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# Tarskiโs result on the undefinability of truth

Assume $\mathbf{L}$ is a logic which is closed under contradictory negation and has the usual truth-functional connectives. Assume also that $\mathbf{L}$ has a notion of open formula with one variable and of substitution. Assume that $T$ is a theory of $\mathbf{L}$ in which we can define surrogates for formulae of $\mathbf{L}$, 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 $T$ canโt be augmented with a truth predicate $\mathbf{True}$ for which the following T-schema holds

$\mathbf{True}(^{{\prime}}\phi^{{\prime}})\leftrightarrow\phi$ |

Assume that the open formulae with one variable of $\mathbf{L}$ have been indexed by some suitable set that is representable in $T$ (otherwise the predicate $\mathbf{True}$ would be next to useless, since if thereโs no way to speak of sentences of a logic, thereโs 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

$\mathbf{Liar}(x)=\neg\mathbf{True}(B_{x}(x))$ |

Now, since $\mathbf{Liar}$ is an open formula with one free variable itโs indexed by some $i$. Now consider the sentence $\mathbf{Liar}(i)$. From the T-schema we know that

$\mathbf{True}(\mathbf{Liar}(i))\leftrightarrow\mathbf{Liar(i)}$ |

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

$\mathbf{True}(\mathbf{Liar}(i))\leftrightarrow\neg\mathbf{True}(\mathbf{Liar(i% )})$ |

which clearly is absurd. Thus there canโt be an extension of $T$ with a predicate $\mathbf{Truth}$ for which the T-schema holds.

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

## Mathematics Subject Classification

03B99*no label found*

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## Corrections

Capital T in Truth? by mathwizard โ

Small correction by poohdov โ

Proof is included! by rspuzio โ

The definition of Liar has bad parenthesis by m_hemaly โ