Truth values and modes

In classical two-valued logic, there are only two truth values and one mode. Every statement is judged as either true or false. While other statements may exist, classical logic excludes them from discourse, doesn’t talk about them, and doesn’t even admit they exist.

While most of us can agree on what “true” or “false” means, there is no single word that clearly expresses the third logical value. Uncertain, undecided, unknown, debatable, and contingent all have meanings or connotations that may be misleading. The clearest understanding of what it means is developed by the behavior of the various logical expressions.

Statements such as “Roses are red”, or “Wanda is a fish” or “1+1 = 2” or “Frodo Lives” are assigned letters, such as P, Q, R, S, and truth values, T, U, and F.

So far, so good. That’s the easy part. Unlike classical two valued logic, the three valued logic allows us to make statements about statements. These are referred to as “modes”, and there are six possibilities:
P is definitely true: Symbolized []P; true when P has the value T; false if it has the values U or F.
P is possibly true: Symbolized <>P; true when P has the value T or U; false if it has the value F.
P is not possibly true: Symbolized ~<>P; true when P has the value F, false if it has the value T or U.
P is not necessarily true: Symbolized ~[]P; true when P has the value F or U, false if it has the value F.
P is equivocal: Symbolized ?P: True when P has the value U, false if it has the value T or F.
P is dichotomous: Symbolized !P: True when P has the value T or F, false if it has the value U.

I have chosen to use the “box” and “diamond” notation of the Lewis systems of modal logic, although there is a difference from these systems, to be discussed later. In the system defined by Lukasiewicz and extended by Tarski, []P is written Lp and <>P is written Mp.

It seems simple and clear enough so far, but there already pitfalls and problems that await the unwary. I will discuss these, next post.

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