Io stimo più il trovar un vero, benché di cosa leggiera, che ’l disputar lungamente delle massime questioni senza conseguir verità nissuna. Galileo Galilei.
Abstract
The existence of a close connection between results on axiomatic truth and the analysis of truth-theoretic deflationism is nowadays widely recognized. The first attempt to make such link precise can be traced back to the so-called conservativeness argument due to Leon Horsten, Stewart Shapiro and Jeffrey Ketland: by employing standard Gödelian phenomena, they concluded that deflationism is untenable as any adequate theory of truth leads to consequences that were not achievable by the base theory alone. In the paper I highlight, as Shapiro and Ketland, the irreducible nature of truth axioms with respect to their base theories. But, I argue, this does not immediately delineate a notion of truth playing a substantial role in philosophical or scientific explanations. I first offer a refinement of Hartry Field’s reaction to the conservativeness argument by distinguishing between metatheoretic and object-theoretic consequences of the theory of truth and address some possible rejoinders. In the resulting picture, truth is an irreducible tool for metatheoretic ascent. How robust is this characterizaton? I test it by considering: (i) a recent example, due to Leon Horsten, of the alleged explanatory role played by the truth predicate in the derivation of Fitch’s paradox; (ii) an essential weakening of theories of truth analyzed in the first part of the paper.
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Notes
There is the temptation, vastly satisfied by the philosophical literature on the argument, to reason in terms of arbitrary base theories. This looks like a hasty move: many results mentioned below that hold of \({\textit{PA}}\) are no more than conjectures for weaker mathematical base theories, for instance. An important feature of the theories that will be introduced in Sects. 2 and 3 is given by the possibility of reasoning in terms of arbitrary base theories. For further details on coding and notational conventions we refer to Leigh and Nicolai (2013) or Halbach (2011). In particular, we only consider natural codings.
Furthermore, \({\textit{TB}}\) is (proof-theoretically) conservative over \({\textit{PA}}\) (for the definition of conservativeness, cf. Sect. 1.4): this is achieved by noticing that any proof of a \(\mathcal {L}_{\textit{PA}}\)-sentence \(\sigma \) in \({\textit{TB}}\) can be transformed in a proof of \(\sigma \) in \({\textit{PA}}\) by replacing the finitely many instances of \({ Tb}\) with a partial truth predicate definable in \({\textit{PA}}\). Despite the conservativity of \({\textit{TB}}\) over \({\textit{PA}}\), it is not the cases that every model of \({\textit{PA}}\) can be expanded to a model of \({\textit{TB}}\). Models of \({\textit{PA}}\) that expand to models of \({\textit{TB}}\) are in fact such that the theory of the model \({ Th}(\mathcal {M})\) (i.e. the set of standard sentences true in \(\mathcal {M}\)) is coded by some \(a\in |\mathcal {M}|\). One can easily find models of PA that do not enjoy this property. For a proof of the (proof-theoretic) conservativity of TB over PA, we refer again to Halbach (2011).
Including the general principle of contradiction — essentially, that the extension of the truth predicate is consistent — whose unprovability was one of the sources of Tarksi’s skepticism on \({\textit{TB}}\). Cf. Tarski (1956a).
Assuming that \(\mathcal L_B\) features only \(\lnot , \wedge ,\forall \) as logical constants.
This is not to say, however, that a theory of truth that counts as compositional has to display axioms in the style of the truth axioms of CT. There may be ways to recover compositionality from other principles, such as reflection principles in the case of typed truth (this was pointed out to the author by Graham Leigh), or disquotation itself in the case of type-free truth. Halbach’s theory PUTB, described for instance in (Halbach (2011), Chap. 19), can in fact (partially) recover compositional axioms of Feferman’s axiomatization of Kripke’s theory of truth KF.
A proof can be found in (Halbach (2011), pp. 104–105).
The standard reference for this form of truth theoretic deflationism is (Horwich (1998), p. 32).
