Logical Musings #6

About this time I transferred from Phoenix College to ASU, and while I was there, I took the opportunity to do some extracurricular library digging. I found a bibliography of the history of symbolic logic in the Journal “Symbolic Logic” and troubled myself to look up references. One of them that I found was an early attempt at a logic of indecision. This didn’t work because it was inconsistent: a calculation done one way and a calculation done another way came up with different results. I believe I know why, now, but It would take some digging (on the order of an archaeological research with a grant and a team of undergraduate laborers) to find the references again. I also looked up the work of C. I. Lewis as published in Symbolic Logic, by Lewis and Langford, 1932, and found the axioms for his systems. I also found his definition for the “Strict implication” he used in his axiomatic systems, and that the had examined the same Lukasiewicz three valued logic I had been working with, and found it deficient because a a couple of important logical laws were missing. I will return to this subject later.

I also took a course on mathematical structure, which considered how mathematical object can be defined, the relations among them and the properties of those relations, and the mathematical operations that can be performed on them. I didn’t think to apply this to mathematical logic at the time, that came later, but it appears that it has not occurred to many logicians or philosophers, either. But this subject lay fallow for a while.

I believe it was after I left ASU that I began seriously flogging the algebra to discover why it was yielding some seriously paradoxical results. Let us take, for example “Charlie is a boy” versus “Charlie is a girl”, and what happens if we give this a truth value of U, for undecided. The algebra gives “Charlie is a boy and Charlie is a girl” a truth value of undecided. But what if these are mutually exclusive condition? Should not the expression be false? This point was raised by Lewis and it was his decision to incorporate the premise that they would be into his logic. As it turns out, the reason is that the “law” of the excluded middle is so easily and often taken for granted that we try to sneak it in anyway, even when the logic explicitly rejects it as a law.

As it turns out, the concept “The fribbles are thurgish” is undecided (to use an absurdity for illustrative purposes) can be mean two different things. It could mean that the thribbles are thurish is either true or false, but we don’t know which, or it could mean that there is an intermediate state between thurgish and not thurgish. Another example? “The door is open”. We could mean that the door is either definitely open or it is definitely closed, but we don’t know which is the case, or it could mean that the door is ajar, an intermediate state between fully open and fully closed. Another one. “it is raining”. We may not know which is the case, or there may be states of the weather that make it difficult to determine which is this the case.

The logic makes it clear that these are two different cases. One of them, which I call the “Dichotomous” sense, can be symbolized, but does not have a single truth value. The other sense, which I call the equivocal sense, can be symbolized and does have the single truth value U. What makes it tricky is that the negation behaves the same in both sases. The negation of a dichotomous statement “It is raining” and evaluated as true or false, would be naturally be “it is not raining”, and evaluated as false or true. If “The door is open” is considered equivocal because the door is ajar, “the door is not open” is also equivocal for the same reason. Without the symbolism to make the distinction clear, these two different kinds of uncertainty can be easily confused. “if it is possible that some proposition is true and it is possible that that proposition is not true then that proposition is equivocal. ” This is a theorem of the logic I use and I can prove it.

For practical purposes, the algebra works the same for either interpretation. What you cannot do, however, is used “undecided” as a logical value on one hand, and then turn around and try to invoke the law of the excluded middle on the other. BLAATTT! REEE!. Violation! If you’re going to say “Charlie is a girl” is undecided, you must also admit Charlie is a girl and Charlie is not a girl (i.e. a boy)” to also be undecided. You don’t have to accept such a contradiction as true. You just have to refrain from claiming it to be impossible. There is no problem with setting up a condition “Charlie is a girl” if and only if “Charlie is not a boy”. Once you know one, you know the other, and the case where they are equivocal slides through because they have the same truth value, even though they mean different things. But if you do insist ahead of time that it is impossible to say both “Charlie is a girl” and “Charlie is not a girl” then you have no business using three valued logic at all and the ordinary two valued case is sufficient.

The logic admits no compromise. Either you accept the law of the excluded middle as a general law, or do not. No, rejection of the law of the excluded middle doesn’t mean that it is never true. It only means that you can’t count on for all kinds of statement. In this particular case and a few others, it bites down hard.

“Mother may I go out to swim? Yes, my darling daughter. Hang your clothes on a hickory limb, but don’t go near the water. “

Your intuition may lead you astray until you have properly trained it. The algebra is consistent.

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