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 2 3 L = 8

6.3 COMPARING STATEMENTS Logically equivalent 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 T T T F T T F T T F F T F F F T F T T F T T T F F T F T F T T F

6.3 COMPARING STATEMENTS Logically equivalent statements Example: B ⊃ C ~ C ⊃ ~ B T T T F T T F T T F F T F F F T F T T F T T T F F T F T F T T F 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 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 T T T T F F T T F F T T T F F T T F F F T F T F F F T F 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: B V C B ⋅ C T T T T T F T F F F T T F F F F F F F F 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 ≡ BA ⋅ ~ B T T T T F F T T F F T T T F F F T F F F T F T F F F T F There are no lines in which both statements are true (where both primary operators have true values).

6.4 Truth Tables For Arguments P1) R ⊃ E P2) ~ R C) ~ E R ⊃ E / ~ R //~ E T T TF T T F FF TT F F T TT FF T F T FT F

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…

Common forms are as follows: Disjunctive Syllogism (DS) 1) p v q 2) ~p C) q

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.

Modus ponens (MP): 1) p  q 2) p. C) q

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.

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.

Constructive Dilemma (CD): 1. (p  q) (r  s) 2. p v r. C. q v s

Destructive Dilemma (DD): 1. (p  q) (r  s) 2. ~q v ~s. C. ~p v ~r

Constructive dilemma: grasp it by the horns  Prove the conjunctive premise false by proving either conjunct false. Example: (p  q) (r  s)

Constructive dilemma: grasp it by the horns  Prove the conjunctive premise false by proving either conjunct false. Example: (p  q) (r  s) T F F F

Destructive dilemma: escape between the horns  Prove the disjunctive premise false Example: p V r

Destructive dilemma: escape between the horns  Prove the disjunctive premise false Example: p V r F F F

Affirming the Consequent (AC): 1) p  q 2) q. C) p

Denying the Antecedent (DA): 1. p  q 2. ~p. C. ~q

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.

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.)

Important rules: Commutativity: p v q is logically equivalent to q v p Double negation: p is logically equivalent to ~~p