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.