2 Chapter 8 Categorical Logic Students will learn to:Recognize the four types of categorical claims and the Venn diagrams that represent themTranslate a claim into standard formUse the square of opposition to identify logical relationships between corresponding categorical claimsUse conversion, obversion, and contraposition with standard form to make valid argumentsRecognize and evaluate the validity of categorical syllogisms
3 Chapter 8 Categorical Logic IntroductionCategorical LogicAll Xs are YsNo Xs are YsSome Xs are YsSome Xs are not Ys.Examples & ApplicationsCategorical ClaimsCategorical ClaimStandard form categorical claimTermPredicate TermNoun/Noun Phrase
5 Chapter 8 Categorical Logic Translation into Standard FormEquivalent Claim“Only”: introduces the predicate of an A claim.Only sophomores are eligible candidates.All eligible candidates are sophomore.“The Only”: introduces the subject of an A claim.Bats are the only true flying mammals.All true flying mammals are bats.Time & Space“Whenever”: often indicates an A or E claim.I always get nervous whenever I take logic exams.All times I take logic exams are times I get nervous.“Wherever”: often indicates an A or E claim.He makes trouble wherever he goes.All places he goes are places he makes trouble.
6 Chapter 8 Categorical Logic Claims about single individualsTranslated to claims about classes.A or E claim.A claim about an X of type Y becomes All/No Ys identical to X are PsAristotle is a logician=All people identical to Aristotle are logicians.Tallahassee is in Florida=All cities identical to Tallahassee are cities in Florida.Claims involving mass nounsTreated as claims about examples of the kind of stuff.Gold is a heavy metal=All examples of gold are heavy metal.
7 Chapter 8 Categorical Logic The Square of OppositionThe SquareContrary ClaimsSubcontrary ClaimsContradictory ClaimsLogical RelationsEmpty Subset ClassesAssumptionUse
8 Chapter 8 Categorical Logic Three Categorical OperationsConversionSwitching the subject and predicate terms.(A) All S are P: All P are S(E) No S are P: No P are S(I) Some S are P: Some P are S(O)Some S are Not P: Some P are not SE and I claims are equivalent to their converses.A and O claims are not.
9 Chapter 8 Categorical Logic Obversion1)Replace the claim with the claim directly across from it on the square or opposition and 2) change the predicate to its complement.(A) All S are P: No S are non-P(E) No S are P: All S are non-P(I) Some S are P: Some S are not non-P(O)Some S are Not P: Some S are non-PComplementary ClassComplementary TermAll categorical claims are equivalent to their obverses.
10 Chapter 8 Categorical Logic Contraposition1)Switch the subject and predicate terms 2) replaces both terms with their complements(A) All S are P: All non-P are non-S(E) No S are P: No non-P are non-S(I) Some S are P: Some non-P are non-S(O)Some S are Not P: Some non-P are not non-SComplementary ClassComplementary TermAll categorical claims are equivalent to their obverses.A and O claims are equivalent to their contrapositions.E and I claims are not.
11 Chapter 8 Categorical Logic Categorical SyllogismsSyllogism: an argument with 2 premises and 1 conclusion.Categorical Syllogism1. All Americans are consumers.2. Some consumers are not democrats.C. Therefore, some Americans are not Democrats.Terms of a syllogismMajor term (P): the term that occurs as a predicate term of the syllogism’s conclusion.Minor term (S): the term that occurs as the subject term of the syllogismMiddle term (M): the term that occurs in both of the premises but not in the conclusion.Validity & the relation between the terms.
12 Chapter 8 Categorical Logic The Venn Diagram Method of Testing For ValidityStepsDiagram premise 1Diagram premise 2Determine if the conclusion can be read from the diagram (valid) or not (invalid).
13 Chapter 8 Categorical Logic Example1. No Republicans are collectivists.2. All socialists are collectivists.C. Therefore, no socialists are Republicans.
14 Chapter 8 Categorical Logic Example1. Some S are not M2. All P are MC. Some S are not P
15 Chapter 8 Categorical Logic Example1. All P are M2. Some S are MC. Some S are P
16 Chapter 8 Categorical Logic Categorical Syllogisms With Unstated PremisesExample: You shouldn’t give chicken bones to dogs. They could choke on them.1. All chicken bones are things dogs could choke on.2. (No things dogs could choke on are things you should give dogs.C. No chicken bones are things you should give dogs.Real Life SyllogismsIt can be useful to replace long phrases with letters.ExampleAll C are DNo D are SNo C are S
17 Chapter 8 Categorical Logic Rules Method for Testing ValidityDistributionClaim Distribution ( )A-claim All (S) are PI-claim Some S are PE-claim No (S) are (P)O-claim Some S are not (P)
18 Chapter 8 Categorical Logic The RulesRule #1The number of negative claims in the premises must be the same as the number of negative claims in the conclusion.Rule #2At least one premises must distribute the middle termAny term that is distributed in the conclusion must be distributed in its premise.ExamplesBreaks Rule #11. No dogs up for adoption at the animal shelter are pedigreed dogs.2. Some pedigreed dogs are expensive dogs.C. Some pedigreed dogs up for adoption at the animal shelter are expensive dogs.
