Is the Role of Truth Merely Expressive?

I worked (a bit) on truth theories in my PhD. I was interested in the question of whether the role of truth is "merely expressive" (as deflationists claimed) or whether it might be "explanatory" (as non-deflationists might claim). One day in 1997, the thought

"I wonder if the T-scheme is conservative over $\mathsf{PA}$?"

popped into my head. (I remember the incident quite well!) If so, there might be an interesting analogy between three kinds of instrumentalism:

(i) Instrumentalism about infinitary objects in mathematics (Hilbert)
(ii) Instrumentalism about mathematical objects in connection with physics (Field).
(iii) Instrumentalism about semantic properties (Field, Horwich).

Debates surrounding the first two eventually turned on whether an extended "ideal" theory $T_I$ was conservative with respect to the underlying "real" one $T_R$. And I knew that, in some cases, the extended theory is not conservative: for Gödelian reasons.

For the case of Hilbertian finitism/instrumentalism, the aim of Hilbert was to show that "Cantor's Paradise" (of infinite sets) was a convenient, but dispensable, instrument for proving finitary combinatorial facts. The hope was to prove Cantorian set theory consistent using just finitary assumptions, perhaps as encoded in $\mathsf{PRA}$ or $\mathsf{PA}$:

In the early 1920s, the German mathematician David Hilbert (1862-1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics. (Richard Zach, 2003, "Hibert's Program", SEP.)

But, it follows from Gödel's results that a consistent finitary theory $T$ of arithmetic can be extended with a suitable amount of Comprehension (one extends induction too), and the result, $T^{+}$, proves theorems in the language of $T$ that $T$ doesn't. The most well-understood examples involve a theory $T$ and its impredicative second-order extension $T^{+}$. For example, if we pass from $\mathsf{PA}$ to second-order arithmetic $\mathsf{Z}_2$. But there are other kinds of example.

It is this insight about increasing power which in part leads to the idea of an "Interpretability Hierarchy", an idea pursued vigorously by Harvey Friedman for several decades. The idea is that, with certain exceptions, mathematical theories form an ascending linear hierarchy of interpretability strength, with weak systems at the bottom (e.g., $\mathsf{Q}$ and $\mathsf{AST}$ (Visser's Adjunctive Set Theory)) and very powerful set theories, with large cardinal axioms, much higher up.

Similarly, for the case of Fieldian nominalism. Consider, for example, Hartry Field's nominalistic reformulation of Newtonian mechanics and gravity in his brilliant monograph,

Field, H. 1980: Science Without Numbers.

(Tragically out of print.) This theory $T$ can be extended with a suitable amount of Comprehension (one extends certain schemes in $T$ too), and the result, $T^{+}$, proves theorems in the nominalistic language that $T$ doesn't.

These Gödelian objections were raised by John Burgess, Yiannis Moschovakis, and Field mentions them in the Appendix of his monograph; I believe similar objections were raised but not published by Saul Kripke; and shortly after, the objections were spelt out in a bit more detail by Stewart Shapiro in his:

Shapiro, S. 1983: "Conservativeness and Incompleteness" (Journal of Philosophy)

So, knowing much of this, I began to see whether the same situation held for truth/semantics, and eventually it became clear that the analogy did hold quite well (I got some help here from John Burgess). In some cases, truth axioms are conservative: e.g., disquotational axioms, $T(\ulcorner \phi \urcorner) \leftrightarrow \phi$, with the sentence $\phi$ used restricted to the object language. And in some cases they are not (e.g., Tarski-style compositional truth axioms). At the time, I had no idea that these technical issues had been investigated by other authors (Feferman, Friedman, Sheard, Cantini, Halbach), though I found out quite quickly after my paper on the topic appeared! My paper on this is:

Ketland, J. 1999: "Deflationism and Tarski's Paradise" (Mind)

("Tarski's Paradise" is a joke, alluding to the analogy mentioned above with "Cantor's Paradise".)

