[FOM] Deflationism

[A.S.Virdi@lse.ac.uk]

On Mon, 20 Mar 2006, I wrote:

> To get what the deflationist wants (namely the ability of truth to
> prove soundness/reflection principles), we need a theory of truth
> that does not conservatively extend the base theory.

On Tues, 21 Mar 2006 Neil Tennant wrote:

>Why should the deflationist want what you claim she wants?
>Can you cite an expression of her alleged want?

Many thanks for your reply! In "Philosophy of Logic" (1970, p. 12), Quine wrote:

   We may affirm the single sentence just by uttering it, unaided by quotation 
   or by the truth predicate;

so committing oneself to the claim that 'p is true' does no more than commit 
oneself to the claim that 'p', i.e. the concept is dispensable in such locutionary acts.
This is typical of deflationism about truth:- the concept is redundant, as signalled (for deflationists)
by the cognitive equivalence of 'p' with 'p is true'. All that truth is doing here is cancelling
the effect of quotation. Straight after this, Quine goes on to say:

   but if we want to affirm some infinite lot of sentences, then the truth 
   predicate has its use.

Truth becomes indispensable in affording us the ability to make certain
types of generalizations. Actually, it is perfectly reasonable to highlight this feature. 
In "Theories of Reference and Truth" (1978, Erkenntnis journal, p. 121), Stephen Leeds wrote:

   It is not surprising that we should have use for a predicate P with the property that " '...' is P" and '...' 
   are always interdeducible. For we frequently find ourselves in a position to assert each sentence in a 
   certain infinite set z (e.g., when all the members of z share a common form); lacking the means to formulate
   infinite conjunctions, we find it convenient to have a single sentence which is warranted precisely when each 
   member of z is warranted. A predicate P with the property described allows us to construct such a sentence: 
   (for all x, if x is a member of z then x has property P). Truth is thus a notion that we might reasonably want to 
   have at hand, for expressing semantic ascent and descent, infinite conjunction and disjunction.

In "Deflating the conservativeness argument" (1999, Journal of Philosophy, pp. 536-537), Hartry Field says:

   it is quite uncontroversial that the notion of truth can be used to make generalizations
   that cannot be made without it...[n]ot only is this generally recognized, it is often 
   cited by deflationists as the main point of the notion of truth.

This exhausts all there is to be said about truth, from the deflationary point of view. This is Michael Williams in his
 "Epistemological Realism and the Basis of Scepticism" (Mind,p. 424): deflationists...

   ...think that when we have pointed to certain formal features of the truth-predicate (notably its
    'disquotational' feature) and explained why it is useful to have a predicate like this (e.g. as a device
    for asserting infinite conjunctions), we have said just about everything there is to be said about truth.

It seems reasonable to suggest then that certain logico-mathematical principles like the 
soundness statement for PA are exactly the kind of statements that the deflationist wants in her
arsenal. "All theorems of PA are true" finitarily expresses an infinite conjunction. To be unable to 
prove this from her theory of truth (plus, of course, her theory of natural number) would be an
embarrassment for her. {Incidentally, Volker Halbach in "Disquoationalism and Infinite Conjunctions" (Mind 1999) 
clarifies that truth-predicated sentences express infinite conjunctions/disjunctions is best 
understood as their being deductively equivalent, i.e. they have exactly the same consequences.
There are a few minor problems with equivocating "express" with "prove" in this context}.

Neil Tennant proceeded:

>And even if you can, might she not be confused in wanting that
>much?---that is, might she not be failing to hold faith with her own main
>claim(s) about the essential point of our use of the truth predicate?

If she argues that truth's raison d'etre is to make fertile generalizations and she is unable to 
prove these on her own terms then surely she must revise her deflationary understanding of
the concept of truth.

On Mon, 20 Mar 2006, I also wrote:

> As soon as the deflationist makes an appeal to the Tarskian axioms, isn't
> she already moving away from her deflationary position? 

On Tues, 21 Mar 2006 Panu Raatikainen replied:

>Well, it is not completely clear what exactly constitutes the essence of 
>deflationism. In this case, there are (at least) two alternative 
>interpretations: (i) T-sentences say all there is to say about truth; (ii) 
>the truth theory should not entail anything substantial; i.e., it must be a 
>conservative extension of the base theory.  
>So, the answer is: in the former interpretation, yes, in the latter, no.

Many thanks for your reply! Firstly, your (i) and (ii) seem not to be mutually exclusive. If the T-sentences say 
all there is to be said about truth then this truth theory should only have insubstantial 
consequences, i.e. it should not contribute any insights beyond those already garner-able 
from the theory of the world to which the T-sentences are being added ;so, in the particular realm 
of logico-mathematics, the truth theory must be a conservative extension of the mathematical base theory.
If deflationism entails that all there is to be said about the nature of truth can be gotten from
the T-sentences AND that its utility derives from its being able to express things like soundness
statements for PA (say) then the deflationist is in trouble; to get to the soundness principle she will need 
to appeal to Tarski's compositional theory of truth. BUT, in so doing, she would have to renege her deflationism! 

However, on On Tues, 21 Mar 2006 Leon Horsten observes:

>Tarski's compositional theory of truth, axiomatically expressed, does 
>not seem to be committed to facts or to a correspondence with facts. 
>Indeed, T(PA) and Tr(PA) do not mention facts at all. So in as much as 
>facts and correspondence to facts are metaphysical notions, it seems 
>that T(PA), Tr(PA) still deserve the label 'deflationary'.

Many thanks for this! It is true that facts and correspondence do not feature in T(PA) nor Tr(PA) but this does not
disqualify Tarski's theory from being a correspondence theory. Tarski has shown how, uncontroversially, 
one gets a *homomorphism* between sentences of the language concerned and 
elements/objects of the associated domain of discourse. Letting L be a first-order language and D a set-theoretical
structure, one can then establish an interpretation function, I, mapping the non-logical constants of L into the 
domain X of D. For example, the truth definition of an atomic sentence with a two-place predicate P is:
D |= P(a1,a2) iff <I(a1), I(a2)> is a member of I(P).


All the best,
Arhat Virdi




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