Ketland 1999: "Deflationism and Tarski's Paradise" (The main point of this paper was to argue thatMind).

*deflationary*truth theories should (and indeed do, usually) satisfy a conservation condition, but

*adequate*theories should satisfy a reflection condition. This reflection condition is Leitgeb's adequacy condition (b) in

Leitgeb 2007: "What Theories of Truth Should be Like (but Cannot be)" (These two conditions arePhil Compass: available here).

*incompatible*, in some scenarios, because of Gödel's incompleteness theorems: hence, d

*eflationary truth theories are inadequate*.

Just before that, but unbeknownst to either of us, Stewart Shapiro published a paper,

Shapiro 1998, "Proof and Truth - Through Thick and Thin" (giving a similar argument against deflationism, based on conservation and reflection. And a few years before that (the similarities were unbeknownst to any of the three of us), Leon Horsten had given some similar arguments in a 1995 article,J. Phil, 1998),

Horsten 1995, "The Semantic Paradoxes, The Neutrality of Truth and the Neutrality of the Minimalist Theory of Truth".But in the 1999 paper, I'd given a "theorem", labelled "Theorem 1". This "theorem" is wrong. Somewhat fortunately, this theorem can be "fixed" by adding certain side conditions (see below). But, as stated, it's wrong. How I ended up with a wrong theorem is moderately interesting.

Still, the wrong "theorem" itself comes from a theorem which is

*not wrong*, which is what I'd started with (I vividly remember guessing it and working out the proof), and which is important in the sense that it's connected to the ensuing debate about the conservativeness/non-conservativeness of truth theories. That theorem is:

$PA$ + the Tarski biconditionals for $L$-sentences is a conservative extension of $PA$.This can be established in a number of ways: e.g., model-theoretically and proof-theoretically. It's clear that the theorem should generalize, because very little of the apparatus of $PA$ is needed for this to hold. In fact, it is sufficient that the base theory $T$ prove the distinctness of distinct expressions. That is,

if $\epsilon_1 \neq \epsilon_2$, then $T \vdash \ulcorner \epsilon_1 \urcorner \neq \ulcorner \epsilon_2 \urcorner$.Unfortunately, I generalized too much! In my 1999 paper, I say that the set of Tarski-biconditionals

$\Delta_{TB} := \{\mathbf{True}(\ulcorner \phi \urcorner) \leftrightarrow \phi \mid \phi \in Sent(L)\}$conservatively extends

*any*theory $T$, and this isn't so. $T$ must satisfy certain conditions. And the proof I give, by model expansion, is somewhat muddled, as it assumes that one can extend the

*domain*of the starting model $\mathcal{M}$ by adding infinitely many (codes of) sentences, and this is not correct.

I noticed the theorem was wrong, when the article appeared in January 1999. For in the same issue of

*Mind*, Volker Halbach published

Halbach 1999, "Disquotationalism and Infinite Conjunctions" (and reading his paper carefully made me notice that my result needed side conditions. Halbach's paper included reference to a wealth of other recent work on axiomatic truth theories of which I'd been completely unaware when I wrote my paper. (I knew only Tarski's original 1936 paperMind),

*Der Wahrheitsbegriff*and worked from there. I didn't know the subsequent work by, e.g., Wang, Mostowksi, Feferman, Friedman & Sheard, Cantini, Halbach

*et al*.)

To give a corrected version, first here's a lemma---one that is useful in thinking about truth predicates (it shows that given a

*finite*set of sentences, then defining truth for these is not difficult, given just a bit of syntax):

Proof: Let the assumptions be as stated, and let $\Psi = \{\psi_1, \dots, \psi_n\}$ be given. Define the following $L$-formula:Lemma: Let $L$ be the usual language of arithmetic and let $L^+$ be the result of adding a new unary predicate symbol $\mathsf{True}$. Let $T$ in $L$ be a theory such that, for anydistinctsentences $\phi, \theta$ in $L$,

$T \vdash\ulcorner \phi \urcorner \neq \ulcorner \theta \urcorner$.Let $\Psi$ be afiniteset of $L$-sentences. Then there is an $L$-formula $\mathsf{True}^{\circ}(x)$ such that,

$T \vdash \mathsf{True}^{\circ}(\ulcorner \psi \urcorner) \leftrightarrow \psi$,for each $\psi \in \Psi$.

