Abstract
The contemporary study of the notion of truth divides into two main traditions: a philosophical tradition concerned with the nature of truth and a logical one focused on formal solutions to truth-theoretic paradoxes. The logical results obtained in the latter are rich and profound but often hard to connect with philosophical debates. In this paper I propose some strategy to connect the mathematics and the metaphysics of truth. In particular, I focus on two main formal notions, conservativity and relative interpretability, and show how they can be taken to provide a natural way to read formally the simplicity of the property and the simplicity of the concept of truth respectively. In particular, I show that, this way, we obtain a philosophically interesting taxonomy of axiomatic truth theories.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Deflationary conceptions of truth are philosophical views generally based on three main tenets: (1) that truth is fully accounted for by means of disquotational principles such as Tarskian biconditionals; (2) that a truth predicate provides a tool for expressing infinite conjunctions and disjunctions through truth generalizations; (3) that truth is unsubstantial.
- 2.
The general picture pursued in this paper is very close to the one advocated for by Fischer and Horsten (2015). The main differences are two. First, I motivate the adoption of a model-theoretic criterion on the basis of metaphysical considerations, while they rely on reflections about the role of truth in model and proof theory. Secondly, they pay much attention to the question of the expressive strength but have no interest for the problem of the substantiality of the concept of truth. Also the notion of relative interpretation is barely mentioned and not defended in detail. Although I have strong sympathy for their approach, in this paper I prefer to focus on different, yet complementary issues.
- 3.
For instance, one could read the above claim like: ‘the concept of courage is a virtue concept’.
- 4.
I give just a couple of disparate references for illustration purposes: Dummett (1959), Burgess and Burgess (2011) and Horsten and Leigh (forthcoming).
- 5.
See Fischer and Horsten (2015).
- 6.
- 7.
In recent years, however, an increasing number of authors has become more sensitive to the differences between talk of the concept and talk of the property of truth. See, for instance, Asay (2013).
- 8.
- 9.
Burgess (1986) and Horsten and Leigh (forthcoming).
- 10.
A notable example is provided by the truth theory CT-, which is, roughly, the theory obtained by turning into axioms the usual Tarskian clauses while allowing for arithmetical induction only. In other words, CT- is a Compositional Theory of truth where the truth predicate does not enter the induction scheme.
- 11.
See Cieslinski et al. (To appear) for a critical discussion.
- 12.
- 13.
To resist this view, one should tinker with the syntax by adopting a view where a truth predicate is only apparently a predicate. The superficial grammar of the language would then be misleading. This idea has been defended for instance, by prosententialists.
- 14.
See Strollo (2013) for a more extensive discussion of the metaphysical significance of expandability of models.
- 15.
- 16.
An example of this approach is provided by the usual reflections on provability predicates. Rosser’s provability predicate, for instance, can be distinguished from other provability predicates for the peculiar axioms governing it, even if it delivers the expected results. Indeed, we can also criticize Rosser’s predicate for not expressing the right concept and having the wrong intension.
- 17.
Notice that we are just considering formal theories with an extensional semantics. If we had a modal semantics at disposal, e.g., we could focus on the modal intension of a certain predicate. Given that our base theory is an arithmetical theory, however, a modal approach is not very suitable. Arithmetical propositions, in fact, are either necessarily true or necessarily false.
- 18.
It might be wondered whether the study of the concept should be related to a language instead of a theory. The problem is that we want to consider a language with a truth predicate, namely with at least a determinate class of interpretations. At the same time, we want to compare the truth concepts embodied in different truth theories. Thus posing some minimal constraints, such as selecting those interpretations validating Tarskian biconditionals (as in Halbach 2001) will not work. In this way we would lose the differences between different truth theories equally satisfying such constraints. Thus there is no alternative to consider different axiom systems.
- 19.
Of course I am assuming that explanatory and justificatory roles are considered with respect to a truth-free base theory. A truth theory can certainly be explanatory or justificatory with respect to truth-theoretical claims.
- 20.
Notice, however, that such strategies can also be criticised by showing that the lack of explanatory and justificatory power does not fit well with proof-theoretic conservativity. See Cieslinski (2015). Cieslinski also criticises the resort to model-theoretic conservativity, but I think that his arguments could be resisted. For matters of space, however, I must put the discussion of these points aside.
- 21.
I focus on explanatory force but parallel considerations can be proposed also for justificatory roles.
- 22.
I am here assuming that the truth predicate ascribes a property in some sense.
