I need to go back to the Principle of the Excluded middle. I say principle, because it's not a law, here: it's a contingent statement which applies to some propositions but not others. It comes in two forms: bivalence (p v ~p), and noncontradiction ~(P & ~P). In classical logic, these are equivalent. Both of these and their equivalence have been frequently challenged by various logicians or philosophers, but few people take these challenges seriously. In this logic, these two formulations are equivalent, but the necessity of the excluded middle may be either asserted or denied. The assertion, "It is necessarily (certainly) the case that either P or not P", or equivalently "It is not possible for both P and not P)" marks a dichotomously uncertain statement, one which must be either true or false: not neither and not both, although it may not be known which is actually the case. [](P v ~P) = ~<>(P & ~P) = !P. The denial, "it is not necessarily the case that either P or not P", or equivalently "It is possible for both P and not P", marks an equivocally uncertain statement, one with the middle truth value. ~[](P v ~P) = <>(P & ~P) = ?P.

When arguments are symbolized, one finds that that the middle may be consistently be either included or excluded, but the logic strictly enforces consistency once the choice is made. It is obviously inconsistent, and in fact results in a genuine contradiction, to allow the use of the middle truth value on the one hand, and then reassert the excluded middle in one of its forms, on the other. Yet the temptation to do so is both insidious and ubiquitous. More than one of the arguments that have been advanced against three valued logic employ just such an argument.

Perhaps more importantly, when I examined the various versions of 3VL, I found that several of them could be expressed in terms I had defined, which made this a more general system. And then, when I was looking at their connectives searching for such a definition, I noticed a certain definition of equivalence and said, "Hey, wait a minute! That's not an equivalence, that's only a biconditional!" Mathematically, an equivalence relation is reflexive, symmetric, and transitive, and these "logical equivalences" were none of those. It should also express the idea that two formulas should have the same truth value, and they didn't do that, either. Only one of them did (I believe it was Kleene's system, the one that had the conditional I had long ago discarded as inadequate). I had a use for that definition, and so I appropriated it.

At this point, I had a partially functional logic. I could establish commutativity, associativity, and the distributive laws for Conjunction (&) and disjunction (v); I had double negation, De Morgan's laws, and the interconversion of the modal functions, and the law of the contrapositive. I had my two types of uncertainty, and their behavior with respect to the other operators. Now I could add properties of equality (If P=Q, then ~P = ~Q), and properties of equality (If P and P=Q then Q; if P = Q and Q=R then P=R), which was an advance.

It also gave me an intepretation for the Lukasiewicz conditional: I could define it as (~P v Q v P = Q), which was curious, but didn't strike me as particularly useful or profound. And then, after I don't remember how long, I noticed that I didn't need a separate definition for equivalence. I could get it by applying necessity (or certainty) to the Lukasiewicz biconditional I was already using. P = Q = [](P <-> Q)

And then, on the basis that what was good for the biconditional was good for the conditional, I decided to define a strict Lukasiewicz conditional, P => Q as [](P -> Q), removing the uncertainty.

Duh. Of course. Obliviously. And the light came on, and suddenly I understood more than I had ever dreamed of, or anyone will believe.

## Monday, May 23, 2005

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