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I am a Math student currently taking my Master's Degree, and last semester I took an introductory course on Mathematical Logic. One of the subjects we covered there was Tarski's truth definition; and at that point, my professor (whose major interests also include philosophy - so I guess he knew what he was talking about) mention that some philosophers praise, almost adore that definition - and also found it very deep.

My question is: why? What are the philosophical questions hidden behind such a natural and expected, if somewhat technical, definition?

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    You can see at least Tarski's Truth Definitions and Truth. The literature on it is huge. Tarski's formal approach to truth is a support for modern form of the correspondence theories of truth, which are "consonant with" common sense views about truth.
    – Mauro ALLEGRANZA
    May 17, 2014 at 18:50

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Tarski's truth definition is very important for philosophical semantics: Since Frege many philosophers interested in the nature of meaning saw a close connection between meaning and truth. After all, the following 'most certain principle' (as Max Cresswell called it) is highly suggestive:

  • For all declarative sentences s, s' and situations w: If s and s' have the same meaning, then s is true in w iff s' is true in w.

So whatever meanings are, it is at least partially clear what meanings do: They determine truth conditions. To get a formal theory of meaning started and formalize this property of meanings philosophers and logicians such as David Lewis, Richard Montague, Max Cresswell and others set out to rigorously define the assignment of truth conditions to possibly complex natural language sentences. This assignment could be conceived as a function from (the syntactic analysis of) sentences to truth conditions. But how to define this function?

This is where Tarski's importance comes in: Tarski showed us how to define such a function by recursion over the complexity of a very simple language (FOL): Define a class of models to interpret the non-logical vocabulary and on this basis recursively define the notion of truth in a model. Lewis et al. solved the problem mentioned above by semantically treating (fragments of) natural languages in the same model theoretic way in which Tarski treated FOL: Define a class of appropriate models that interpret the lexical items of the language and on this basis define the notion of truth of sentences in a model.

Tarski's definition was of course restricted to a quite impoverished language that could not represent a host of natural language phenomena such as intensionality, vagueness etc. But it provided the paradigm that could easily be combined with a more appropriate notion of model, essentially that stemming from higher-order modal logic.

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  • Worth adding in relation to the question 'why?', is that Davidson thought the main ingredients of the Tarski definition - speakers, sentences, truth conditions, and so forth - provided a basis for a new metaphysics (The Method of Truth in Metaphysics). For those interested, Baker and Hacker provided a very extensive critique of the entire concept of truth conditions in Language, Sense and Nonsense, which for me is the locus classicus of this philosophical movement.
    – adrianos
    May 19, 2014 at 13:58
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I'd suggest the relationship is the otherway around. Its well-known that though Philosophy gets results, the results tend to be inconclusive, or rather they lead other natural questions; thus, one finds that truth is elusive.

Its the formalisation of this cliche or adage, that Tarski & Godel achieved with their new methods in mathematical logic that showed their methods had broke new ground in this field that made it significant to mathematics and logic.

Of course 'paradoxical' results like this had already been discovered by Godel - his incompleteness theorems, and by Russell - the set that doesn't contain itself.

It was probably this set results that accelerated the development of analytic philosophy from the positivist doctrine that had arisen from the Vienna Circle. It aligns itself with formal methods, mathematical logic & science and grammar.

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    But Tarski's theory is philosophically interesting; see Douglas Patterson (editor), New Essays on Tarski and Philosophy (2008) and Douglas Patterson, Alfred Tarski: Philosophy of Language and Logic (2012).
    – Mauro ALLEGRANZA
    May 17, 2014 at 19:20
  • @Allegranza:I wasn't asserting that Tarskis results/oeuvre isn't interesting, but placing his work in the larger philosophical context. After all I said that 'their methods had broke new ground' in mathematical logic; and this is philosophy in so far as logic is part of philosophy.
    – Mozibur Ullah
    May 17, 2014 at 20:16
  • The first book you reference looks at:"(1) How are we to understand truth, one of the notions in terms of which logical consequence is explained? What is it that is preserved in valid inference, or that such inference allows us to discover new claims to have on the basis of old? (2) Among what kinds of things does the relation of logical consequence hold? (3) Given answers to the first two questions, what is involved in the consequence relationship itself? What is the preservation at work in 'truth preservation?' (4)Finally, what do truth and consequence so construed have to do with meaning?"
    – Mozibur Ullah
    May 17, 2014 at 20:18
  • I agree with you; my suggestion is that Tarski's approach (very fruitful in mathematical logic and related phil issues) is also important in a more general context; it gives support and elucidate the common sense view about truth as correspondance with the facts.
    – Mauro ALLEGRANZA
    May 18, 2014 at 9:13
  • @Allegranzo: its hard to argue against the correspondance theory of truth, which I think was first considered by Aristotle; but one might want to ask how does one establish what the facts are, which is also a common-sense view - and which one might lead to the notion of the experimental sciences; and there is of course the BHK correspondance - where the one justifies the correspondance with an account of why it matches the facts - the intuitionistic version of Tarskis definition.
    – Mozibur Ullah
    May 18, 2014 at 15:19
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The best explanation of Tarski's truth theory and it's significance for philosophy and semantics is in Kirkham's book Theories of Truth. It is too long to summarize here, but it has to do with Tarski's endorsement of a doctrine called physicalism; the idea that all facts, even facts about semantics, can be reduced to physical facts. There is a whole chapter about it.

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