can anyone understand this?

The biconditional has another useful function beyond translating "if and only if" propositions. Since the biconditional is true whenever the truth values of the component parts are the same, the biconditional can be used to determine whether or not two propositions are logically equivalent; that is, it can show if two propositions have identical truth values. Consider the example from the previous lesson, in which it was stated that B ≡ ~~B. The truth table for this is: B ~B ~~B B ≡ ~~B T F T T F T F T This biconditional is always true, so B and ~~B are seen to be logically equivalent. A proposition that is true for every row in the truth table is called a tautology. In other words, tautologies are propositions that are true by logical structure. The compound proposition B ≡ ~~B is thus a tautology. Other important tautologies are p ⊃ p and p ∨ ~p. So we can now say more briefly that the biconditional of logically equivalent propositions is a tautology. When a proposition is false for every row in the truth table, you have a self-contradiction. Self-contradictions are propositions that are false by logical structure, such as p • ~p. Consider the two propositions p ⊃ q and p • ~q, along with their biconditional. We will do this truth table (and every one from now on) without guide columns, simply placing the truth values immediately below the variables p and q and working out the truth value of the compound propositions, finishing with the ≡ sign.

KEY POINT The biconditional can be used to test for equivalence. If the biconditional of two statements is a tautology, then the statements are equivalent.

( p ⊃ q ) ≡ ( p • ~q ) T T T F T F F T T F F F T T T F F T T F F F F T F T F F F F T F Because this biconditional is a self-contradiction, we can say that p ⊃ q contradicts p • ~q.

When the biconditional of two propositions is a tautology, the propositions are logically equivalent. When it is a self-contradiction, the propositions are contradictory. A tautology is a proposition that is true by logical structure. A self-contradiction is a proposition that is false by logical structure.