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CHAPTER 18 WHY THREE- AND FOUR-VALUED LOGIC DON T WORK
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4VL Number 3
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Although I m not aware of any generally available publication in which he documented the fact, I have it on good authority that Codd subsequently revised his 4VL tables again, thus:
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NOT t a i f f i a t
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OR t a i f
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t a i f t t t t t a a a t a i i t a i f
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t a i f t a i f a a i f i i i f f f f f
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Here the change is in the table for NOT NOT a and NOT i are now defined to return i and a, respectively, instead of (as formerly) a and i. What effect does this change have Well, again it s easy to see that (NOT p) OR q still fails to give the definition we d like for implication, and ((NOT p) OR q) AND ((NOT q) OR p) still fails to give the definition we d like for equivalence. However, De Morgan s Laws do now work, and I think it not unlikely that this fact was Codd s justification for defining his third 4VL the way he did. But De Morgan s Laws aren t everything, of course. I have a more formal criticism of Codd s third 4VL. Returning for a moment to three-valued logic, it s easy to see that Codd s 3VL truth tables for NOT, OR, and AND reduce to those for two-valued logic if we simply delete the rows and columns corresponding to the third truth value a. However, no analogous property holds for Codd s third 4VL. To be specific, if we delete the rows and columns for the fourth truth value i from the 4VL truth tables for NOT, OR, and AND, we do not obtain the corresponding 3VL tables; to be more specific still, we re left with the fact that NOT a is defined to return the fourth truth value i, a truth value that doesn t exist at all in Codd s 3VL.
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Some Questions of Intuition
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Most of this chapter has been concerned with various formal properties of the logics under discussion. In this final section, however, I want to raise some questions of a more intuitive nature. Recall that the A-mark is supposed to denote a value that s missing because it s unknown, while the I-mark is supposed to denote a value that s missing because it doesn t apply, and the two truth values a and i are introduced as corresponding truth values. Now, I deliberately didn t try to explain previously what it might mean for a and i to correspond to A-marks and I-marks, respectively. That s because I m not sure I can! It s quite difficult to find a clear statement on the matter in the RM/V2 book. However, let me give it a shot. Let X, A, and I be variables (of the same type, so they can be tested for equality), and let X have some genuine (i.e., unmarked ) value while A is A-marked and I is I-marked. Then I think the following, at least, are true statements (though, frankly, it s hard to be sure):
CHAPTER 18 WHY THREE- AND FOUR-VALUED LOGIC DON T WORK
The following expressions all evaluate to i: X = I A = I I = I The following expressions both evaluate to a: X = A A = A So a ( missing and applicable ) is what we get if we ask if an A-mark is equal to anything other than an I-mark, and i ( missing and inapplicable ) is what we get if we ask if an I-mark is equal to anything at all. Observe in particular, therefore, that (like null in SQL) nothing, not even the A-mark itself, is equal to the A-mark, and likewise for the I-mark. Given the foregoing state of affairs, I now come to my questions: First, what intuitive as opposed to, possibly, formal justification is there for defining NOT (missing and applicable) to be equivalent to missing and inapplicable Certainly the equivalence doesn t seem to make much sense in ordinary colloquial English. Likewise, what intuitive justification is there for defining NOT (missing and inapplicable) to be missing and applicable In a similar vein, what intuitive justification is there for defining a OR i to be a and a AND i to be i I don t think formal justifications, even if there are any, will be sufficient to persuade the punters that these rules make sense (remember that all of these matters eventually have to be explained to the na ve end user ). I have a related question, too. Let X and Y be variables of type truth value. In Codd s 4VL, the legal values of X and Y are precisely t, a, i, f. But, of course, each of X and Y might be either A-marked or I-marked. Suppose X has the value t but Y is A- or I-marked. Then what s the value of X OR Y It must be t t OR anything is always t but I strongly suspect that Codd would say it has to be a or i (though again I can t find a clear statement on the matter in the RM/V2 book).
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