19 Chapter 8 Categorical Logic Breaks Rule #21. All pianists are keyboard players.2. Some keyboard players are percussionists.C. Some pianists are not percussionists.Breaks Rule #31. No mercantilists are large land owners.2. All mercantilists are creditors.C. No creditors are large landowners.
20 Chapter 8 Categorical Logic Recap1. The four types of categorical claims include A, E, I, and O.2. There are Venn diagrams for the four types of claims.3. Ordinary English claims can be translated into standard form categorical claims. Some rules of thumb for such translations are as follows:a. “Only” introduces the predicate of an A-claim.b. “The only” introduces the subject term of an A-claim.c. “Whenever” means times or occasions.d. “whenever” means places or locations.4. Square of opposition displays contradictions, contrariety, and subcontrariety among corresponding standard-form claims,5. Conversion, obversion, and contraposition are three operations that can be performed on standard-form claims; some are equivalent to the original and some or not.6. Categorical syllogisms are standardized deductive arguments; we can test them for validity by the Venn diagram method or by the rules method-the latter relies on the notions of distribution and the affirmative and negative qualities of the claims involved.
21 Chapter 09 Truth functional Logic Students will learn to:Understand the basics of truth tables and truth-functional symbolsSymbolize normal English sentences with claim letters and truth-functional symbolsBuild truth tables for symbolizations with several lettersEvaluate truth-functional arguments using common argument formsUse the truth-table and short truth-table methods to determine whether an argument is truth-functionally validUse elementary valid argument forms and equivalences to determine the validity of arguments.
22 Chapter 09 Truth functional Logic IntroductionBasic ConceptsTruth functional logicTruth functional claimsApplicationsSet theoryFoundation of mathematicsElectronic circuitsAnalysis of argumentsBenefits of learning truth functional logicLearning about the structure of language.Learning what it is like to work in a precise, nonmathematical system of symbols.Learning how to communicate better.
23 Chapter 09 Truth functional Logic Truth Tables and Truth-Functional SymbolsClaims & Claim VariablesClaim variableAny claim is either true or false (but not both).Truth TablesOne variable table & Two Variable TablePTFPQTF
24 Chapter 09 Truth functional Logic NegationA negation is false when the claim being negated is true, otherwise it is true.Corresponds with “not” and is symbolized by ~Claim variableAny claim is either true or false (but not both).Truth Table for NegationP~PTF
25 Chapter 09 Truth functional Logic ConjunctionA conjunction is true only if both of its conjuncts are true, otherwise it is false.Corresponds with “and” and is symbolized by &.“But’, “while”, “even though” and other phrases also form conjunctions.Truth Table for ConjunctionPQP&QTF
26 Chapter 09 Truth functional Logic DisjunctionA disjunction is false only if both of its disjuncts are false, otherwise it is true.Corresponds with “or” and is symbolized by v.Truth Table for DisjunctionPQP v QTF
27 Chapter 09 Truth functional Logic Conditional ClaimAntecedent: the “A” in “If A then B.”Consequent: The “B” in “If A then B.”A conditional claim is false if any only if its antecedent is true and its consequent is false.A conditional corresponds to “if…then…” and is symbolize by “”.PQP QTF
28 Chapter 09 Truth functional Logic CombinationsPQ~P~PQTF
29 Chapter 09 Truth functional Logic Constructing TablesFormula for determining the number of rows: r=2N, where r is the number of rows in the table and n is the number of claims.Constructing at tableAlternate Ts and Fs in the right most column.Alternate pairs of Ts and Fs in the next column to the left.Alternative sets of four Ts and four Fs in the next column to the left .Alternate sets of 8 Ts and 8 Fs and so on until all rows for the claim variables are filledThe top half of the left most column will always be all s and the bottom half will be all Fs
30 Chapter 09 Truth functional Logic Three Variable TablePQRTF
31 Chapter 09 Truth functional Logic More on Constructing TablesParenthesesExample: If Paula goes to work, then Quincy and Rogers get the day off.Symbolized as P (Q&R).The parentheses are neededThe truth value of a compound claim depends entirely upon the truth of its parts.If the parts are themselves compounded, their truth values depends on the truth value of the parts, and so on.Constructing the tableThe reference columns are those for variables.The table provides a truth functional analysis of the claim.
