Reflecting on believability: on the epistemic approach to justifying implicit commitments

Philosophical Studies (2024) 181:3135–3163 https://doi.org/10.1007/s11098-024-02215-6 Reflecting on believability: on the epistemic approach to justifying implicit commitments Maciej Głowacki1 · Mateusz Łełyk1 Accepted: 23 August 2024 / Published online: 7 November 2024 © The Author(s) 2024 Abstract By definition, the implicit commitment of a formal theory Th consists of sentences that are independent of the axioms of Th, but their acceptance is implicit in the acceptance of Th. In Cieśliński (2017, 2018), the phenomenon of implicit commitments was studied from the epistemological perspective through the lenses of the formal theory of believability. The current paper provides a comprehensive proof-theoretic analysis of this approach and compares it to other main theories of implicit commitments. We argue that the formal results presented in the paper favour the believability theory over its main competitors. However, there is still a fly in the ointment. We argue that in its current formulation, the theory cannot deliver all the goods for which it was defined. In particular, being amenable to a generalised conservativeness argument, it does not support the view that the notion of truth is epistemically light. At the end of the paper, we discuss possible ways out of the problem. Keywords Implicit commitments · Believability · Deflationism · Truth · Reflection principles · Epistemology of mathematics 1 Introduction: implicit commitments The notion of implicit commitments of a mathematical theory became central for many inquiries in the philosophy of mathematics. Although such commitments may belong to various categories (e.g. ontological commitments) in this paper we focus on commitments of formal theories, that can be expressed in their languages. Thus we B Maciej Głowacki Mateusz Łełyk 1 Department of Philosophy, University of Warsaw, Krakowskie Przedmieście 3, 00-047 Warsaw, Poland 123 3136 M. Głowacki, M. Łełyk aim to elucidate the following problem (the Problem of Implicit Commitments (PIC): (PIC) Which sentences in the language of a mathematical theory Th should one accept upon accepting Th? In this context of particular interest are sentences which, although unprovable in Th, are still somewhat implicitely implied by it. For theories Th capable of expressing some basic arithmetic of natural numbers, the incompleteness phenomenon provides many candidates for such implicit commitments of Th, first of which is the consistency assertion for Th expressing “Th does not prove ⊥” (where ⊥ is a fixed contradiction). The second source of examples are various reflection principles for Th, which aim at grasping the informal statement, “All theorems of Th are true.” Since the truth predicate for the full language of Th is not available in Th, this is normally expressed via the collection of sentences. For example, it may be expressed by the local reflection for Th, denoted Rfn(Th), that is the collection of sentences of the form If Th proves ‘φ’, then φ, for every sentence φ in the language of Th separately. Recently, the reflection principles became the object of intense research in the philosophy of mathematics and the theory of truth. It is widely believed that accepting (at least some) theories trigger such nontrivial commitments. Often it is claimed that it would be irrational for an agent to accept a theory Th while not accepting its consistency or soundness (see, e.g. Horsten, 2021, p. 9). But the epistemic situation of such an agent is quite peculiar: in accepting a set of sentences, she is committed to some further statements that are not derivable from this set due to the phenomenon of incompleteness. Providing the philosophical justification for the move from accepting Th to the acceptance of, say, Rfn(Th) is one of the fundamental problems of the epistemology of mathematics (see Dean, 2015; Cieśliński, 2017; Łełyk & Nicolai, 2022; Horsten, 2021).1 This inquiry also has nontrivial consequences for the debate on deflationism in the theory of truth. Some deflationists (see, e.g. Horwich, 1990; Cieśliński, 2017) claim that the meaning of the notion of truth for a language L is fully characterised by the collection of disquotational axioms for L . The immediate problem with this thesis is that the theory consisting of uniquely these sentences (and some background theory of syntax; such a theory is denoted TB(L ) in the literature) is too weak to prove many interesting properties of truth, such as the compositional clauses. One possible deflationist response to this objection is based on the observation that, although TB does not prove the compositional clauses for a truth predicate, the extension of the theory with uniform reflection principles proves them (see Halbach, 2001; Horsten & Leigh, 2017). Hence, it may be argued that the truth is indeed characterised by the 1 It should be pointed out that in the literature there are two different readings of commitments in this context. The stronger one, present e.g. in Dean (2015), is that in accepting a theory Th one is obliged to accept certain statements unprovable in Th. The weaker one, proposed e.g. in Horsten (2021), is that in accepting a theory Th one is entitled or justified in acceptance of unprovable statements. In the paper, we want to remain neutral with respect to the reading of commitments, but we believe that our considerations can shed light on both of them. 123 Reflecting on believability: on the epistemic approach... 3137 weak theory TB and the deflationists have access to compositional clauses via the implicit acceptance of the (iterated) reflection principles over TB(L ). This strategy depends on the claim that one is entitled to accept the uniform reflection over theory Th if one accepts the theory itself. In this paper, we are concerned with formal approaches to justifying implicit commitments and delineating their scope. Our main goal is to present new results on the theory of believability, proposed by Cezary Cieśliński in (2017), and use them to compare the theory with other formal approaches to the problem of commitments. In Sect. 2.2, we sketch the landscape of available philosophical explanations of the phenomenon of implicit commitments and explain the benefits of the axiomatic, formal approach. Later, in Sect. 2.3, we describe two main such approaches: truththeoretical and epistemic. We propose a way of comparing them and point toward possible problems with the truth-theoretic approach. In Sect. 4, we present technical results concerning the theory of believability that point out the versatility and conceptual innocence of this theory and contrast it to well-known results in the theory of truth. We provide a detailed analysis of the proof theory of the internal theory of believability and some of its extensions, including a result on the feasible definability of this theory in the theory of iterated reflection over Th. We generalise our results to the theory of un (...truncated)