Around the same time, Stewart Shapiro developed more or less the same argument, in his

Shapiro, S. 1998: "Proof and Truth - Through Thick and Thin" (Journal of Philosophy)

This kind of objection---the "conservativeness argument", as it's been called---to deflationism argues that if deflationism is understood as claiming that the role of truth is "merely expressive", then it must require conservativeness of one's truth theory; but then this is inconsistent with the manifest non-conservativeness of truth axioms, in certain cases. More recently, some authors have followed Shapiro and myself in calling truth theories "deflationary" or "non-deflationary" depending on whether they are conservative or not, and explicitly restricting to conservative truth theories if they wish to defend deflationism. This issue is complicated and usually depends on whether one allows inductive reasoning using the truth predicate.

The argument that Shapiro and I gave can be summarized like this (see this M-Phi post "Reflective Adequacy and Conservativeness" (17 March 2013)):

(P1) A truth theory is deflationary only if conservative over suitably axiomatized theories $B$.
(P2) A truth theory is reflectively adequate only if it combines with $B$ to prove "all theorems of $B$ are true".
(P3) For many cases of $B$, reflective adequacy implies non-conservativeness.
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(C) So, deflationary truth theories are reflectively inadequate.

For a similar formulation and discussion, see also:

Armour-Garb, B. 2012: "Challenges to Deflationary Theories of Truth" (Philosophy Compass)

The second premise (P2) corresponds to Hannes Leitgeb's adequacy condition (b) in his:

Leitgeb, H. 2007: "What Theories of Truth Ought to be Like (But Cannot Be)" (Philosophy Compass)

Leitgeb formulated (P2) as follows:

"(b) If a theory of truth is added to mathematical or empirical theories, it should be possible to prove the latter true" 

Leitgeb adds that this is "uncontroversial".

Paul Horwich has an interesting interview with 3am Magazine, titled "Deflationism and Wittgenstein". In it, he makes a couple of crucial points concerning his own form of deflationism, which make it clear that he endorses some form of instrumentalism:

Thus truth is not as profound a phenomenon as has often be assumed. Its role, even in philosophy, must be merely expressive rather than explanatory.

The idea here is that the single sentence:

(1) for any proposition $x$, if $A$ asserts $x$, then $x$ is true 

"re-expresses" the scheme:

(2) if $A$ asserts that $p$, then $p$. 

It is certainly the case that recursively enumerable theories $S$ in some language $L$ meeting certain conditions can be re-axiomatized as a finite set of axioms, by introducing a satisfaction predicate $Sat(x,y)$. This result was originally given by S.C. Kleene (see here). See,

Craig and Vaught 1958: "Finite Axiomatizability using Additional Predicates" (JSL)

for a strengthened version of Kleene's results.

In philosophy of logic, this point appeared quite a while later, in W.V. Quine's Philosophy of Logic (1970) and then

Leeds, S. 1978: "Theories of Reference and Truth" (Erkenntnis). 

Quine was probably not advocating the deflationary view, and was instead endorsing Tarski's semantic conception of truth. It is a serious error to insist that Tarski was a "deflationist". He argued strongly against the deflationary view of this time, "the redundancy theory". Most contemporary arguments against deflationism are due, in fact, to Tarski. E.g., both the problem of generalizations and the non-conservativeness of axiomatic truth are there in his classic:

Tarski, A. 1936: "Der Wahrheitsbegriff in den formalisierten Sprachen" (Studia Philosophica)

But Leeds was putting forward the deflationary view, and many others have followed suit, including Horwich and Field: the deflationary claim is that the reason that languages contain a truth predicate is so that schemes like (2) can be "re-expressed" as single sentences, like (1). Hence the slogan that

"Truth is merely a device for expressing schematic generalizations"

The presence of "merely" is crucial. Remove the "merely", and the above becomes a theorem of mathematical logic! (As mentioned above.) But with "merely" added, it becomes an interesting philosophical claim.

