Sunday, September 28, 2014

A second post on Tarski
This post presupposes the previous one!

Field’s reconstruction of Tarski: the link between reference and truth
In his deservedly often-cited “Tarski’s Theory of Truth”[i], Hartry Field argued that Tarski failed to reduce the notion of truth “to other nonsemantical notions”, but that what Tarski actually did was reduce the notion to other semantical notions.
 This is an interesting reading, although I am not certain that Tarski had any such aim. What supports Field’s reading is that  Tarski (1933) begins by describing the ordinary language notion of truth, and that this is where Tarski puts forward the T-Schema[ii] , an instance of which is the famous “Snow is white” is true iff snow is white; what perhaps counts against it is that at the end of that section (section §1) Tarski seems to give up the project of clarifying the ordinary language notion of truth entirely:
“If these observations [concerning the inconsistency of the ordinary language notion – HP] are correct, the very possibility of a consistent use of the expression ‘true sentence’ which is in harmony with the laws of logic and the spirit of everyday language seems to be very questionable, and consequently the same doubt attaches to the very possibility of constructing a correct definition of this expression. [emphasis in original][iii]
In any case, one of Tarski’s purposes, which he achieved, was certainly to find a way to define in Meta-Bob a predicate that has the same extension as the ordinary language semantical predicate “true in Bob” (no matter which interpreted formalized language ‘Bob’ might be); however, the predicate(s) FORM(x),  we get by Tarski’s method do not express the same concept as the semantical predicate “true in Bob”[iv]. In Fregean language, FORM(x) has the right Bedeutung, but not the right Sinn. But by giving two of his papers the titles “The Concept of Truth in Formalized Languages” and “The Semantical Concept of Truth” [emphasis added] Tarski certainly gave the impression that his formal results somehow do capture the Sinn of “true in Bob” (that is that they capture the concept of truth), and not just the extension of that everyday language semantical predicate.
A Technical Section
To explain all this, I need to say something about how one defines FORM(x) following Tarski’s recipe. Simplifying outrageously by pretending that all the predicates of Bob are monadic[v] (the fact that this is in general not the case is the reason that all the mathematical ingenuity in Tarski’s paper was required), the easiest way to do this is to imagine that we construct Meta-Bob in two stages.
(Stage 1) First we start with the vocabulary of Bob, and we add (unless it is already in Bob) enough apparatus to do logical syntax (but not semantics!), that is, logical means for quantifying over strings of symbols such as the letter capital S followed by the letter n followed by the letter o followed by the letter w, and over finite sequences of such strings. We also define such syntactic predicates as “x (a string of symbols) is a wff (well formed formula) of Bob” and “x (a string of wffs) is a proof in Bob”,  “x is a theorem of Bob”, etc.
(Stage 2) We temporarily add to Meta-Bob a primitive predicate Ref(x, y) which we will read as “x (a wff of Bob with one free variable) refers to y”. The interpretation of Ref should be clear from the following example: if x is “F(v)” and  a is an entity in the range of the quantifiers of Bob, then  Ref(x, a) if and only if F(a). Next, we start to create an inductive definition of “Ref” by adding to the axioms (the axioms of Bob plus whatever axioms we need to do syntax and set theory in Meta-Bob) as follows:
For each atomic predicate (say, “Glub(x)”) of Bob, we add as an axiom (y)[Ref(“Glub” followed by “(“ followed by “x” followed by “)”, y)   F(y)]. In words, “Glub(x)” refers to y if and only if Glub(y), and similarly for all the other atomic predicates of Bob. Note that there are only finitely many atomic predicates in what Tarski calls “formalized languages”, and hence only finitely many of these axioms. These axioms are the basis clauses of the inductive definition.
 (Stage 3) We add further axioms recursively extending the definition of “Ref” to molecular formulas, for example an axiom saying that if x is the result of writing the disjunction symbol “v” between inscriptions w and z, then Ref(x, y)   Ref(x, w) v Ref(x,z), and an axiom saying that if x is the result of writing the negation symbol “-” before w, then Ref(x,y)  ≡ -Ref(x, w),
AND (important!) we add to the axioms the statement, duly formalized, that for all w, if v is a variable and w is z preceded by an existential quantifier (Ev) and z contains no free variables different from v, then {Ref(w,1)     (Ex) Ref(z,x)} & {(Ref(w,0)  ≡ -(Ex)(Ref(z,x)}. This amounts to adopting the convention that a sentence (a wff with no free variable) “refers to” 1 if it is true and to 0 if it isn’t.
