TRUTH TABLES

1. TRUTH TABLES continued

2. 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.

3. 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.

4. Truth Table for 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.

5. Truth Table for 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.

6. Truth Table for 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.”

7. 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.

8. 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).

9. 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 pv q is given by ~(pv q) ≡ ~p ^ ~q. “Not p or q” is equivalent to “not p and not q.”

10. Truth Table for 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.)