Abstract:^[1]

Tarski's Undefinability Theorem showed (under some plausible assumptions) that no language can contain its own notion of truth. This deeply counterintuitive result launched several generations of research attempting to get around the theorem, by carefully discarding some of Tarski's assumptions.

In Hartry Field's Saving Truth from Paradox, he considers and discards a theory of truth based on Łukasiewicz logic, calling it the "most successful theory so far" (out of very very many considered by that point in the book) and enumerating its many virtues. The theory is discarded due to a negative result due to Greg Restall, which shows that a version of the theory with quantifiers must be $ω$ -inconsistent.

We consider this abandonment too hasty. The success of Łukasiewicz logic lies in postulating "between values" of truth, starting at $\frac{1}{2}$ -valued sentences positioned precisely between true and false, and ending in the whole continuum of real values $[0, 1]$ . Why stop there? We propose a variant of the theory based on nonstandard analysis, which avoids Greg Restall's paradox by introducing truth values between the real numbers, such as $\frac{1}{ϵ}$ .

Tarski's Theorem

Tarski investigated the subject of truth predicates. The idea is to have a $T (‘ ‘ s ")$ which is true if $s$ is a true sentence, and false otherwise. ( $T$ is a function that acts on numbers, and $‘ ‘ s "$ is the Gödel encoding of $s$ , a number which encodes the symbol string of $s$ .) This is formalized as Tarski's T-Schema. For each sentence $s$ , we add this axiom to the formal system:

$T (‘ ‘ s ") ⟺ s$

Working in classical logic, Tarski shows that T-Schema is incompatible with a technical assumption called the diagonal lemma. The diagonal lemma says that for any predicate $P (.)$ which is definable, we can construct a sentence $s$ such that the following holds:

$s ⟺ P (‘ ‘ s ")$

That is: $s$ says precisely " $P$ is true of $‘ ‘ s "$ ". (This is equation is different from the previous schema, because it applies to any predicate $P$ instead of just $T$ , but only some special $s$ rather than all sentences).

This sort of self-referential sentence might sound illegal, but a major part of Gödel's contribution in the proofs of his incompleteness theorems was to show that it is actually quite difficult to rule out: the diag...

Market Logic

Market Logic