Note that even if Bob’s primitive predicates were all monadic, Bob would in general still contain polyadic defined predicates, e.g. “F(x) v y=0” and “F(x) & -F(y)”, so we need a further clause in our inductive definition for existential quantification of a polyadic formula, but formulating such a clause requires more of Tarski’s technique than we shall explain. Suffice it say that Tarski constructs Ref (his “sat”), as a relation between formulas and sequences of objects.
(Stage 4) Now comes the crucial stage: Given that “Ref” has been inductively defined, it is a straightforward application of techniques familiar to logicians from Frege on to turn the inductive definition of “Ref(x,y)” into a explicit definition of a two-place predicate, say SAT(y,x) [Read: “y satisfies x”[vi]] provided Meta-Bob has strong enough set theoretic axioms; and if it doesn’t we just add them. And voilà, we have a predicate of Meta-Bob whose definition involves only the atomic predicates of Bob and (if necessary) the primitive predicates of our favorite set theory, but no non-logical predicates that weren’t already in Bob[vii]  - a predicate that provably satisfies all the clauses in the inductive definition of Ref! So we can drop Ref from our list of primitive predicates, drop the axioms that contained it, and use  SAT(y,x) instead of Ref(x,y), since we know that all the old axioms for Ref are theorems of Meta-Bob, once we simply replace “Ref( x,y)” with SAT(y,x). And, again voilà!, it can be verified that if we take FORM (x) to be “x is a wff with no free variables & SAT(1,x)”, then for each sentence s of Bob, the result of substituting it for “s” and its structural-descriptive name for “s*” in FORM(s*≡ s will be a theorem of Meta-Bob - Convention T is satisfied!
What Field observed
What Field observed, and what I hoped to bring out clearly by breaking up the definition of SAT (Tarski’s formal predicate for “satisfaction”) into stages in the way I did, was that (1) the “truth predicate” (in the case of Bob, the predicate FORM(x)) is defined in terms of the “reference predicate” SAT, and (2) the construction mimics the process of  defining truth in Bob in terms of reference in Bob (i.e. reference restricted to formulas of Bob); (3) reference in Bob (Field’s “denotation”) is defined, or rather SAT,  a relation coextensive with the converse of reference in Bob is defined,  by turning an inductive definition into an explicit definition by well known means; and (3) the basis clauses in the inductive definition “define” reference in the case of atomic (undefined) predicates, or rather, specify the extension of reference in the case of such predicates, by a finite list of cases. From this, Field concluded that Tarski reduced reference to what Field called “primitive denotation” (denotation for atomic predicates), but that he failed to reduce “primitive denotation” to nonsemantical notions (which, on Field’s reading of Tarski(1933) was Tarski’s aim), but only listed the extensions of the primitive predicates, and that what remains to do after Tarski is to “naturalize” primitive denotation, that is, to define reference for atomic predicates in terms acceptable to a physicalist philosopher. A finite list of clauses like “in Bob, ‘w’ followed by ‘h’ followed by ‘i’ followed by ‘t’ followed be ‘e’ refers to all and only white objects” is not a reduction at all, let alone one acceptable to a “physicalist” like Field.
(to be continued)

[i] Hartry Field,  “Tarski's Theory of Truth”, The Journal of Philosophy, Vol. 69, No. 13 (Jul. 13, 1972), 347-375.
[ii] Schema (2) on p. 155 of Tarski(1933).
[iii] Ibid, p.165.
[iv] See my “A Comparison of Something with Something Else,” New Literary History 17.1 (Autumn 1985), 61-79. Repr. in my Words and Life (Cambridge, MA: Harvard University Press, 1994), 330-350.
[v] I shall also, less outrageously, simplify by assuming Bob contains no names, only predicates, and, since the universal quantifier can be defined in terms of the existential quantifier [(x)F(x)  ≡ -(Ex)-F(x)], I shall assume the primitive notation of Bob has only the existential quantifier.
[vi] Instead of speaking of a formula as referring to (a sequence of) objects, Tarski speaks of (the sequence of) objects as satisfying the formula; I have imitated Tarski’s language here, which is why “Ref(x,y)” has suddenly become “SAT(y,x)”. Note that “Ref(x,y)” gets dropped from the notation of Meta-Bob in the course of my “stages”.
[vii] N.B. the definition of REF depends only on the logical and syntactical axioms we added, but not on any “semantical” axioms!