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The Unknown Truth Value and A-marks
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There s one point that might be bothering you in connection with the foregoing discussion. To be specific, you might be thinking that the fact that p IFF p gives unknown, not true, if p is unknown is no big deal; after all, isn t it rather like the situation we already have in SQL according to which the comparison v = v gives unknown, not true, if v is null Well not to mention the fact that I have major problems with the fact that null = null gives unknown anyway the two situations aren t the same at all. In the RM/V2 book, Codd talks about what he calls A-marks, which are essentially the same as SQL-style nulls, and he makes it very clear that A-marks aren t values. At the same time, he makes it very clear that the third truth value definitely is a value; as the RM/V2 book puts it, t, f, ... and a are actual values,
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CHAPTER 18 WHY THREE- AND FOUR-VALUED LOGIC DON T WORK
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and should not be regarded as marked values. Now, by marked values here, Codd means, among other things, A-marks, which he had previously stated explicitly aren t values at all; thus, marked values is actually a contradiction in terms. I don t really want to pursue that point here, though I do think that careful analysis of the term marked value leads inexorably to the position that many-valued logics don t solve the problem of missing information at all. But never mind that; the point is, as Codd does make clear, the third truth value and A-marks are logically different things. It follows that to confuse them is a logical mistake. The following is an edited version of a discussion of this point from my book An Introduction to Database Systems, 8th edition (Addison-Wesley, 2004):3 It s important to understand that an A-mark (an SQL null, if you prefer) and a (the unknown truth value) aren t the same thing. Indeed, this state of affairs is an immediate consequence of the fact that a is a value, while an A-mark isn t a value at all. But let s be a little more specific. Suppose X is a variable of type truth value. Then X must have one of the values t, a, or f. Thus, the statement X has the value a means the value of X is known to be the unknown truth value, a. By contrast, the statement X is A-marked means X has no value at all (because we don t know what value to assign to X, and therefore we don t assign a value to it but mark it instead). To repeat, to confuse A-marks and a (or nulls and UNKNOWN, in SQL terms) is a logical mistake. Indeed, using an A-mark instead of a to represent the third truth value is exactly as serious an error as using an A-mark instead of 0 to represent zero! As Hugh Darwen and I have argued in many places (see, e.g., my paper On the Notion of Logical Difference, 4 in the present book), logical mistakes are big mistakes and, sadly, this particular big mistake has been committed both by the SQL standard and by Codd himself, in the RM/T paper.
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How Many 3VLs Are There
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You might have been a little surprised by my earlier use of the phrase three-valued logics in general. How many 3VLs are there Isn t there just one Well, in a sense there is just one. Let s think about the connectives again. As I said earlier, there are exactly as many distinct connectives as there are distinct truth tables, and of course this remark is just as true of 3VL as it is of 2VL. Since 3VL has three truth values, there are 3 * 3 * 3 = 27 monadic 3VL connectives, because each of the three possible inputs t, a, and f can map to each of the three possible outputs t, a, and f. And there are 39 = 19,683 possible dyadic 3VL operators, as the following table suggests:
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3. Honesty compels me to point out that the text quoted, if carefully analyzed, fails the coherency test. To say that a variable has no value is to say, by definition, that the variable is not a variable after all as can easily be seen by checking the definition of the term variable. The truth is, the very act of trying to state precisely what nulls and 3VL are all about is (or should be) sufficient to show why the idea is not exactly coherent. As a consequence, it s hard to explain it coherently, too. In fact, the entire topic illustrates perfectly something I ve elsewhere called The Principle of Incoherence (sometimes referred to, unkindly, as The Incoherent Principle) viz., it s hard to criticize something coherently if what you re trying to criticize is itself not very coherent in the first place.
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