32 Chapter 09 Truth functional Logic Three Variable Example TablePQRQ&RP-->(Q&R)TF
33 Chapter 09 Truth functional Logic Truth Functional EquivalentDefinedExamplePQ~PPQ~P v QTF
34 Chapter 09 Truth functional Logic Symbolizing Compound ClaimsTruth functional structureTruth functionally equivalent“If” and “only if”“If” introduces the antecedent of a conditional.Sam will buy the popcorn if Sally buys the ticketsIf Sally buys the tickets, then Sam will buy the popcorn.P, if Q = Q P“Only if” introduces the consequent of a conditional.Sam will buy the popcorn only if Sally buys the tickets.If Sam buys the popcorn, then Sally buys the tickets.P only if Q = P Q“If and only If” combines “if” and “only if”Sam will go if and only if Sally goes.If Sam goes, then Sally will go and if Sally goes, then Sam will go.P if and only if Q = (P Q) & (Q P)
35 Chapter 09 Truth functional Logic Necessary & Sufficient ConditionsNecessary ConditionA is necessary for B= “If A is the case, then B can be the case” or “if A is not the case, then B cannot be the case.”The necessary condition is the consequent of the conditional.Oxygen is necessary for human life=If there is human life, then there is oxygen.P is necessary for Q = Q P“Only if” introduces the necessary condition.Sufficient ConditionA is sufficient for B= “If A is the case, then B must be the case.”Earning a 60 or better is sufficient to pass this class = if a person earns a 60 or better, then they pass the class.P is sufficient for Q = P QSufficient conditions are not necessary conditions, and vice versa.
36 Chapter 09 Truth functional Logic Necessary and sufficient ConditionIf A is necessary and sufficient for B, then B cannot occur without A and if A occurs, then B must occur.“If and only if”A person is a bachelor if and only if he is an unmarried man=if a person is a bachelor then he is an unmarried man and if a person is an unmarried man, then he is a bachelor.P is necessary and sufficient for Q = (PQ) & (Q P)Ordinary LanguageFast & LooseYou can watch television only if you clean your room.Intended: If you clean your room, then you can watch TV.Actual: If you watch TV, then you have cleaned your room.
37 Chapter 09 Truth functional Logic UnlessP unless Q = if not Q, then P = ~Q P= P v QBill will go unless Sally goes= If Sally does not go, then Bill will go=Sally will go or Bill will go.EitherEither indicates a disjunction.Either P and Q or R= (P&Q) v RP and either Q or R = P & (Q v R)Truth Functional ArgumentsValidityAn argument is valid if and only if the truth of the premises guarantees the truth of the conclusion.It does not matter whether the premises are actually true or not.
38 Chapter 09 Truth functional Logic Valid Truth Functional Argument PatternsModus Ponens (Valid)If P, then QPTherefore QModus Tollens (Valid)Not QTherefore not PChain Argument (Valid)If Q, then RTherefore If P, then R
39 Chapter 09 Truth functional Logic Invalid Truth Functional Argument PatternsAffirming the Consequent (Invalid)If P, then QQTherefore PDenying the Antecedent(Invalid)Not PTherefore Not QUndistributed Middle(Invalid)If R, then QTherefore If P, then R
40 Chapter 09 Truth functional Logic Truth Table Test for ValidityPresent all the possible circumstances for an argument by building a truth table for it.Look to see if there are any circumstances in which all the premises are true and the conclusion is false.If there is even a single row in which all the premises are true and the conclusion is false, then the argument is invalid.Otherwise the argument is valid.
41 Chapter 09 Truth functional Logic ExampleArgument: If the Saints beat the Forty-Niners, then the Giants will make the playoffs. But the Saints won’t beat the Forty-Niners. So the Giants won’t make the play-offs.Symbolized:P -->Q~P~QPQ~PP Q~QTF
42 Chapter 09 Truth functional Logic ExampleArgument: We’re going to have large masses of arctic air (A) flowing into the Midwest unless the jetstream (J) moves south. Unfortunately, there’s no chance of the jet stream going south. So you can bet there’ll be arctic air flowing into the Midwest.SymbolizedA v J~JAAJA v J~JTF
43 Chapter 09 Truth functional Logic ExampleArgument: If Scarlet is guilty of the crime, then Ms. White must have left the back door unlocked and the colonel must have retired before ten o’clock. However, either Ms. White did not leave the back door unlocked, or the colonel did not retire before ten. Therefore, Scarlet is not guilty of the crime.S= Scarlet is guilty of the crime.W= Ms. White left the back door unlocked.C=The colonel retired before ten o’clock.SymbolizationS-->(W&C)~W v ~C~SSWC~W~CW&CS--> (W&C)~W v ~C~STF
44 Chapter 09 Truth functional Logic Short Truth Table MethodThe idea behind this method is that if an argument is invalid, then the argument must have at least one row in which all the premises are true and the conclusion is false.The method is to look directly for such a row by trying to make all the premises true and the conclusion false at the same time.In some cases neither the conclusion nor the premises forces an assignment.In such cases trial and error must be used.It must be kept in mind that it only takes one row in which the premises are all true and the conclusion is false to make an argument invalid.To be valid, an argument must have a true conclusion in every row in which the premises are all true.ExampleArgument:P-->Q~Q-->R~P-->RFor ~P -->R to be false, ~P must be true (P must be false) and R must be false.Assuming P is false, P-->Q is true when Q is true or false.Assuming R is false, ~Q-->R is true when ~Q is false, so Q must be assumed to be true.This row makes the premises all true and the conclusion false, which proves the argument to be invalid.PQRP-->Q~Q~Q-->R~P~P-->RFT
45 Chapter 09 Truth functional Logic The MethodTry to assign Ts and Fs to the letters in the symbolization so that all the premises come out true and the conclusion comes out false.There may be more than one way to do this, any one will do to prove the argument to be invalid.If it is impossible to do this, the argument is valid.