- 23.
I have in mind ontological explanations, but merely epistemic explanations would exhibit the same basic features.
- 24.
Fischer and Horsten (2015) pair a demand for model-theoretic conservativity with a particular constraint on the expressive power of a language including a truth predicate. Although their proposal is certainly valuable and interesting, I should stress the discrepancy with respect to my approach. The main difference is that here we are concerned with the unsubstantiality of the concept of truth, not with its expressive power. Although a deflationist might have reasons to claim that a truth predicate increases the expressive power of a base language, in principle, such a claim might also be independent from the unsubstantiality of the concept. Even if the relations between conceptual unsubstantiality and expressive power should be investigated, the two have different nature so that we should offer different accounts of such ideas.
- 25.
I take PA in its usual first order formulation. See Horwich (1998) for the relevant details.
- 26.
- 27.
See, for instance, Halbach (2011).
- 28.
See Cieslinski (2015) for a discussion of similar worries.
- 29.
See Fischer (2014).
- 30.
- 31.
Notably, both theories (and, a fortiori, TB- and UTB-) have no significant speed up over PA. At the same time, although TB- and UTB- are acceptable by crude deflationists, lacking speed up, they are hard to defend with instrumental considerations.
- 32.
See Halbach (2001).
- 33.
PTtot is often called PT-. Here I prefer to avoid this label to avoid possible confusions.
- 34.
See Halbach (2001) for details. PTtot has also non elementary speed up over PA.
- 35.
- 36.
Cieslinski, Łelyk, Wcislo (To appear).
References
Asay, Jamin. 2013a. The Primitivist Theory of Truth. Cambridge: Cambridge University Press.
———. 2013b. Primitive truth. Dialectica 67 (4): 503–519.
———. 2014. Against truth. Erkenntnis 79 (1): 147–164.
Burgess, John P. 1986. The truth is never simple. Journal of Symbolic Logic 51 (3): 663–681.
Burgess, Alexis G., and John P. Burgess. 2011. Truth. Princeton: Princeton University Press.
Cieslinski, Cezary. 2015. The innocence of truth. Dialectica 69 (1): 61–85.
Cieslinski, C., M. Łelyk, and B. Wcislo. To appear. Models of PT- with Internal Induction for Total Formulae.
Dummett, Michael. 1959. Truth. Proceedings of the Aristotelian Society 59 (1): 141–162.
Edwards, Douglas. 2013. Truth as a substantive property. Australasian Journal of Philosophy 91 (2): 279–294.
Field, Hartry. 1999. Deflating the conservativeness argument. Journal of Philosophy 96 (10): 533–540.
Fischer, Martin. 2009. Minimal truth and interpretability. Review of Symbolic Logic 2 (4): 799–815.
———. 2014. Truth and speed-up. Review of Symbolic Logic 7 (2): 319–340.
Fischer, Martin, and Leon Horsten. 2015. The expressive power of truth. Review of Symbolic Logic 8 (2): 345–369.
Halbach, Volker. 2001. How innocent is deflationism? Synthese 126 (1–2): 167–194.
———. 2011. Axiomatic Theories of Truth. Cambridge: Cambridge University Press.
Hofweber, Thomas. 2000. Proof-theoretic reduction as a philosopher’s tool. Erkenntnis 53 (1–2): 127–146.
Horsten, Leon. 1995. The semantical paradoxes, the neutrality of truth and the neutrality of the minimalist theory of truth. In The Many Problems of Realism (Studies in the General Philosophy of Science: Volume 3), ed. P. Cartois. Tilburg: Tilberg University Press.
Horsten, Leon, and Graham E. Leigh. forthcoming. Truth is simple. Mind 184.
Horwich, Paul. 1998. Truth. Oxford: Clarendon Press.
Ketland, Jeffrey. 1999. Deflationism and Tarski’s paradise. Mind 108 (429): 69–94.
Niebergall, Karl-Georg. 2000. On the logic of reducibility: Axioms and examples. Erkenntnis 53 (1–2): 27–61.
Shapiro, Stewart. 1998. Proof and truth: Through thick and thin. Journal of Philosophy 95 (10): 493–521.
Strollo, Andrea. 2013. Deflationism and the invisible power of truth. Dialectica 67 (4): 521–543.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Strollo, A. (2018). Making Sense of Deflationism from a Formal Perspective: Conservativity and Relative Interpretability. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-93342-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93341-2
Online ISBN: 978-3-319-93342-9
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)