From: William J. Rapaport (rapaport@buffalo.edu) Date: Sep. 12, 2009 Subj: Propositional Logical Equivalences Here's another example of how to show that 2 propositions are logically equivalent. (I can't type the triple-bar symbol for "is logically equivalent to", so I'll use the made-up word "equiv". And I'm using "-" for the negation sign.) Show (p ^ q) equiv -(-p v -q): First, construct a truth-table for (p^q): p q (p^q) T T T T F F F T F F F F Second, construct a truth-table for -(-p v -q) [Here, I'll write the computed truth-values under the principal connective of each molecular proposition]: 1 2 3 4 5 = (3v4) 6 = -5 p q -p -q (-p v -q) -(-p v -q) T T F F F T T F F T T F F T T F T F F F T T T F Because the output columns of both truth tables are identical, we can say that, for all rows of the truth tables, the truth value of (p^q) is identical to the truth value of -(-p v -q), i.e., tval(p^q)=tval(-(-p v -q)). Note, by the way, that this is also a proof that conjunction(^) can be defined in terms of negation(-) and inclusive disjunction(v). In other words, if we have negation and disjunction, we don't really "need" conjunction.