Chapter 7 Evaluating Deductive Arguments II: Truth Functional Logic Invitation to Critical Thinking First Canadian Edition Joel Rudinow Vincent E. Barry Mark Letteri

© 2008 by Nelson, a division of Thomson Canada Limited 7-2 Truth Functional Logic  Known as symbolic logic  Uses logical operators  Uses truth tables  Useful in translating claims into categorical statements  Useful in determining validity

© 2008 by Nelson, a division of Thomson Canada Limited 7-3 Truth Tables Sample truth table Modus Ponens PQ P  Q PQ P  Q TTT TFF FTT FFT all possible combinations of truth value for all components of a truth functional compound statement corresponding truth value of the compound statement

© 2008 by Nelson, a division of Thomson Canada Limited 7-4 Logical Operators Represent the relationships between the “truth values” of the compound sentences we make with them. Truth functional analysis is simply a way of keeping track of this.  Symbols ~ = not (negation) & = and (conjunction)  if, then (conditional) v = either or (disjunction)

© 2008 by Nelson, a division of Thomson Canada Limited 7-5 Logical Operators Negation  Negation simply reverses the truth value of the component statement to which it is applied.  The symbol “~” represents the logical operator negation.  Thus “~P” or “not P” represents the negation of P.  Negation operates on a single component statement, P. Since P is either true or false (not true), our truth table for negation required only two lines.  Negation operates on a single component statement, P. Since P is either true or false (not true), our truth table for negation required only two lines. P = The weather is great. ~P = The weather is not great. P~P TF FT

© 2008 by Nelson, a division of Thomson Canada Limited 7-6 Logical Operators Conjunction  The components of a conjunction are called “conjuncts”.  “P” stands for the first conjunct and the letter “Q” stand for the second, and the symbol “&” represents the logical operator conjunction.  Compound statements are based on conjunction. Since P and Q may be either true or false, our truth table for conjunction will require four lines. PQ P & Q TTTTTTTTTTTT TFFTFFTFFTFF FTFFTFFTFFTF FFFFFFFFFFFF The weather is great and I wish you were here.

© 2008 by Nelson, a division of Thomson Canada Limited 7-7 Logical Operators Disjunctions:  Component statements are called “disjuncts”.  At least one of the disjuncts is true (possibly both).  The letters “P” and “Q” represent the two disjuncts and the symbol “v” represents the operator disjunction.  Thus “P v Q” represents the statement “Either P or Q”. PQ P v Q TTTTTTTTTTTT TFTTFTTFTTFT FTTFTTFTTFTT FFFFFFFFFFFF Either you party or you study.

© 2008 by Nelson, a division of Thomson Canada Limited 7-8 Logical Operators Conditionals:  Compound statements in which the “if” statement is the antecedent and the “then” statement is the consequent  The consequent depends on the truth of the antecedent and follows the antecedent  “P” represents the antecedent and “Q” represents the consequent. The symbol “  ” represents the logical operator implication.  Thus “P  Q” represents the conditional “If P then Q”. “only if” has the effect of reversing the conditional relationship between the antecedent and consequent. “only if” has the effect of reversing the conditional relationship between the antecedent and consequent. PQ P  Q PQ P  Q TTTTTTTTTTTT TFFTFFTFFTFF FTTFTTFTTFTT FFTFFTFFTFFT If you study, then you are likely to do better on the quiz.

© 2008 by Nelson, a division of Thomson Canada Limited 7-9 Argument Forms Deductively Valid modus ponens: based on one hypothetical statement and the affirmation of its antecedent. (1)P  Q (2)P (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.

© 2008 by Nelson, a division of Thomson Canada Limited 7-10 Argument Forms Deductively Valid modus tollens: based on one hypothetical statement and the denial of its consequent. (1)P  Q (2b) ~Q (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.

© 2008 by Nelson, a division of Thomson Canada Limited 7-11 Argument Forms Deductively Valid hypothetical syllogism: based on two hypothetical statements as premises, where the consequent of the first is the antecedent of the second. (1)P  Q (2d) Q  R (3d) P  R 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.

© 2008 by Nelson, a division of Thomson Canada Limited 7-12 Argument Forms Deductively Valid disjunctive syllogism: based on a disjunction and the denial of one of its disjuncts (1)P v Q (2)~P (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.

© 2008 by Nelson, a division of Thomson Canada Limited 7-13 Argument Forms Deductively Invalid fallacy of denying the antecedent: based on a hypothetical statement and the denial of its antecedent (1)P  Q (2c) ~P (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.

© 2008 by Nelson, a division of Thomson Canada Limited 7-14 Argument Forms Deductively Invalid fallacy of affirming the consequent: based on a hypothetical statement and the affirmation of its consequent. (1)P  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.

© 2008 by Nelson, a division of Thomson Canada Limited 7-15 Constructive Dilemma  An argument form or strategy combining hypothetical and disjunctive premises that seeks to prove its point by showing that it is implied by each of two alternatives, at least one of which must be true.