46 Chapter 8 Examples Categorical Logic 8-115. Every voter is a citizen, but some citizens are not residents. Therefore, some voters are not residents.1. All voters are citizens.2. Some citizens are not residents.C. Some voters are not residents.Invalid.
47 Chapter 8 Examples Categorical Logic 8-125. A few compact disc players use 24X sampling, so some of them must cost at least fifty dollars, because you can’t buy a machine with 24X sampling for less than $50.1:Some compact disc players are players that use 24x sampling.2: No players that use 24x sampling are players that cost under $50C: Some compact disc players are not players that cost under $50.ValidOrP1: Some compact disc players are players that use 24X sampling.P2: All players that use 24X sampling are players that cost more that $50.C: Some compact disc players are players that cost more than $50.
48 Chapter 8 Examples Categorical Logic I was talking to Bill the other day and he told me that he is a runner. People who run, at least if they have any sense, own at least one pair of running shoes. So, I’m sure that Bill has a pair of running shoes.P1: All people identical to Bill are people who run.P2: All people who run are people who have/own running shoes.C: All people identical to Bill are people who have/own running shoes.
49 Chapter 8 Examples Categorical Logic P1: All people identical to Bill are people who run.P2: All people who run are people who have/own running shoes.C: All people identical to Bill are people who have/own running shoes.
50 Chapter 8 Examples Categorical Logic It is often said that all creatures with blood are either cold-blooded or warm-blooded. It is well known that every non-mammal is a non-cat. Of course, it is also known that All mammals are non cold-blooded things. So, it must be concluded that not a single cat is cold blooded. The same is true of dogs.P1 (before contraposition): All non-mammals are non-cats.P2 (before obversion): All mammals are non cold-blooded things.P1: All cats are mammals.P2: No mammals are cold-blooded things.C: No cats are cold-blooded things.
51 Chapter 8 Examples Categorical Logic P1: All cats are mammals.P2: No mammals are cold-blooded things.C: No cats are cold-blooded things.
52 Chapter 8 Examples Categorical Logic It is well known from biology that not a single mammal is a creature that lacks a developed spine. Spines are, of course, composed of bone and contain an important part of the nervous system. So, it is obvious that all creatures with spines have some sort of nervous system. It can be concluded that each mammal has some sort of nervous system.P1 (before obversion): No mammals are creatures without developed spines.P1: All mammals are creatures that have developed spines.P2: All creatures that have developed spines are creatures that have some sort of nervous system.C: All mammals are creatures that have some sort of nervous system.
53 Chapter 8 Examples Categorical Logic P1: All mammals are creatures that have developed spines.P2: All creatures that have developed spines are creatures that have some sort of nervous system.C: All mammals are creatures that have some sort of nervous system.
54 Chapter 9 Examples Truth Functional Logic Translations#1. If the first party fails to fulfill the contract, then the second party is entitled to a refund or a replacement product of equivalent value. The first party failed to fulfill the contract, so either the second party will receive a refund or a replacement product.P= The first party fails to fulfill the contract.Q= The second party is entitled to a refund.R= The second party is entitled to a replacement product.TranslationP1: P-->(Q vR)P2: PC: Q v R
55 Chapter 9 Examples Truth Functional Logic #2. The payment of fees is sufficient to become a member of the club. Either Bill will pay his fees or he will not and he will do something else. Unless he becomes a member of the club, he will do something else. Bill didn’t do something else, so he is in the club.P= Payment of fees.Q= Become a member of the clubR= Do something else.P1: P-->QP2: P v (~P & R)P3: ~Q-->RP4: ~RC: Q
56 Chapter 9 Examples Truth Functional Logic P1: PvQP2: ~PC: Q-->PPQ~PP v QQ-->PTF