Wednesday, October 7, 2009

What's wrong with Tarski's definition of application?

Tarski's definition of truth depends on a portion which is, essentially, a disjunctive definition of application. As Field has noted in 1974, unless that definition of application is a naturalistically acceptable reduction, Tarski has failed in the project of reducing truth to something naturalistically acceptable. Field thinks the disjunctive definition of application is no good, but his argument that it is unacceptable is insufficient. I shall show why the definition is no good.

In the case of English (or, more precisely, the first order subset of English), the definition is basically this:

  1. P applies to x1, x2, ... (in English) if and only if:
    • P = "loves" and x1 loves x2, or
    • P = "is tall" and x1 is tall, or
    • P = "sits" and x1 sits, or
    • ...
The iteration here is finite and goes through all the predicates of English.

Before we handle this definition, let's observe that this is a case of a schematic definition. In a schematic definition, we do not give every term in the definition, but we give a rule (perhaps implicitly by giving a few portions and writing "...") by which the whole definition can be generated.

Now consider another disjunctive definition that is generally thought to be flawed:

  1. x is in pain if and only if:
    • x is human and x's C-fibers are firing, or
    • x is Martian and x's subfrontal oscillator has a low frequency, or
    • x is a plasmon and x's central plasma spindle is spinning axially, or
    • ...
Why is this flawed? There is a simple answer. The rule to generate the additional disjuncts is this: iterate through all the natural kinds K of pain-conscious beings and write down the disjunct "x is a K and FK(x)" where FK(x) is what realizes pain in Ks. But this definition schema is viciously circular, even though the infinite definition it generates is not circular. If all the disjuncts were written out in (2), the result would be a naturalistically acceptable statement, with no circularity. However, the rule for generating the full statement—the rule defining the "..." in (2)—itself makes two uses of the concept of pain (once when restricting the Ks to pain-conscious beings and the other when talking of what realizes pain in Ks). Thus, giving the incomplete (2) does not give one understanding of pain, since to understand (2) one must already know what the nature of pain is. (The same diagnosis can be made in the case of Field's nice example of valences. To understand which disjuncts to write down in the definition in any given world with its chemistry, one must have the concept of a valence.)

Now, the Tarskian definition of application has the same flaw, albeit this flaw does not show up in the special cases of English and First Order Logic (FOL). The flaw is this: How are we to fill in the "..." in (1)? In the case of English we give this rule. We iterate through all the predicates of English. For each unary predicate Q, the disjunct is obtained by first writing down "P =", then writing down a quotation mark, then writing down Q, then writing down a quotation mark, then writing down a space followed by "and x1" flanked by spaces, then writing down Q. Then we iterate through all the binary predicates expressible by transitive verbs, and write down ... (I won't bother giving the rule—the "love" line gives the example). We continue through all the other ways of expressing n-ary predicates in English, of which there is a myriad.

Fine, but this is specific to the rules of English grammar, such the subject-verb-object (SVO) order in the transitive verb case. If we are to have an understanding of what truth and application mean in general, we need a way of generating the disjuncts that is not specific to the particular grammatical constructions of English (or FOL). There are infinitely many ways that a language could express, say, binary predication. The general rule for binary predication will be something like this: Iterate through all the binary predicates Q of the language, and write down (or, more generally, express) the conjunction of two conjuncts. The first conjunct says that P is equal to the predicate Q, and the second conjunct applies Q to x1 and x2. We have to put this in such generality, because we do not in general know how the application of Q to x1 and x2 is to be expressed. But now we've hit a circularity: we need the concept of a sentence that "applies" a predicate to two names. This is a syntactic sense of "applies" but if we attempt to define this in a language independent way, all we'll be able to say is: a sentence that says that the predicate applies to the objects denoted by the names, and here we use the semantic "applies" that we are trying to define.

It's worth, to get clear on the problem, to imagine the whole range of ways that a predicate could be applied to terms in different languages, and the different ways that a predicated could be encapsulated in a quoted expression. This, for instance, of a language where a subject is indicated by the pattern with which one dances, a unary predicated applied to that subject is indicated by the speed with which one dances (the beings who do this can gauge speeds very finely) and a quote-marked form of the predicate is indicated by lifting the left anterior antenna at a speed proportion to the speed with which that predicate is danced. In general, we will have a predicate-quote functor from predicates to nominal phrases and an application functor from (n+1)-tuples consisting of a predicate plus n nominal phrases to sentences. Thus, the Tarskian definition will require us to distinguish the application functor for the language in order to form a definition of truth for that language. But surely one cannot understand what an application functor is unless one understands application, since the application functor is the one that produces sentences that say that a given predicate applies to the denotations of given nominal phrases.

A not unrelated problem also appears in the fact that a Tarskian definition of the language presupposes an identification of the functors corresponding to truth-functional operations like "and", "or" and "not". But it is not clear that one can explain what it is for a functor in a language to be, say, a negation functor without somewhere saying that the functor maps a sentence into one of opposite truth value. And if one must say that, then the definition of truth is circular. (This point is in part at least not original.)

The Tarskian definition of truth can be described in English for FOL and for English. But to understand how this is to be done for a general language requires that one already have the concept of application (and maybe denotation—that's slightly less obvious), and we cannot know how to fill out the disjuncts in the disjunctive definition, in general, without having that concept.

Perhaps Tarski, though, could define things in general by means of translation into FOL. Thus, a sentence s is true in language L if and only if Translation(s,L,L*) is true in L*, where L* is a dialect of FOL suitable for dealing with translations of sentences of L (thus, its predicates and names are the predicates and names take from L, but its grammar is not that of L but of FOL). However, I suspect that the concept of translation will make use of the concept of application. For instance, part of the concept of a translation will be that a sentence of L that applies a predicate P to x will have to be translated into the sentence P(x). (We might, alternately, try to define translation in terms of propositions: s* translates s iff they express the same proposition. But if we do that, then when we stipulate the dialect L* of FOL, we'll have to explain which strings express which propositions, and in particular we'll have to say that P(x) expresses the proposition that P applies to x, or something like that.) The bump in the carpet moves but does not disappear.

None of this negates the value of Tarski's definition of truth as a reduction of truth to such concepts as application, denotation, negation (considered as a functor from sentences to sentences), conjunction (likewise), disjunction, universal quantification and existential quantification.

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