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As you can see, IFF is essentially just equality, as that operator applies to truth values it returns true if its operands are equal and false if they aren t. As for IF, in this case the truth table needs some explanation, since (unlike all of the other dyadic ones we ve seen so far) it isn t symmetric. It s meant to be read as follows. Let operands p and q have truth values as indicated down the left side and across the top of the table, respectively; then the expression IF p THEN q has truth value as indicated in the body of the table. More specifically, IF p THEN q is false if and only if p is true and q is false; more specifically still, IF p THEN q is logically equivalent to (NOT p) OR q which illustrates the point that the connectives aren t all primitive; in fact, as you probably know, all 20 connectives in 2VL can be expressed in terms of suitable combinations of NOT and either AND or OR.
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CHAPTER 18 WHY THREE- AND FOUR-VALUED LOGIC DON T WORK
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I remark in passing the point isn t directly germane to the main argument of this chapter, but it s interesting that the 20 2VL connectives can actually all be expressed in terms of just one primitive: viz., either NOR or NAND. NOR (also known as the Peirce arrow and usually written as a down arrow, ) is a dyadic connective that evaluates to true if and only if both of its operands are false; that is, the expression p q is equivalent to NOT (p OR q). NAND (also known as the Sheffer stroke and usually written as a vertical bar, | ) is a dyadic connective that evaluates to false if and only if both of its operands are true; that is, the expression p|q is equivalent to NOT (p AND q).1 It s an interesting exercise, here left to the reader, to show that either of these connectives can serve as a generating connective for the whole of 2VL.
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Tautologies, Identities, and Contradictions
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Let me now point out explicitly what you probably know already: viz., that p and q are equivalent if and only if each implies the other. In symbols: ( p IFF q ) IFF ( ( IF p THEN q ) AND ( IF q THEN p ) ) This overall expression is an example of a tautology, or in other words an expression that s guaranteed to evaluate to true, regardless of the truth values of any operands involved (here p and q). What s more, it s a tautology of the following special form: x IFF y Tautologies of this form can be taken as identities, in the sense that the expressions x and y are clearly logically identical they re guaranteed to have the same truth value, regardless of the truth values of any operands they might involve. And, very importantly, such identities can be used in transforming expressions: Any expression involving x can be transformed into a logically equivalent expression by replacing all appearances of x by y. Again as you probably know, such transformations (among others) lie at the heart of the query rewrite process that s performed by relational DBMS optimizers. Now, we can, of course, write down on paper any logical expression we like. Here are a few examples (I ve numbered them for purposes of subsequent reference): 1. IF p THEN q 2. IF ( p AND q ) THEN p 3. IF p THEN ( p OR q ) 4. p OR NOT p 5. p IFF ( NOT ( NOT p ) ) 6. ( p IFF q ) IFF ( ( IF p THEN q ) AND ( IF q THEN p ) ) 7. p AND NOT p
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1. Elsewhere (e.g., in my book An Introduction to Database Systems, 8th edition, Addison-Wesley, 2004), I ve said the Sheffer stroke corresponds to NOR, not NAND. I don t accept full responsibility for this mistake, however! the logic text I was using at the time got it wrong, too.
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