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Two serious problems arise immediately. In ordinary discourse, and in 2VL, the expression IF p THEN p ( p implies p ) is clearly true for all p; in fact, it s a tautology. For example, if it s raining, then it s raining is clearly a true statement. But with IF as just defined, IF p THEN p is unknown ( the third truth value ) if p in turn has truth value unknown. Note: Since we ve effectively just defined IF p THEN p to be equivalent to (NOT p) OR p, therefore, we have the situation that this latter expression, which is a simple and well-known tautology in 2VL, isn t a tautology in 3VL. It follows that not all tautologies in 2VL are tautologies in 3VL. Likewise, not all contradictions in 2VL are contradictions in 3VL (consider p AND NOT p, for example). It seems to me that these are facts that ought to give us some pause. In ordinary discourse, and in 2VL, the expression p IF AND ONLY IF p ( p is equivalent to p ) is clearly true for all p (indeed, it s another tautology). For example, it s raining if and only if it s raining is clearly a true statement. But with IFF as just defined, p IF AND ONLY IF p is unknown ( the third truth value ) if p in turn has truth value unknown, and remarks similar to those of the previous bulleted paragraph apply again here. In connection with the second of these points, incidentally, I note that according to page 138 of Nicholas Rescher s book Many-Valued Logic (McGraw-Hill, 1969) the standard text on the subject, hereinafter referred to as the Rescher book equivalence is required to be reflexive,
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CHAPTER 18 WHY THREE- AND FOUR-VALUED LOGIC DON T WORK
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meaning that p IF AND ONLY IF p is indeed required to be true for all p; but this requirement is violated by 3VL as here described, if p has truth value unknown (i.e., a). Now, one way to address the foregoing difficulties might be to define implication and equivalence differently, in order to guarantee that IF p THEN p and p IF AND ONLY IF p do indeed evaluate to true for all p:
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With these definitions, the expressions IF p THEN p, p IF AND ONLY IF p, and ( p IFF q ) IFF ( ( IF p THEN q ) AND ( IF q THEN p ) ) are indeed all tautologies, as required. Sadly, however, this one isn t: ( IF p THEN q ) IFF ( ( NOT p ) OR q ) What are the consequences of this state of affairs One is that users are likely to make mistakes in formulating queries (because they probably think that this last expression is a tautology, or in other words that the identity does hold). Another is that the optimizer is likely to make incorrect expression transformations, for essentially the same reason. Either way, the system is likely to deliver wrong answers. Note: As you might be aware, I ve made the foregoing point in different ways in many other writings. I could sum it up by saying that the pragmatic problem with three-valued logics in general is not that they can t be defined and implemented in a consistent way, but rather that the answers they say are correct are not always correct in the real world. A couple of simple examples are discussed in my book Database in Depth: Relational Theory for Practitioners (O Reilly Media Inc., 2005), pages 54 56. What s more, a literally infinite number of analogous examples can be constructed to illustrate the same overall point. The nub of the matter is this: All bets are off! You can never trust a system that s based even if, unlike today s SQL systems, it s correctly based on some 3VL, because you never know whether what the system is telling you is true. In my view, this state of affairs is a complete showstopper.
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