Reflective Adequacy and Conservativeness

In Catarina's previous post, "Desiderata for theories of truth", discussing a recent talk in Amsterdam by Leon Horsten on truth, Catarina notes:

Initially, the discovery that some respectable axiomatic truth theories were conservative over their base theories was viewed as an argument in favor of deflationist conceptions of truth, in the sense that adding a truth predicate did not add anything of ‘substance’ to a theory. But it seems that the demand that a theory of truth be conservative goes well beyond deflationist concerns, and is sufficiently neutral so as to count as a condition of material adequacy for any theory of truth about a given topic/domain S, on a par with Tarski’s T-schema.

[The result that certain systems of truth axioms -- e.g., the restricted T-sentences, of the form

$T \ulcorner \phi \urcorner \leftrightarrow \phi$,

where $\phi$ is a sentence of the object language -- are conservative can be found in Tarski's original 1936 monograph on truth, Der Wahrheitsbegriff. In general, the set of all instances of the T-scheme, where $\phi$ may itself contain the truth predicate, will be inconsistent if the overall metatheory implements self-reference or diagonalization, because one can define a sentence $\lambda$ equivalent (in the theory of syntax) to $\neg T \ulcorner \lambda \urcorner$. In his 1936 monograph and his 1944 article, "The Semantic Conception of Truth and the Foundations of Semantics", Tarski argues against the redundancy theory of truth, which was an earlier incarnation of deflationism. The result that under certain circumstances the axioms will be non-conservative can also be found in the Postscript to Tarski 1936.]

However, the point I would emphasize is that hoping for conservativeness is not really neutral: it is specifically a deflationary demand: more exactly, a necessary condition for such a theory to count as deflationary:

Conservativeness Condition
A truth theory $TR_L$ in metalanguage $ML$ for object language $L$ is deflationary only if its "combination" $TR_L(B)$ with suitably axiomatized object language theories $B$ in $L$ is conservative.

One might accept or reject this conceptual analysis of deflationism. For example, Stewart Shapiro (1998) and I (1999) proposed it, and think it is one way of explicating the demand that truth be "non-substantial" -- the basic instrumentalist constraint. And, for example, others have suggested the condition needn't be accepted: for example, Halbach 2001, "How Innocent is Deflationism" (Synthese). However, there is a second adequacy condition worth considering - namely Reflective Adequacy.

Reflective Adequacy
A truth theory $TR_L$ in metalanguage $ML$ for object language $L$ is reflectively adequate only if its "combination" $TR_L(B)$ with suitably axiomatized object language theories $B$ in $L$ proves the reflection principle "all $L$-theorems of $B$ are true".

This corresponds to Leitgeb's adequacy condition (b) in his 2007 Philosophy Compass paper, "What Theories of Truth Should be Like (but Cannot be)", on adequacy conditions for truth theories. The problem is that it is not difficult to show that, in a fairly general setting:

Reflective adequacy is inconsistent with conservativeness.

For example, let $L$ be the first-order language of arithmetic, and let $ML$ be $L_T$: that is, $L$ extended with a primitive predicate $Tx$ intended to express "$x$ is true in $L$". Let $TR_L$ consist in Tarski's compositional axioms for truth, assuming that the syntax of $L$ has been coded into $L$. Let $B$ be $PA$, Peano arithmetic. Then the result of "combining" $TR_L$ with $PA$, with full induction for all $L_T$-formulas, does prove "All $L$-theorems of $PA$ are true". Let me indicate this, somewhat loosely,

(a) $TR_L(PA) \vdash \forall x(Sent_L(x) \wedge Prov_{PA}(x) \to Tx)$

It follows that this combined theory proves $Con(PA)$. That is,

(b) $TR_L(PA) \vdash Con(PA)$

It follows that this combined theory $TR_L(PA)$ is non-conservative with respect to $PA$.

This argument against deflationism is sometimes called the Conservativeness Argument, and was given in 1998 and 1999 by Shapiro and yours truly:

Shapiro, Stewart. 1998: "Proof and Truth - Through Thick and Thin", Journal of Philosophy 95.
Ketland, Jeffrey. 1999: "Deflationism and Tarski's Paradise", Mind 108. 

And an earlier version was given by Leon Horsten in 1995:

Horsten, Leon. 1995: "The Semantical Paradoxes, the Neutrality of Truth and the Neutrality of the Minimalist Theory of Truth", in P. Cortois (ed.) 1995, The Many Problems of Realism. 

(This article is difficult to locate if one's library does not have this book, and is not online.)

In the Shapiro and Ketland articles, it is noted that certain axioms for truth are, under certain circumstances, conservative. And also that certain axioms for truth are, again under certain circumstances (usually connected to induction), non-conservative. In particular, if one wishes to reason from a theory $B$ to its reflection principle "all theorems of $B$ are true", the result is bound to be non-conservative if $B$ is a theory to which Godel's incompleteness theorems apply. It is argued by both of us too that the deflationist ought to accept the conservativeness condition; and that, in general, an adequate theory of truth ought to be reflectively adequate. But, of course, these desiderata are incompatible.

One can give a semi-regimented formulation of this philosophical argument as follows:

(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 theorem 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.

The technical result here is (P3). Premises (P1) and (P2) are philosophical explications on the concepts of "deflationism" and "adequacy". There is still a certain amount of imprecision to this, and some caveats in its formulation. (I gloss over these, but they include clarifying what "combining" means precisely when theories contain axiom schemes: when one considers object theories which are infinitarily axiomatized; and issues to do with "disentangling" the syntactical entities from the objects of the object language theory, usually numbers and sequences.)

However, caveats aside, it is close to being a valid philosophical argument, and one that requires the deflationist either to reject the conservativeness condition (P1) or to reject the reflective adequacy condition (P2).