16RAA (3)
176
Answers to Chapter 3 Exercises
Exercise 3.4.1 .VxPx -1.- 3x - Px
~~ -l ~.m
5150 (a) 1 2 3 3 2 2
(b) 1 2 3 2 3 1
A A A 331 2,4 RAA (3) SVI 1,6 RAA (2)
A A A
2' VE 3,4 RAA(2) 1,5 3E(3)
'v'x(Px-+Q) -it- 3xPx-+ Q
(a) I 2 I 4 1,4 1,2 1
'v'x(Px-+Q) t- 3xPx-+ Q 'v'x(Px-+Q) 3xPx Pa-+ Q Pa Q Q 3xPx-+Q
3,4 -+E 2,5 3E(4) 6 -+1(2)
3xPx-+Q ~ 'v'x(Px-+Q) 3xPx-+Q (I ) Pa (2) 3xPx (3)
A A 231
(b) 1 2 2
---
SISS
A A I ' VE A
177
Answers to Chapter 3 Exercises
7 7 7
~ >; 0~. ~-~ 0~
5156 1 2 2 2 2
( 1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Q Pa- + Q 'v'x(Px-+ Q)
1.3 -+E 4 -+1(2) 5 VI
-
(4) (5) (6)
1,2 1 1
'v'xPx
A A
2VE 3 vI 4 VI 5 -+1(2) A 7 ' v' E 8 vi 9 ' v1
10-+1(7) 1,6,11SimDll
A A A 3 &E 3 &E 431 531 6.7 &1 2.8 3E (3) 1.9 3E (2)
(b) 1 1 1 4
3xPx& 3xQxI- 3xy(Px& Qy) 3xPx& 3xQx 3xPx 3xQx Pa
A l &E l &E A
---
(a) 1 2 3 3 3 3 3 3 2 1
3xy(Px& Qy) -i~ 3xPx& 3xQx 3xy(Px& Qy) ~ 3xPx& 3xQx (I ) 3xy(Px& Qy) 2 () 3y(pa& Qy) Pa& Qb 3 () Pa (4) (5) Qb 3xPx (6) 3xQx (7) 3xPx& 3xQx 8 () 3xPx& 3xQx (9) 3xPx& 3xQx ( 10)
5157
178
Answers to Chapter 3 Exercises
5 4,5 4,5 4,5 1,4 1
(5) (6) (7) (8) (9) ( 10)
SI60
P-+3xQx~J- 3x(P-+ Qx)
(a) 1 2 3 3 2 2 2 1,2 2 10 10 1,2 1
P-+3xQxJ- 3x(P-+Qx) P-+3xQx ( 1) - 3x(P-+ Qx) 2 () P -+ Qa (3) 3x(P-+ Qx) (4) - (p -+ Qa) (5) - Qa 6 P& () P (7) (8) 3xQx - Qa (9) ( 10) Qa ( 11) ( 12) ( 13)
(b) 1 2 3 2,3 2,3 3 1
3x(P-+ Qx) ~ P -+ 3xQx 3x(P-+ Qx) P P -+ Qa
---
mmm ~~t 666
Qb Pa& Qb 3y(pa& Qy) 3xy(Px& Qy) 3xy(Px& Qy) 3xy(Px& Qy)
Qa 3xQx p ~ 3xQx p ~ 3xQx
A 4,5 &1 631 731 3,8 3E(5) 2,9 3E(4)
A A
A 331 2,4 RAA (3) 5 Neg-+ 6 &E
1,7 -+E 6&E A 9,10RAA(2) 8,113E( 10) 2,12RAA(2)
A A A 2,3 - +E 431 S-+1(2) 1,6 3E (3)
Exercise 3.4. 2
T40 1 2
.- 'v'x( Fx - + Gx) - + ('v'xFx - + 'v'xGx)
(1) (2)
'v'x(Fx-+Ox) VxFx
A [for - +1] A [for - +1]
179
Answers to Chapter 3 Exercises
I 2 1,2 1,2 1
(3) (4) (5) (6) (7) (8)
T42 I 2 2 2 2 2 2 2 2 10 2,10 10 1
r 3x(Fxv Ox) H 3xFxv 3xOx 3x(Fxv Ox) - (3xFxv 3xOx) - 3xFx& - 3xOx - 3xFx - 3xOx 'v'x- Fx 'v'x- Ox - Fa - Oa Fav Oa Oa 3xFxv 3xOx 3xFxv 3xOx 3x(Fxv Ox) -+ (3xFxv 3xOx) 3xFxv 3xGx 3xFx Fa Fav Ga 3x(Fxv Gx) 3x(Fxv Gx) 3xFx-+ 3x(Fxv Gx)
---
-