$\mathsf{True}^{\circ}(x) := \bigvee \{x = \ulcorner \psi \urcorner \wedge \psi \mid \psi \in \Psi\}$Let $\psi_i \in \Psi$ and consider $\mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner)$. That is,

$(\ulcorner \psi_i \urcorner = \ulcorner \psi_1 \urcorner \wedge \psi_1) \vee \dots \vee (\ulcorner \psi_i \urcorner = \ulcorner \psi_n \urcorner \wedge \psi_n)$Note that, if $i \neq j$, then

$T \vdash \ulcorner \psi_i \urcorner \neq \ulcorner \psi_j \urcorner$So, if $i \neq j$,

$T \vdash \neg(\ulcorner \psi_i \urcorner = \ulcorner \psi_j \urcorner \wedge \psi_j)$So,

$T \vdash \bigwedge \{\neg(\ulcorner \psi_i \urcorner = \ulcorner \psi_j \urcorner \wedge \psi_j) \mid j \neq i\}$But, by the definition of $\mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner)$, we have:

$T, \mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner) \vdash (\ulcorner \psi_i \urcorner = \ulcorner \psi_1 \urcorner \wedge \psi_1) \vee \dots \vee (\ulcorner \psi_i \urcorner = \ulcorner \psi_n \urcorner \wedge \psi_n)$So, then

$T, \mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner) \vdashand so,

(\ulcorner \psi_i \urcorner = \ulcorner \psi_i \urcorner \wedge \psi_i)$

$T, \mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner) \vdash \psi_i$By similar reasoning,

$T, \psi_i \vdash \mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner)$So,

$T \vdash \mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner) \leftrightarrow \psi_i$QED.

Here then is one modification of "Theorem 1" which is ok:

Proof. We will suppose that $T \cup \Delta_{TB}$ proves some $L$-sentence $\phi$, and then "convert" the proof into a proof in $T$ of $\phi$.Theorem: Let $L$ be the usual language of arithmetic and let $L^+$ be the result of adding a new unary predicate symbol $\mathsf{True}$. Let $T$ in $L$ be a theory such that, for anydistinctsentences $\phi, \theta$ in $L$,

$T \vdash \ulcorner \phi \urcorner \neq \ulcorner \theta \urcorner$.Then $T \cup \Delta_{TB}$ conservatively extends $T$ (for $L$-sentences).

So, suppose that

$T \cup \Delta_{TB} \vdash \phi$,with $\phi \in Sent(L)$. Let

$P = (\theta_0, \dots, \theta_k)$be such a proof. If no axioms from $ \Delta_{TB}$ occur in $P$, then $P$ is a proof in $T$ of $\phi$ and we are done.

So suppose that the following are axioms of $\Delta_{TB}$ occurring in $P$

$\mathsf{True}(\ulcorner \psi_1 \urcorner) \leftrightarrow \psi_1$(I.e., these are T-sentences occurring in the proof $P$.)

$\dots$

$\mathsf{True}(\ulcorner \psi_n \urcorner) \leftrightarrow \psi_n$

Next let $\Psi = \{\psi_1, \dots, \psi_n\}$. By the above Lemma, we may define $\mathsf{True}^{\circ}(x)$, and then we have, by the Lemma above, that

$T \vdash \mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner) \leftrightarrow \psi_i$for each $\psi_i \in \Psi$.

Next, given any $\theta_i$ in the proof $P$, we define $(\theta_i)^{\circ}$ to be the result of replacing each occurrence of a subformula $\mathsf{True}(t)$ in $\theta_i$ by $\mathsf{True}^{\circ}(t)$. We obtain the sequence:

$P^{\circ} = ((\theta_0)^{\circ}, \dots, (\theta_k)^{\circ})$Substitution preserves deducibility, and so this sequence is a

*proof*

*using the translations of*$T$-

*sentences as axioms*. However, each $T$-sentence

$\mathsf{True}(\ulcorner \psi_i \urcorner) \leftrightarrow \psi_i$occurring in the proof $P$ is translated to a

*theorem*of $T$, namely,

$\mathsf{True}^{\circ}(\ulcorner \psi_i \urcorner) \leftrightarrow \psi_i$So, if we extend $P^{\circ}$ by inserting the missing subproofs (in $T$) of these theorems, we obtain then a new sequence $P^{\ast}$. This is then a proof of $\phi$ in $T$, as required. QED.

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