## Wednesday, August 27, 2008

A More Direct Attack

So, as I noted last time, there are many possible ways to fit the predicate "True" into a language. Which of the many ways is correct? What does that even mean? What makes one better than another? Or, are all of them OK? Is it merely a matter of taste, since all of them capture the normal everyday uses of the concept of truth? In other words, is "truth" merely a concept that might be used in slightly different ways by different people, meaning there is no right answer?

I also noted last time that these logics do not seem to automatically solve the other problems I'm worried about; in particular, they do fairly little in the way of providing a foundation of mathematics.

The goal is to create a logic that can talk about its own semantics; a logic that can reference any mathematical ideas that were needed in describing the logic in the first place. Tarski started the whole line of research in this way; he asked, if a logic could define its own notion of truth, then what? (And he showed that, given some additional assumptions, such logics turn out to be contradictory.) But the theories I referenced last time, and those like them, have diverged somewhat from this line of inquiry... instead of providing a logic capable of defining its own truth predicate based on the semantic notions a human would employ in explaining the logic to another, these theories simply put the truth predicate in the language to begin with, and attempt to state rules that make it behave as much like our concept of truth as possible without allowing Tarski's contradiction.

That's totally different!

So, a more direct approach would be to try to create a logic that can define itself; a logic that can develop its notion of truth from more basic concepts, rather than declaring it as given. I suspect that such an attempt would not result in the same multitude of possible solutions.

The Logic of Definitions

On a related train of thought-- I have come up with another idea for a logic. The idea was based on the observation that many of the proposed solutions to defining truth were actually proposals for a theory of definitions in general; truth was then treated as a defined term, using the dfollowing definition, or some variant:

" "X" is true" means "X".

The revision theory treats this as a rule of belief revision: if we believe "X", we should revise our belief to also accept " "X" is true". The supervaluation theories claim that the above definition is incomplete (aka "vague"), and formulate a theory about how to work with incomplete definitions. (But, see this blog for a note on the ambiguity of the term "supervaluation".)

So, I have my own ideas about how to create a logic of definitions.

--The base logic should be normal classical logic. It could simply be first-order logic if that is all you need in order to define your terms; it could be arithmetic if that is needed; or it could be axiomatic set theory, if you need it to define what you want to define. In other words, the basic theory will depend on what you want to talk about with your definitions, but the logic that theory is set in will be classical.
--The logic of the defined concepts, however, will be violently non-classical. A defined predicate may be neither true nor false in some cases, because its definition simply fails to say anything one way or the other. It could be both true and false in other cases, when the definition implies both.

This is, of course, only a rough outline of the logic... it differs from most treatments of definitions (such as revision theory and supervaluation theory) by allowing not just statements that are assigned no value, but also statements that get both truth values.