Chapter 7 Evaluating Deductive Arguments II: Truth Functional Logic Invitation to Critical Thinking First Canadian Edition

Chapter 7Chapter 7Evaluating Deductive Arguments II:Evaluating Deductive Arguments II:

Truth Functional LogicTruth Functional Logic

www.criticalthinking1ce.nelson.comwww.criticalthinking1ce.nelson.com

Invitation to Critical ThinkingInvitation to Critical ThinkingFirst Canadian EditionFirst Canadian Edition

Joel RudinowJoel RudinowVincent E. BarryVincent E. Barry

Mark LetteriMark Letteri

© 2008 by Nelson, a division of Thomson Canada Limited© 2008 by Nelson, a division of Thomson Canada Limited 7-7-22

Truth Functional LogicTruth Functional Logic Known as symbolic logicKnown as symbolic logic Uses logical operatorsUses logical operators Uses truth tables Uses truth tables Useful in translating claims into Useful in translating claims into

categorical statementscategorical statements Useful in determining validityUseful in determining validity


Truth TablesTruth TablesSample truth tableSample truth tableModus PonensModus Ponens PP QQ PPQQ

TT TT TTTT FF FFFF TT TTFF FF TT

all possible combinations of truth value for all components of a truth functional

compound statement

corresponding truth value of the compound statement


Logical OperatorsLogical OperatorsRepresent the relationships between the “truth values” Represent the relationships between the “truth values”

of the compound sentences we make with them. of the compound sentences we make with them. Truth functional analysis is simply a way of keeping Truth functional analysis is simply a way of keeping

track of this. track of this. SymbolsSymbols

~~ = not (negation) = not (negation)&& = and (conjunction)= and (conjunction)

if, then (conditional)if, then (conditional)vv = either or (disjunction) = either or (disjunction)


Logical OperatorsLogical OperatorsNegationNegation Negation simply reverses the truth value of the component Negation simply reverses the truth value of the component

statement to which it is applied.statement to which it is applied. The symbol “~” represents the logical operator negation. The symbol “~” represents the logical operator negation. Thus “~P” or “not P” represents the negation of P.Thus “~P” or “not P” represents the negation of P. Negation operates on a single component statement, P. Negation operates on a single component statement, P.

Since P is either true or false (not true), our truth table for Since P is either true or false (not true), our truth table for negation required only two lines. negation required only two lines.

P = The weather is great.

~P = The weather is not great.

P ~PT FF T


Logical OperatorsLogical OperatorsConjunctionConjunction The components of a conjunction are called “conjuncts”. The components of a conjunction are called “conjuncts”. ““P” stands for the first conjunct and the letter “Q” stand for P” stands for the first conjunct and the letter “Q” stand for

the second, and the symbol “&” represents the logical the second, and the symbol “&” represents the logical operator conjunction.operator conjunction.

Compound statements are based on conjunction. Since P Compound statements are based on conjunction. Since P and Q may be either true or false, our truth table for and Q may be either true or false, our truth table for conjunction will require four lines.conjunction will require four lines.

PP QQ P P && Q QTT TT TTTT FF FFFF TT FFFF FF FF

The weather is great and I wish you were here.


Logical OperatorsLogical OperatorsDisjunctions:Disjunctions: Component statements are called “disjuncts”.Component statements are called “disjuncts”. At least one of the disjuncts is true (possibly both).At least one of the disjuncts is true (possibly both). The letters “P” and “Q” represent the two disjuncts The letters “P” and “Q” represent the two disjuncts

and the symbol “and the symbol “vv” represents the operator ” represents the operator disjunction.disjunction.

Thus “P v Q” represents the statement “Either P Thus “P v Q” represents the statement “Either P oror Q”.Q”.

PP QQ P P vv QQTT TT TTTT FF TTFF TT TTFF FF FF

Either you party or you study.


