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)
(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]
[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!
