TRUTH TABLES continued

Recall: A truth table is a listing of all possible combinations of the individual statements as true or false, along with the resulting truth value of the compound statements. Truth tables are an aide in distinguishing valid and invalid arguments.

Number of Rows If a compound statement consists of n individual statements, each represented by a different letter, the number of rows required in the truth table is 2n.

T F Truth Table for p  q p q p  q Recall that conditional is a compound statement of the form “if p then q”. Think of a conditional as a promise. If I don’t keep my promise, in other words q is false, then the conditional is false if the premise is true. If I keep my promise, that is q is true, and the premise is true, then the conditional is true. When the premise is false (i.e. p is false), then there was no promise. Hence by default the conditional is true. p q p  q T F

T F Truth Table for p  q p q p  q Another way to think about this is with the Law of Detchment. In order for the conditional statement to be true, a true hypothesis must lead to a true conclusion. p q p  q T F

T F Truth Table for q  p p q q  p Here we have the converse, or if q then p. Notice that the second and thirds rows switch place as we are “going backwards.” p q q  p T F

Equivalent Expressions Equivalent expressions are symbolic expressions that have identical truth values for each corresponding entry in a truth table. Hence ~(~p) ≡ p. The symbol ≡ means equivalent to. p ~p ~(~p) T F

Negation of the Conditional Here we look at the negation of the conditional. Note that the 4th and 6th columns are identical. Hence p ^ ~q is equivalent to ~(p  q). p q ~q p ^ ~q p  q ~(p  q) T F

De Morgan’s Laws The negation of the conjunction p ^ q is given by ~(p ^ q) ≡ ~p v ~q. “Not p and q” is equivalent to “not p or not q.” The negation of the disjunction p v q is given by ~(p v q) ≡ ~p ^ ~q. “Not p or q” is equivalent to “not p and not q.”

T F Truth Table for q ↔ p p q q ↔ p Here we have a biconditional, or p if and only if q. Recall that for a biconditional to be true the original conditional statement and it converse must be true, (see cases 1 & 4.) p q q ↔ p T F