Tuesday, September 23, 2014

First of a series of posts on Tarski

I am working on a paper which will include this material. It will take several posts just to share the part about Tarski and the philosophical significance of his work.  The footnotes are (obviously) incomplete.

Truth: Tarski’s contribution and its interpretation
In “The Concept of Truth in Formalized Languages”[i] [henceforth “Tarski (1933)”], Tarksi immeasurably advanced our grasp of the formal logic of the predicate “True sentence of L”, where L is a formalized language, and by doing so he provided powerful new tools for mathematical logicians (something which is under-appreciated by philosophers), as well as material for philosophers to assimilate and interpret.  The tools for mathematical logicians that I refer to include techniques that were later adapted to formalize the statement that a set is definable over a given collection of sets, with parameters from that collection and quantifiers ranging over that collection (tools that Gödel employed in the construction of the hierarchy of  “constructible sets” that he used to prove the consistency of the Axiom of Choice and the Generalized Continuum Hypothesis[ii]), and that have been employed in many contexts ever since; the material for philosophers to assimilate and interpret includes the T-Schema or truth schema in section §1 of Tarski (1933) (not to be confused with “Convention T”), Convention T, and the “truth definitions”, as they have come to be called, that are required to meet that criterion of adequacy[iii]. Although a minority of philosophers,  including Popper, saw Tarski (1933) as putting forward a correspondence theory of truth, other philosophers have linked Tarski(1933) to “deflationary” theories of truth, while the particular techniques Tarski employed[iv] led Hartry Field[v] (correctly in my opinion) to see Tarkski as finding an intimate link between reference (or, in Tarski’s - and Field’s - terminology, “denotation”) and truth; an insight that Field unfortunately linked to an unworkable “naturalization project” (to use Burge’s term), and later abandoned completely. The next two sections elaborate on these remarks.
Tarski’s Criterion of Adequacy
Following Tarski, Let us pretend that we have a particular interpreted formal language in mind, call it “Bob”. I choose an obvious proper name, “Bob”, and not “L”, because “L” looks like a variable – a variable for quantifying over formal languages – and it is essential to Tarski’s approach that truth definitions are constructed one language at a time. Tarksi (rightly) does not attempt to construct a definition of “true in L” when L is a variable over all languages, and doing so would immediately lead to paradoxes. We will assume that Bob has a finite number of primitive descriptive predicates and individual constants, etc., and, to avoid any appearance of circularity or question-begging, that none of these is interpreted as referring to semantical properties or entities, such as truth or reference or “correspondence” or facts or states of affairs. Let “Meta-Bob” be a second formal language in which it is possible to quantify over inscriptions (sequences of symbols) in the language Bob and permit meta-Bob to be set-theoretically more powerful than Bob in the ways needed for Tarski’s purposes, but, importantly, not to have any primitive descriptive predicates or names other than those of Bob itself. A “truth predicate” for Bob is a formula FORM(x) of meta-Bob with one-free variable, say “x”. I avoid the custom of abbreviating the formula as “True-in-Bob” (or even worse, as “True-in-L”), because doing so obscures the very issues I wish to raise. Of course, in ordinary language there is such a predicate as “true in Bob”, with no hyphens, but that predicate invoves the ordinary language notion of truth, a Tarski claimed is inconsistent, and that is the reason that we need a formal replacement for it.  “Defining truth in Bob”, in Tarski’s sense, means (1) finding a FORM(x) that is true of all and only those inscriptions x  in Bob that are true in Bob; and, in addition, (2) reformulating this requirement in a way that does not involve using the everyday language word “true” or any related notion. Tarski’s solution to this latter desideratum, defining “material adequacy” of a truth definition without using the word “true”,  is the famous Convention T, which we state by saying:
(T) FORM(x) is a materially adequate truth predicate for Bob just in case
For all sentences s of Bob, s iff  FORM(s*) is a theorem of Meta-Bob, where s* is a structural-descriptive name of the inscription s.
[Tarski’s notion of a “structural-descriptive name” of an inscription is easiest to explain via an example: if s is “Snow is white”, then the structural-descriptive name s* is “The letter capital S followed by the letter n followed by the letter o followed by the letter w followed by space followed by the letter I followed by the letter s followed by space followed by the letter w followed by the letter h followed by the letter i followed by the letter t followed by the letter e”.[vi]]
- Thus, pretending that “Snow is white” is a sentence of Bob:
Convention T requires that FORM(x) be defined in Meta-Bob in such a way that
“Snow is white iff FORM(The letter capital S followed by the letter n followed by the letter o followed by the letter w followed by space followed by the letter i followed by the letter s followed by space followed by the letter w followed by the letter h followed by the letter i followed by the letter t followed by the letter e) is provable from Meta-Bob’s axioms
- Readers who have gone through life believing that “Convention T” is a Davidsonian “T-sentence” or that it is an instance of “disquotation” will be shocked to observe that “true” does not occur in Convention T! (Although abbreviating Form(x) as “True-in-Bob(x)”) can make this hard to see, which is why I didn’t do that.

[i]  COMPLETE REF ("The Concept of Truth in Formalized Languages"...)
[ii] Gödel, Kurt (1940). The Consistency of the Continuum Hypothesis. Annals of Mathematics Studies 3. Princeton, N. J.: Princeton University Press. ISBN 978-0-691-07927-1.
[iii]  In both the German and the original (1933) Polish of Tarski’s “Der Wahrheitsbegriff in den formalisierten Sprachen”, “Convention T” is called a “criterion” and not a “convention”. “Convention T” comes from the 1956 English translation, T being the first letter of the English word “true” and W being the first letter of the German word “Wahr” [The Polish word for "true" is "prawda", pronounced pravda]
[iv] For a self-contained account, see http://plato.stanford.edu/entries/tarski-truth/
[v] Hartry Field,  “Tarski's Theory of Truth”, The Journal of Philosophy, Vol. 69, No. 13 (Jul. 13, 1972), pp. 347-375.
[vi] See Tarski(1933), p. 157

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