Volker Halbach has suggested, for instance, to understand ‘expressing’ as ‘replacing an infinite list of sentences with a single one’, a sort of finite reaxiomatization. For instance, PA+ all instances of \({\textit{Rfn}}_{{\textit{PA}}}\) proves the same arithmetical sentences as \({\textit{TB}}+{\textit{GR}}_{\textit{PA}}\). Halbach’s example seems to suggest that, given a background (deflationary acceptable) axiomatization of the truth predicate, the single sentence \(\sigma \) capturing the infinite set \(S\), when added to the base theory, should prove the same sentences in the base language as the base theory plus \(S\) (Halbach 1999). Richard Heck has shown, on the other hand, that if instead of just infinite conjunctions one considers both infinite conjunctions and disjunctions, then the equivalence breaks down (Heck 2004). Furthermore, although this strong interpretation may not be completely satisfactory, it may be also observed that it is assumed in the discussion of less controversial cases. It seems in fact that one of the reasons behind the rejection of a purely disquotational truth predicate lies in the deductive weakness of theories in the style of TB. This point was clearly made by Tarski (1956a). More recently, the monographs by Halbach and Horsten also highlight this point (Halbach 2011; Horsten 2012).
Cfr. (Shapiro (1998), p. 497).
In Kotlarski et al. (1981) it was shown in fact that a nonstandard countable model of \({\textit{PA}}\) can be expanded to a model of \(\textit{CT}\mathpunct \upharpoonright \) if and only if it is recursively saturated. The claim that not every model of \({\textit{PA}}\) can be expanded to a model of \(\textit{CT}\mathpunct \upharpoonright \) thus follows from the observation that there are models of \({\textit{PA}}\) that are not recursively saturated. The conservativeness of \(\textit{CT}\mathpunct \upharpoonright \) follows from the fact that every consistent first-order theory has a recursively saturated model. A new, fascinating proof of the conservativeness of \(\textit{CT}\mathpunct \upharpoonright \) is contained in Enayat and Visser (2013).
I.e. once we accept the truth of its axioms: cfr. Sect. 2.4.
More precisely, let \(U\) and \(V\) be theories in predicate logic with identity and in a relational language. A relative translation of \(\mathcal {L}_ U\) into \(\mathcal {L}_ V\) can be described as a pair \((\delta , F)\) where \(\delta \) is a \(\mathcal {L}_V\)-formula with one free variable—the domain of the translation—and \(F\) is a (finite) mapping that takes \(n\)-ary relation symbols of \(\mathcal {L}_ U\) and gives back formulas of \(\mathcal {L}_ V\) with \(n\) free variables. The translation extends to the mapping \(\rho \):
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\((R(x_1,\ldots ,x_n))^\rho :\leftrightarrow F(R)(x_1,\ldots ,x_n)\);
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\(\rho \) commutes with propositional connectives;
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\( (\forall x\varphi (x))^\rho :\leftrightarrow \forall x(\delta (x)\rightarrow \varphi ^\rho )\quad \text {and}\quad (\exists x\varphi (x))^\rho :\leftrightarrow \exists x(\delta (x)\wedge \varphi ^\rho )\).
An interpretation \(K\) is then specified by a triple \(( U,\rho , V)\) such that for all sentences \(\sigma \) of \(\mathcal {L}_\mathsf U \),
We notice that this definition entails \(V\vdash \exists x\,\delta (x)\).
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My emphasis.
For instance:
Therefore, we might try to defend the conservativeness of his theory of truth over a base thery of sentence types rather than logic. But what is a suitable theory of sentence types? There are good reasons for picking Peano arithmetic, although other theories might be suitable as well. Because the discussion so far has mainly focused on Peano arithmetic (which has some nice features as a base theory), I will concentrate on it; similar points, however, can be made for several other theories (like ZF). (Halbach 2001a, p. 182)
Other choices are possible, and perhaps more motivated: in Nicolai (2014) it is considered a theory of finite sets, which naturally captures the informal development of the syntax of \({B}\) in terms of finite trees. We recall that, although \(Q\) is extremely weak, there is an interpretation of Buss’ theory \(S^{1}_{2}\) in it on a definable cut. \(S^{1}_{2}\) is sufficiently strong and extremely efficient for an intensional formalization of the syntax of \(B\). This is the preferred choice in Nicolai (2015).
In particular, by a straightforward metatheoretic induction on the complexity of an arbitrary formula \(\varphi (v_i)\), we prove
for all formulas \(\varphi (v_i)\) of the base language.
