This is a continuation of a series of blog posts on a three valued logic, and employs concepts and notations previously described,
beginning with “The Learned Professors”
Classical logic has one form of negation; a three valued logic has at least three.
~P signifies “It is not the case that P”, and takes a value of F where P has a value of T and T where P has a value of F. The question of what to do when P has a value of U naturally arises. The standard negation gives this a value of U. This sounds reasonable, since when “the cat is black” is uncertain or unknown, so is “the cat is not black”.
However, it is also possible to define a strong negation, in which statements with the value of U are assigned a value of F. This sense of negation is used in intuitionistic logic. I do not use a separate symbol for it, since this can be represented by ~<>P.
Similarly, a weak negation can be defined, in which statements with the value of U are assigned a value of T. Likewise, this can be represented by ~P. In evaluating expressions, these are evaluated in order right to left so that one closest to the valuable is evaluated first.
It may be observed that ~<>P has the same table as ~P, and ~P as <>~P. This is one of the oldest and earliest expressions of modal logic and dates to Aristotle. “Not possible” is equivalent to “necessarily (or certainly) not”, and “Not necessarily” is equivalent to “possibly not”.
It gets a little trickier when the other modes are considered. !P = !~P; The negation of a dichotomously uncertain statement is also dichotomously uncertain, and ?P = ?~P, the negation of an equivocally uncertain statement is also equivocally uncertain. However, the negation of claim that a statement is equivocally uncertain is equivalent to a claim that the statement is dichotomously uncertain, ~!P = ?P, and the negation of a claim that a statement is dichotomously uncertain is equivalent to a claim that the statement is equivocally uncertain. Since ordinary language does not distinguish equivocal and dichotomous uncertainty, it is easy to shift definitions and thus arrive at contradiction and confusion. This is a cases where the symbolism helps clarify the discussion, and aids in the distinction between use and mention of a claim. Also, !!P and !?P, but ~?!P and ~??P, or in words, the claim that a statement is dichotomous (either true or false) is itself dichotomous, and the claim a statement is equivocal (possibly true and possibly false) is also dichotomous. The claim that a statement is dichotomously uncertain may not equivocal, and the claim that a statement is equivocally uncertain may not itself be equivocal. Natural language does not make these distinctions or enforce these rules, but in order to avoid contradiction and confusion, the three valued logic must. This is more or less equivalent to a law of the excluded fourth. Three valued logic, like two-valued logic, is a tool with its own uses and limitations, not a cure-all for all philosophical or logical conundrums.