# Equivalence and Tarski

I am currently writing a paper on Tarski's Semantic Concept of Truth. His T-schema is as follows:

'X' is true if, and only if, 'p'

Where 'p' is a sentence such as "snow is white" and 'X' is the name of a sentence. If snow is white is used as an example (as Tarski did), the sentence reads:

"Snow is white" is true if, and only if snow is white.

Now, my question is to do with equivalence. Tarski calls all instances of this schema equivalences of the form 'T'. It is my opinion that Tarski's schema is not a true equivalence as it is not expressed as a relation between two things, but a sentence and it's name, and the relation is not clearly expressed. (Bela von Juhos made this objection to Tarski's theory - 1937 - and Adrian Heathcote has also made the same objection). The consequences of treating Tarski's theory as a true equivalence seem to be grave for Tarski's theory... I'm wondering what other people think of this, it also made me wonder whether a biconditional is the same as an equivalence? I'm a little confused... Let me know if more information or clarification is necessary. I found it an interesting topic for discussion :)

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Why isn't it an equivalence? On the left we have a state of affairs, the truth of some sentence, and on the right we also have a state of affairs, that some proposition holds. – Benedict Eastaugh Jun 28 '12 at 4:26
In multi-valued and infinite valued logic, A=B does not necessarily imply that (A->B) as well as (B->A), nor that (A<->B), where "=" means that A and B have the same truth value. See here for example en.wikipedia.org/wiki/…. – Doug Spoonwood Jun 28 '12 at 17:49
My earlier comment points out how an equivalence may not be a bi-conditional. – Doug Spoonwood Sep 23 '12 at 21:17

Tarski pioneered the crisp separation between syntax and semantics that ushered in modern mathematical logic. The definition of truth provides the vital link between syntax and semantics that is at the heart of Model Theory. There is no element of controversy about the value of Tarski's definition.

The definition is neither a biconditional nor an equivalence in the usual sense. It is not a biconditional because biconditional is a syntactic notion. I would not describe it as an equivalence because it does not meet the criteria for an equivalence relation. However, that is a weak argument: a number of mathematical terms, such as congruence, have meanings that are context-dependent.

Tarski's definition of truth, in its mathematical version, is a definition that looks much like any other mathematical definition.

Note that the above answer does not attempt to say anything about any philosophical content that the definition may have.

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I have to wonder how many of Tarski's ideas came from St. Lesniewski. – Doug Spoonwood Aug 28 '13 at 23:28
I think of them as being in essentially different fields. But to the degree that Tarski was interested in philosophy, perhaps a lot. – André Nicolas Aug 28 '13 at 23:43

Perhaps formalising the T-schema will help clarify this point. Given a truth predicate $T$ and a Gödel coding for formulae $\varphi$ of some formal language $\mathcal{L}$, the T-schema is given as follows:

$$T(\ulcorner \varphi \urcorner) \longleftrightarrow \varphi.$$

So on the left-hand side of the biconditional we have that the predicate $T$ holds of a code for a sentence, and on the right-hand side that some formula $\varphi$ holds. So the equivalence here is between the truth of $\varphi$ and $\varphi$ obtaining, not between the code of $\varphi$ and $\varphi$ itself.

In Tarski's original formulation each sentence had a name, rather than a Gödel code, but this is less perspicuous since it involves extending our language in a rather unparsimonious way. Using codes for sentences to name them expresses more directly the relationship between the sentence and how we refer to the sentence.

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In the example 'snow is white' the conditionals are implied. That is to say the truth-function (snow is white) is elementary. For example the sentence S: 'the chair has two legs' could be evaluated as A: 'the char has one leg' and B: 'the char has another leg'. A sentence is true if and only if the evaluation of a sentence and the fundamental truth functions leads to T. S is true if (A and B) are true. This, one could say, is the idea of logic analysis itself. We analyze the component of every sentence, so that the expression becomes elementary. The idea is that this analysis is just like the analysis of mathematical expressions.

What the conditions of the individual truth functions is, is not the concern of logic. Wittgenstein's tractatus was an attempt at a complete description of the equivalence of language and the world, and how we form logical statements. Tarski's program was much more constrained as far as I remember it.

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I like this answer. But, as soon as you take this seriously, you'll come to realize that Tarski would do better to have another example for people interested in classical logic. "Snow is white" can get analyzed into "patch A of snow is white, patch B of snow is white, ..., patch Z of snow is white." Not all of those patches of snow are white, therefore, snow is not white. A better example, in my opinion, would be "a clean sheet of blank computer paper is white." Of course, that also will fail such a method eventually, but it does work better. – Doug Spoonwood Sep 23 '12 at 21:27