6 3 Truth tables for propositions L of

6. 3 Truth tables for propositions L = # of lines n = # of different simple propositions L = 2 n EXAMPLE: consider the statement, (A ⋅ B) ⊃ C A, B, C are three simple statements 23 L=8

6. 3 COMPARING STATEMENTS Logically equivalent statements have the same truth value on each line under their main operators.

6. 3 COMPARING STATEMENTS Logically equivalent statements Example: B⊃ C ~C⊃ ~ B TTT FTTFT TFFFT FTTTF FTF TFTTF

6. 3 COMPARING STATEMENTS Logically equivalent statements Example: B⊃ C ~C⊃ ~ B TTT FTTFT TFFFT FTTTF FTF TFTTF Note: to compare the truth values for the main connectives, you must use the same combination of truth-value possibilities for both statements, as you go down the rows. . . I’ve circled the values for C to illustrate this here…

6. 3 COMPARING STATEMENTS Logically contradictory statements have opposite truth values on each line under their main operators.

6. 3 COMPARING STATEMENTS Logically contradictory statements Example: B⊃ C B ⋅ ~C TTT TFF TTTF FTT FFFT FTF FFTF Note: to compare the truth values for the main connectives, you must use the same combination of truth-value possibilities for both statements, as you go down the rows. . .

6. 3 COMPARING STATEMENTS Logically consistent statements Two pairs of statements are logically consistent if there is at least one line on which the truth values for the main operators are both true.

6. 3 COMPARING STATEMENTS Logically consistent statements Example: BVC B⋅ C TTT TTF TF F FTT FF F FF T There is at least one line where both statements are true at the same time.

6. 3 COMPARING STATEMENTS Logically inconsistent statements Two pairs of statements are logically inconsistent if there is no line on which the truth values for the main operators are both true.

6. 3 COMPARING STATEMENTS Logically inconsistent statements Example: A≡B TTT A TF FT TFF TT TF FFT FF FT FTF FF TF ⋅ ~B There are no lines in which both statements are true (where both primary operators have true values).

6. 4 Truth Tables For Arguments P 1) R ⊃ E P 2) ~ R R ⊃ E / ~ R // ~ E C) ~ E T T T FT FT T F F FT TF F T T TF FT F TF TF

6. 6 Argument Forms and Fallacies An Argument Form is an arrangement of statement variables and operators so that uniformly substituting statements in place of variables results in arguments Common forms…

6. 6 Argument Forms and Fallacies Common forms are as follows: Disjunctive Syllogism (DS) 1) p v q 2) ~p C) q

6. 6 Argument Forms and Fallacies Common forms are as follows: Disjunctive Syllogism (DS) 1) p v q 2) ~p C) q Bob will either get a raise or quit his job. Bob won’t get a raise. Therefore, he’s going to quit.

6. 6 Argument Forms Modus ponens (MP): 1) p q 2) p. C) q

6. 6 Argument Forms Modus tollens (MT): 1) p q 2) ~q. C) ~p If you break your leg, then I will buy you ice cream. I didn’t buy your ice cream, so it’s clear you didn’t break your leg.

6. 6 Valid Forms Pure hypothetical syllogism (HS): 1. p q 2. q r. C. p r If the world population continues to grow, then cities will become hopelessly overcrowded. If cities become hopelessly overcrowded, pollution will become intolerable. Therefore, if world population continues to grow, then pollution will become intolerable.

6. 6 Valid Forms: Dilemmas Constructive Dilemma (CD): 1. (p q) • (r s) 2. p v r. C. q v s

6. 6 Valid Forms: Dilemmas Destructive Dilemma (DD): 1. (p q) • (r s) 2. ~q v ~s. C. ~p v ~r

6. 6 Refuting Dilemmas Constructive dilemma: grasp it by the horns Prove the conjunctive premise false by proving either conjunct false. Example: (p q) • (r s)

6. 6 Refuting Dilemmas Constructive dilemma: grasp it by the horns Prove the conjunctive premise false by proving either conjunct false. Example: (p q) • (r s) TFF F

6. 6 Refuting Dilemmas Destructive dilemma: escape between the horns Prove the disjunctive premise false Example: p V r

6. 6 Refuting Dilemmas Destructive dilemma: escape between the horns Prove the disjunctive premise false Example: p V r F F F

6. 6 Formal Fallacies Affirming the Consequent (AC): 1) p q 2) q. C) p

6. 6 Formal Fallacies Denying the Antecedent (DA): 1. p q 2. ~p. C. ~q

6. 6 Argument Forms Note on Invalid Forms: Any substitution instance of a valid argument form is a valid argument. However, this result does not apply to invalid forms.

6. 6 Argument Forms An argument will be invalid if it is a substitution instance of that form and it is not a substitution instance of any valid form… Sometimes by making a substitution into an invalid form, you end up with a form that is valid for some independent reason. (For example, because the conclusion is tautologous. )

6. 6 Argument Forms Important rules: Commutativity: p v q is logically equivalent to q v p Double negation: p is logically equivalent to ~~p