Tarski’s result on the undefinability of truth

Assume $𝐋$ is a logic which is under contradictory negation and has the usual truth-functional connectives. Assume also that $𝐋$ has a notion of formula with one variable and of substitution. Assume that $T$ is a theory of $𝐋$ in which we can define surrogates for formulae of $𝐋$, and in which all true instances of the substitution relation and the truth-functional connective relations are provable. We show that either $T$ is inconsistent or $T$ can’t be augmented with a truth predicate $\mathrm{𝐓𝐫𝐮𝐞}$ for which the following T-schema holds

$$\mathrm{𝐓𝐫𝐮𝐞}{(}^{\prime }{\varphi }^{\prime })\leftrightarrow \varphi$$

Assume that the formulae with one variable of $𝐋$ have been indexed by some suitable set that is representable in $T$ (otherwise the predicate $\mathrm{𝐓𝐫𝐮𝐞}$ would be next to useless, since if there’s no way to speak of sentences of a logic, there’s little hope to define a truth-predicate for it). Denote the $i$:th element in this indexing by $B_i$. Consider now the following open formula with one variable

$$\mathrm{𝐋𝐢𝐚𝐫}(x)=\mathrm{\neg }\mathrm{𝐓𝐫𝐮𝐞}(B_x(x))$$

Now, since $\mathrm{𝐋𝐢𝐚𝐫}$ is an open formula with one free variable it’s indexed by some $i$. Now consider the sentence $\mathrm{𝐋𝐢𝐚𝐫}(i)$. From the T-schema we know that

$$\mathrm{𝐓𝐫𝐮𝐞}(\mathrm{𝐋𝐢𝐚𝐫}(i))\leftrightarrow \mathrm{𝐋𝐢𝐚𝐫}(𝐢)$$

and by the definition of $\mathrm{𝐋𝐢𝐚𝐫}$ and the fact that $i$ is the of $\mathrm{𝐋𝐢𝐚𝐫}(x)$ we have

$$\mathrm{𝐓𝐫𝐮𝐞}(\mathrm{𝐋𝐢𝐚𝐫}(i))\leftrightarrow \mathrm{\neg }\mathrm{𝐓𝐫𝐮𝐞}(\mathrm{𝐋𝐢𝐚𝐫}(𝐢))$$

which clearly is absurd. Thus there can’t be an of $T$ with a predicate $\mathrm{𝐓𝐫𝐮𝐭𝐡}$ for which the T-schema holds.

We have made several assumptions on the logic $𝐋$ which are crucial in order for this proof to go through. The most important is that $𝐋$ is closed under contradictory negation. There are logics which allow truth-predicates, but these are not usually closed under contradictory negation (so that it’s possible that $\mathrm{𝐓𝐫𝐮𝐞}(\mathrm{𝐋𝐢𝐚𝐫}(i))$ is neither true nor false). These logics usually have stronger notions of negation, so that a sentence $\mathrm{\neg }P$ says more than just that $P$ is not true, and the proposition that $P$ is simply not true is not expressible.

An example of a logic for which Tarski’s undefinability result does not hold is the so-called Independence Friendly logic, the semantics of which is based on game theory and which allows various generalised quantifiers (the Henkin branching quantifier, etc.) to be used.

Title	Tarski’s result on the undefinability of truth
Canonical name	TarskisResultOnTheUndefinabilityOfTruth
Date of creation	2013-03-22 13:49:19
Last modified on	2013-03-22 13:49:19
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	14
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 03B99
Related topic	IFLogic