Fa - + Ga Fa Ga VxGx 'V'xFx - + 'V'xGx 'v'x( Fx - + Gx) - + ('v'xFx - + 'v'xGx)
15 16 17 17 17 16 22 23 23 23 22 15
(22) (23) (24) (25) (26) (27) (28)
3xGx Ga Fav Ga 3x(Fxv Gx) 3x(Fxv Gx) 3xGx-+ 3x(Fxv Gx) 3x(Fxv Gx)
i 've 2VE 3,4 -+E S' v1 6 -+1(2) 7 -+1( I )
A [for-+1] A [forRAA] 20M 3 &E 3 &E 4QE 5QE 6VE 7VE A [for 3E on I ] 8,10vE 9,11RAA (2) 1,123E( 10) 13-+1( I ) A [for - +1] A A [for 3E on 16] 17v I 1831 16,193E ( 17) 20-+1( 16) A A [for 3E on 22]
23vI 2431
22,253E(23) 26-+1(22) Dll 15,21,27Sirn
180
Answers to Chapter 3 Exercises
(29) (30) T46 1 2 2 2 2 2 2 1,2 9 2 9 1,2 1
T59 1 2 2 2 5 1,5 1,5 1 9 1,9 1,2 1,2 1,2 1,2 1,2 1,2 1
3xFxv 3xOx-+ 3x(Fxv Ox) 3x(Fxv Ox) H 3xFxv 3xOx
J- (3xFx-+ 3xGx) -+ 3x(Fx-+ Gx) 3xFx-+ 3xGx ( 1) - 3x(Fx-+ Gx) (2) 'v'x- (Fx- + Gx) (3) (4) (5) (6) (7) (8) (9) ( 10) ( 11) ( 12) ( 13) ( 14)
( 1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) ( 13) ( 14) ( 15) (16) (17)
- 3x(FxH P) 'v'X- (FxH P) - (FaH P) P 'v'xFx Fa P -+ Fa Fa-+ P FaHP - (Fa-+ P) Fa& - P FaP 'v'xFx P 3x(FxH P)
28 -+1( 15) 14,29HI
A [for - +1] A [for RAA]
2QE 3'v'E 4 Neg~ 5&E 631 1,7 -+E A [for3Eon8] 5&E 9,10RAA(2) 8,113E(9) 2,12RAA(2) 13-+1( I )
A A
2QE 3 ' v' E A 1,5 SP 6 ' v' E 7 -+1(5) A 8,9 HI 4,10RAA (9) II Neg-+ 12&E 12&E 13VI 1, 15BP 14,16RAA (2)
(
(
(
(
~
:
UH
eH
(
8mqm : 8 q : 8H qH : (
eo
qO
8AqA :
:
eH
~
8AqA :
. d
8 q
d
d
80
8
qO ~
qHA: 8 q : 8HA
t
lelet
(
( (
(
(
t ~ r
(
(
181
Answers to Chapter 4 Rxerci ~ ~
-~NU
Answers to Chapter 4 Exercises
- (Fa& Ga& fb & Gb& Fc& Gc) - (Fa& - Ga) - Fa & - (Ga& Gb & (Fb& - (Ga& Gb )
- Fa & - (Oa&Ob&Gc & (Fb&--(Oa& Ob&Gc & (Fc& - (Oa&Gb&Gc ) - (OaH .Ha& - Fa) - (Oa& ObH (Ha& - Fa) v ( Hb& - fOb - (Oa& Ob& GcH (Ha& - Fa) v ( Hb& - fOb ) v (Hc & - Fc
u
~
.u u
ib lib iiib ivb vb vib viib viiib ixb xb
tl
T F F T F T T T F F
~
.~ -
Exerdse 4.1.2
Exerdse 4.2
U:{a,b} F:{a}
G:{ }
Fa& fb -+ Ga& Gb.- (Fa-+ Ga) & (fb -+ Gb) U: {a, b} F:{a}
G:{b}
Fav fb -+ Gav Gb.- (Fa-+ Ga) & (fb -+ Gb) Samemodelasii (Fav fb ) & (Gav Gb) .- (Fa& Ga) v (fb & Gb) Samemodelasi (Fav Ga) v (fb v Gb) .- (Fa& fb ) v (Ga& Gb)
183
Answers to Chapter 4 Exercises v
Same model as i