Logical OperatorsLogical OperatorsConditionals:Conditionals: Compound statements in which the Compound statements in which the “if”“if” statement is the antecedent statement is the antecedent

and the and the “then”“then” statement is the consequent statement is the consequent The consequent depends on the truth of the antecedent and follows The consequent depends on the truth of the antecedent and follows

the antecedentthe antecedent ““P” represents the antecedent and “Q” represents the consequent. P” represents the antecedent and “Q” represents the consequent.

The symbol “The symbol “” represents the logical operator implication. ” represents the logical operator implication. Thus “P Thus “P Q” represents the conditional “If P then Q”.Q” represents the conditional “If P then Q”.

““only if” has the effect of reversing the conditional relationship only if” has the effect of reversing the conditional relationship between the antecedent and consequent.between the antecedent and consequent.

PP QQ P PQQTT TT TTTT FF FFFF TT TTFF FF TT

If you study, then you are likely to do better on the quiz.


Argument FormsArgument FormsDeductively ValidDeductively Valid

modus ponens:modus ponens: based on one hypothetical based on one hypothetical statement and the affirmation of its statement and the affirmation of its antecedent.antecedent.

(1)(1) PPQQ(2)(2) PP (3) Q(3) Q

If good water exists on Earth, then adequate support for life exists on Earth.

Good water exists on Earth.

Adequate support for life exists on Earth.



modus tollens:modus tollens: based on one hypothetical based on one hypothetical statement and the denial of its statement and the denial of its consequent.consequent.

(1)(1) PPQQ(2b) ~Q(2b) ~Q(3b) ~P(3b) ~P

If life exists on Mars, then adequate support for life exists on Mars.

No adequate support for life exists on Mars _______________________________

No life exists on Mars .



hypothetical syllogism:hypothetical syllogism: based on two based on two hypothetical statements as premises, where hypothetical statements as premises, where the consequent of the first is the antecedent the consequent of the first is the antecedent of the second.of the second.

(1)(1) PPQQ(2d) Q(2d) QRR(3d) P(3d) PRR

If life exists on Mars, then adequate support for life exists on Mars.

If adequate support for life exists on Mars, then an astronautical mission to Mars is feasible.

If life exists on Mars, then an astronautical mission to Mars is feasible.



disjunctive syllogism:disjunctive syllogism: based on a based on a disjunction and the denial of one of its disjunction and the denial of one of its disjunctsdisjuncts

(1)(1) P v QP v Q(2)(2) ~P~P(3) Q(3) Q

Either the battery is dead or there is a short in the ignition switch.

The battery is not dead.

There is a short in the ignition switch.


Argument FormsArgument FormsDeductively InvalidDeductively Invalid

fallacy of denying the antecedent:fallacy of denying the antecedent: based on a hypothetical statement and based on a hypothetical statement and the denial of its antecedentthe denial of its antecedent

(1)(1) PPQQ(2c) ~P(2c) ~P(3c) ~Q(3c) ~Q

If a figure is square then it has four sides.

This figure (a rhombus) is not a square.

This figure (a rhombus) does not have four sides.


Argument FormsArgument FormsDeductively InvalidDeductively Invalid

fallacy of affirming the consequent:fallacy of affirming the consequent: based on a hypothetical statement and the based on a hypothetical statement and the affirmation of its consequent.affirmation of its consequent.

(1)(1) PPQQ(2a) Q(2a) Q

(3a) P(3a) P

If a figure is square, then it has four sides.This rhombus has four sides.This rhombus is square.


Constructive DilemmaConstructive Dilemma

An argument form or strategy combining An argument form or strategy combining hypothetical and disjunctive premises that hypothetical and disjunctive premises that seeks to prove its point by showing that it seeks to prove its point by showing that it is implied by each of two alternatives, at is implied by each of two alternatives, at least one of which must be true.least one of which must be true.

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Chapter 7 Evaluating Deductive Arguments II: Truth Functional Logic Invitation to Critical Thinking First Canadian Edition