TRUTH TABLES continued Recall A truth table is
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 2 n.
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. p q T T F F T F T T
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. p q T T F F T F T T
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. ” p q q p T T F F T F T
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 T F ~p ~(~p) F T T F
Negation of the Conditional • Here we look at the negation of the conditional. • Note that the 4 th and 6 th columns are identical. • Hence p ^ ~q is equivalent to ~(p q). p q ~q p ^ ~q p q ~(p q) T T F F T F T F F F 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. ”
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. ) p q q↔p T T F F T F T F F T
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