(Fa-+Ga) v (Fb-+Gb) r (Fav fb ) -+ (Gav Gb) U:{a,b} F:{a,b} G:{a} (Fa-+ Ga) v (fb - + Gb) I- (Fa& fb ) -+ (Ga& Gb) Samemodelasi Fa& FbH Ga& Gbl- (FaH Ga) & (fbH Gb) Samemodelasii Fav fb H Gav GbI- (FaH Ga) & (fb H Gb) U:{a,b} F:{a}
P is FALSE
(Fa& fb ) H PI - (FaH P) & ( fbH P) U:{a,b} F:{a}
Pis TRUE
(Fav fb ) H PI- (FaH P) & (fbH P) Samemodelasix (FaH P) v (fbH P) I- (Fav fb ) H P Samemodelasx (FaH P) v (fbH P) I- (Fa& fb ) H P xiii
U: {a}
F:{ }
G:{ }
H: {a}
Fa-+ Ga, Ga-+ HaI- Ha-+ Fa F:{ } H: { } G:{ } Fa-+ Ha, Ha-+ GaI- Fa& Ga U: {a}
U: {a, b} F:{a}
G:{a,b} H: {b}
Fa v fObH Oa & Ob, - Fa - + Ha) & ( Fb- + Hb ~ Ha v Hb - +- Oa v - Gb
U:{a}
F:{a}
G:{a}
Gav - Ha, Ga& Fa.- - Ha
H:(a}
no
A
no
{
{
{ ( !
q
{
{
q {
l
~ !
qO
{ (
{ eo
{
01
t81
S
Chapter
185
Answersto Chapter4 Exercises vii
U:{a, b} T:{(a, b)}
- (Taa& Tab) & - (Tba& Thb) ..- - (Taav Tab) & - (Thav Thb) viii
U:{a.b,c}
F:{(a.b), (h,c),(a.a),(c,a)}
ix
U:{a.b,c}
F:{(a.b), (h,c), (c,a)}
G:{(a,b)}
x
U:{a.b}
F:{(a,a)}
G:{(a,b), (b,a) }
Exerdse4.4 i
U: {m, n}
a: m
b: m
c: n
d: n
m= m, n= n ~ m= n
F: {a, b} U: {a, b, c} [ Fa & Fa) & ~ ) v Fa & Fb) & a$b) v Fa & Fc) & a#C)] v [ Fb & Fa) & ~ ) v Fb & Fb) & ~ ) v Fb & Fc) & ~ )] v [ Fc & Fa) & ~ ) v Fc & Fb) & ~ ) v Fc & Fc) & t * )] I- Fa& Fb& Fc
F: {(a, b), (a. c) } U: {a, b,c } Faa H ~ ) & ( FabH ~ ) & ( FacH ~ v Fha H b:I:a) & ( FbbH b#b) ( FbcH b#c)) v ( FcaH c:#:a) &: ( FcbH ~ ) ( FccH t * )) t& & & & & & & &
( Faa& Faa - + a=a) & ( Faa& Fab - + a=b) & ( Faa& Fac - + a=c) ( Fab& Faa - + b=a) & ( Fab& Fab - + b=b) & ( Fab& Fac - + b=c) ( Fac& Faa - + c=a) & ( Fac& Fab - + c=b) & ( Fac& Fac - + c=c) ( Fha& Fha - + a=a) & ( Fha& Fbb - + a=b) & ( Fha& Fbc - + a=c) ( Fbb& Fha - + b=a) & ( Fbb& Fbb - + b=b) & ( Fbb& Fbc - + b=c) ( Fbc& Fha - + c=a) & ( Fbc& Fbb - + c=b) & ( Fbc& Fbc - + c=c) ( Fca& Fca - + a=a) & ( Fca& Fcb - + a=b) & ( Fca&Fcc - + a=c) ( Fcb& Fca - + b=a) & ( Fcb& Fcb - + b=b) & ( Fbb& Fbc - + b=c) ( Fcc& Fca - + c=a) & ( Fcc& Fcb - + c=b) & ( Fbc& Fbc - + c=c)
Answers to Chapter 4 Exercises
186
Exerdse 4.5.2
'v'xyz(Fxy& Fyz-+ Fxz), 'v'x3yFxy r 3xFxx U: N. F: {(m,o) : m n } a: zero
187
This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact
[email protected] 