It is not entirely clear what "re-expresses" means. If "re-expresses" means that one can infer one from the other, then although, in fact, one can infer all instances of (2) from (1) (in some fixed language, assuming disquotation sentences $T(\ulcorner \phi \urcorner) \leftrightarrow \phi$ with $\phi$ in the $T$-free language), one cannot infer (1) from all instances of (2), for compactness reasons. There has been an interesting mini-literature on this topic, and important papers on the topic are:

Halbach, V. 1999: "Disquotationalism and Infinite Conjunctions" (Mind)

Heck, R. 2005: "Truth and Disquotaiton" (Synthese)

Even if the "re-expression" component of deflationism could be clarified and sustained, we still need to understand what "explanatory" comes to, and whether the principles governing truth (or a truth predicate) are never explanatory. In mathematics, there are some proofs considered non-explanatory and some considered explanatory. In mathematical logic, semantic reasoning is sometimes used. After all, one makes use of notions like

"$\phi$ is true in the structure $\mathcal{A}$".

The following example is of the kind given in Stewart Shapiro's "Proof and Truth ..." cited above:

Question: If $G$ is a Godel sentence for $\mathsf{PA}$, it is true. Why is it true?

Answer: Because each axiom of $\mathsf{PA}$ is true, and derivations preserve truth; therefore all theorems of $\mathsf{PA}$ are true. In particular, $G \leftrightarrow \neg Prov_{\mathsf{PA}}(G)$ is a theorem of $\mathsf{PA}$, and is therefore true. So, $G$ is true if and only if $G$ is not a theorem of $PA$. But since all theorems of $\mathsf{PA}$ are true, it follows that if $G$ is a theorem of $\mathsf{PA}$, then $G$ is true. So, $G$ is not a theorem of $\mathsf{PA}$. And therefore $G$ is true.

I should note that one can respond to this by insisting that consistency (rather than soundness) is sufficient to obtain the conclusion that $G$ is true. Then the question turns into one about why one should think $\mathsf{PA}$ is consistent.

Furthermore, one can formalize the truth theory within some object language, containing a good theory of truth bearers and a truth predicate governed by certain axioms, and examine its properties and behaviour. We might extend a theory we accept, such as $\mathsf{PA}$, with a truth predicate $T(x)$, with compositional truth axioms. The result is what Volker Halbach calls $\mathsf{CT}$ in his monograph

Halbach, V. 2011:  Axiomatic Theories of Truth.

It seems to me that, within $\mathsf{CT}$, the role of truth is not merely "expressive". For

• $\mathsf{CT}$ proves "All theorems of $\mathsf{PA}$ are true";
• $\mathsf{CT}$ proves new arithmetic facts beyond what $\mathsf{PA}$ does (in particular, coded consistency facts). 

This means that the assumption, "All theorems of $\mathsf{PA}$ are true", used in the above reasoning for the truth of $G$ for $\mathsf{PA}$ can be proved by extending $\mathsf{PA}$ with the compositional truth axioms (with full induction).

Any such truth theory is therefore non-conservative, at least with respect to a large class of of "base theories" ("object language theories", in Tarski's terminology). From this, surely the role of the truth predicate is not "merely expressive". And, if that's right, then the resulting truth theory is non-deflationary.

Later in the interview, Horwich suggests:

Absent a demonstration of this – absent some evidently good theory in which truth is not just a device of generalization – then to speak of deflationism robbing us of a valuable tool is to beg the question.

But we do, at least plausibly, have a demonstration of this: the compositional theory $\mathsf{CT}$.

[Updates (16,17 August). I added some more material related to the reflective adequacy condition on truth theories (Hannes Leitgeb's condition (b) in his article about adequacy conditions on truth theories). I added a link to the paper by Richard Heck on T-sentences and the issue of the role of disquotational truth axioms in the "re-expression"of schematic generalizations.]