The argument is as follows. We reason in CTD \(^{+}\)[B]: assuming, \({\textit{Bew}}_B(\ulcorner \xi \urcorner )\), we get \( {\textit{Sat}}(a,\ulcorner \xi \urcorner )\) and thus \(\xi \) by the Tarski-biconditionals. But also \(\lnot \xi \). So \(\lnot {\textit{Bew}}_B(\ulcorner \xi \urcorner )\).
The same argument straightforwardly applies to TBD[B] and UTBD[B].
For the details (Leigh and Nicolai (2013), Theorem 3.11). The main strategy behind the conservativeness proof was first noticed by Volker Halbach in 2010.
The same holds for \({\textit{CTD}}[{B}]+{AxT}_B\).
Cf. (Horwich (1998), pp. 68–70).
Cf. (Shapiro (1998), p. 499).
In (Leigh and Nicolai (2013), §1.4) the role of induction schemata in this context is explained in full detail.
This new version of the argument was also suggested to the author by Richard Heck and Jeffrey Ketland in private communication.
\({\textit{CTD}}^+[{B}]+{\textit{AxT}}_B\), if \(B\) is schematically axiomatized.
The details of the interpretation are straightforward as the syntactic part of \(B^*\) is essentially ‘collapsed’ in \(B\). To carry out the translation of the induction axioms in the mixed language a sufficient amount of induction is required to be available in \(B\). However, it seems harmless for the present argument to assume that \(B\) contains \(\Sigma _1\)-induction.
Namely the theory, we recall, that is defined as CT with arithmetical induction only.
In CT, for instance, one can only prove the truth of all axioms of PA by applying compositional truth axioms to a suitable instance of the induction schema of CT:
for all \(\varphi (v)\) of \(\mathcal {L}_{\textit{PA}}\), where \({ num}(\cdot )\) is the PA-definable function that sends each number to its numeral. Quantification in the instance above is not over syntactic objects but over all numerals.
We might for instance treat schematic variables as ‘second-order’ variables, as we proceed when we move from \({\textit{PA}}\) to \({\textit{ACA}}_0\), or to the method devised by Craig and Vaught (1958).
For the sake of the present argument, indeed, it can well be just the modal logic K formulated in \(\mathcal {L}_B\).
We notice that \(F^*\) is formulated in the modal logic K, as well as the theory TD[B] was formulated in the same logic as \(B\), that is classical logic.
There are some further worries related to Horsten’s theory \(F\). A first worry concerns the non uniform treatment of modal notions: intensional notions, either epistemic or alethic, seem to belong to a single syntactic category. If one opts for a predicate for knowledge, then one should also opt for a predicate for possibility. \(F\) and \(F^*\), as we have seen, feature by contrast an operator for possibility (or necessity) and a predicate for knowledge. But if one treats also possibility as a predicate, then one has to revise the entire setting as a typed truth predicate does not suffice anymore — or, in the worse scenario, one might fall pray of paradoxes resulting from the interaction of modal notions. A second worry concerns the solution to the paradoxes that is adopted in this context: even in the traditional analysis of knowledge as justified true belief, a finite number of iterations of the truth (or satisfaction) predicate is generally required. In \(F\) and, a fortiori in \(F^*\), this is not possible.
This direction is due to Richard Heck.
We refer to Pudlák’s paper or to Hájek and Pudlák (1993) for the proof.
As an anonynous referee has pointed out, a similar justification of the proposed adequacy requirement has been independently proposed by Fischer and Horsten (2015).
See Sect. 2.4 and the remark below for the assumptions on \(B\).
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Acknowledgments
This work was supported by the Art and Humanities Research Council UK AH/H039791/1 and by the Analysis Trust. I would like to thank Martin Fischer, Volker Halbach, Richard Heck, Leon Horsten, Jeffrey Ketland and two anonymous referees for their comments and suggestions.
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Nicolai, C. Deflationary truth and the ontology of expressions. Synthese 192, 4031–4055 (2015). https://doi.org/10.1007/s11229-015-0729-x
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DOI: https://doi.org/10.1007/s11229-015-0729-x