Tarski's semantic conception of truth states: $X$ is true iff p (where $p$ is a sentence, and $X$ is the name of the sentence $p$ to which the truth predicate applies).

However, in logic, to express the idea that a sentence $p$ is true, we just say $v(p)=1$.

Does Tarski's semantic conception of truth have any use in formal logic? Put differently, do logicians always implicitly mean $v(p)=1$ iff $p$ when they speak about the truth of $p$?

  • $\begingroup$ Well... there's the completeness theorem. $\endgroup$ – Asaf Karagila Jan 27 '16 at 23:47
  • $\begingroup$ Thanks for the reply, but it is not exactly what I asked. What I want to know is whether or not logicians mean $v(p)=1$ iff $p$ when they speak about the truth of $p$? $\endgroup$ – user60264 Jan 28 '16 at 0:33
  • $\begingroup$ Please provide some context or references. Your question is unanswerable as it stands. $\endgroup$ – Rob Arthan Jan 30 '16 at 1:34

In propositional logic the "semantic values" available for (classical) propositional letters : $p,q,\ldots$ are just two: TRUE and FALSE.

Thus, nothing changes if instead of true and false we use $1$ and $0$.

Things are different with more complex languages, like first order logic.

In this case, to say that a f-o arithmetical sentence like:

$\exists x \ (x < 0)$

is true in the domain $\mathbb N$ of natural numbers amounts to, according to Tarski's definition, to assert that there is a number $n$ (i.e. an element of $\mathbb N$) such that $n$ is less than the number zero, and this is a legitimate (i.e. meaningful) assertion, and it is false.

See First-order languages and structures.


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