THE IMPLICIT COMMITMENT FOR ARITHMETICAL THEORIES AND ITS SEMANTIC CORE

According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept re ection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions in the foundation of mathematics which consider a speci c theory S as self-justifying and doubt the legitimacy of any principle that is not derivable S: examples are Tait’s nitism and the role played in it by Primitive Recursive Arithmetic, Isaacson’s thesis and Peano Arithmetic, Nelson’s ultra nitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reection. The analysis we propose is as follows: when accepting of system S, we are bound to accept a xed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justi cation given for our acceptance of S in which, for instance, may or may not appear classical re ection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of di erent arithmetical theories.


P
The acceptance of a system formalizing some portion of mathematics is the outcome of a complex justi catory process that is constrained by philosophical and ontological attitudes, in uenced by pragmatical considerations (fruitfulness, generality), and also hospitable to aesthetical ones (simplicity, elegance). The acceptance of a formal system S, therefore, encompasses the possibility that some of the components of this justi catory process are not expressible or even formalizable in the language of S and that some crucial constituents of this acceptance are only left implicit by the process itself. Examples abound: just to remain on the formal side, for instance, soundness assertions for S involving the notion of truth are not expressible in the language of S, while most of their truth-free surrogates are not provable in S.
We have come to the core idea underlying the notion of implicit commitment: when accepting a theory S, we are also bound to embrace a cluster of formal or semi-formal assertions that are not immediately available in S itself. 1 Historically, the notion of implicit commitment for formal systems of arithmetic, and for mathematical theories more in general, emerged in the work of logicians and philosophers already in the 50's and 60's of the last century. Their main concern is clearly expressed in a later work by Solomon Feferman: To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called re ection principles [12, p. 1].
The articulation of this version of implicit commitment was crucial for the analysis of predicativism and, in particular, for the identi cation of predicatively de nable sets of natural numbers. 2 In one well-known Feferman's formulation, the limit for the generation of such sets is articulated in terms of iterations of uniform re ection principles and predicative comprehension over Peano Arithmetic (see [10,11,30]). This project was re ned and reshaped several times by Feferman in the past decades, moving from iterations of rami ed analysis [11] to more succinct formulations such as the re ective closure of the starting theory involving a primitive notion of truth (see §3-4).
At any rate, no matter what formulation of Feferman's hierarchy of systems one chooses, the resulting picture of implicit commitment will entail that in accepting a starting theory S one is committed to statements that are not provable in S itself. Nevertheless, as we shall see later in the paper, it has recently become clear that the inclusion of re ection principles for the starting theory among the claims we are implicitly committed to when accepting it, although integral part of Feferman's foundational program, may clash with other philosophical standpoints. Therefore, we opt for a more neutral and more general formulation of the implicit commitment thesis: 1 In what follows, we will always deal with systems S formulated in a language LS extending the language of arithmetic L = {0, S, +, ×, ≤}. 2 For an overview of this debate, see [13].
(ICT) In accepting a formal systems S one is also committed to additional resources not available in the starting theory S but whose acceptance is implicit in the acceptance of S. 3 As it is formulated, (ICT) has the advantage of re ecting many instances of disagreement in the existing literature over what these 'additional resources' should amount to. Several authors, following Feferman, allow for conceptual resources that are not immediately available in S due to familiar Gödelian phenomena, such as consistency statements and re ection principles. Some of these authors go even further and argue that, once soundness extensions of S are admitted, they should be formulated by explicitly resorting to a notion of truth and not merely by implicitly referring to it via schemata [12,46,26,5]. Other authors maintain instead that whatever we are committed to when endorsing the system S should already be expressible in the very language of S [49]. And nally, proponents of a more drastic view, like Jean-Yves Girard, even deny that re ection on our acceptance of S may have any epistemological value because it relies on a pre-existing agreement on what axioms and rules should be believed to be true [19].
It's important to notice, nonetheless, that the positions just sketched are extracted from works that are not directly concerned with a clari cation of the notion of implicit commitment, but mainly with the notion of truth in the context of truth-theoretic de ationism. 4 A direct analysis of the notion of implicit commitment, however, is much needed. As Horsten and Leigh put it: philosophers of mathematics have hitherto largely failed to investigate the notion of implicit commitment, and have not spent much philosophical energy on analysing our warrant for re ection principles [23, p. 32].
the well-known theses by William Tait and Daniel Isaacson, according to which nitary mathematics coincides with what can be proved in Primitive Recursive Arithmetic (PRA) and rst order Peano Arithmetic (PA) respectively. Dean observes that both theses can be understood as suggesting that PRA and PA are epistemically stable, "in the sense that there exists a coherent rationale for accepting [these systems] which does not entail or otherwise oblige a theorist to accept statements which cannot be derived from [their] axioms" [6, p. 53]. Although we are not primarily interested in analyzing the notion of epistemic stability introduced by Dean, we will be concerned with some of its e ects: the predicativist à la Feferman and the rst-orderist à la Isaacson -who takes PA to delimit the boundaries of nite mathematics, although both assigning a privileged status to Peano Arithmetic, will do so on di erent grounds and this will heavily a ect their stance on (ICT). For Feferman there will be a recognizable set of statements that are not derivable in PA, while being part of our implicit commitment to it; for Isaacson, by contrast, the additional resources hinted at in (ICT) will likely be non-existent. This determines a form of relativity of implicit commitment with respect to the acceptance of one's preferred arithmetical system that will be a recurring theme of this paper.
In what follows, we propose an alternative analysis of the notion of implicit commitment for arithmetical theories. On our account, implicit commitment exhibits a variable and invariant component. We maintain, with Dean, that the set of principles de ning the implicit commitment with respect to the acceptance of a theory S is relative to the justi cation given for this very acceptance. However, we also argue -contra Dean -that the acceptance of a system does involve an explicit soundness extension in the form of a xed set of semantic principles, which we call 'semantic core'. The relative aspect of implicit commitment thus takes the form of di erent, possible extensions of this ubiquitous core. In other words, we will claim that there is a fundamental body of 're ection' principles formulated through the notion of truth -such as the claim that all non-logical axioms of the accepted theory S are true -that are part of the implicit commitment relative to any reasonable justi cation o ered for the acceptance of a theory S. What extends such a kernel is variable and constrained by the justi cation given by the idealized mathematician.
Here is a sketch of the structure of the paper. In the next section, we brie y discuss Dean's critical analysis of (ICT) in relation to Tait's and Isaacson's theses and we extend his remarks to Nelson's ultra nitism. We claim that Dean's analysis does not lead to a dismissal of (ICT) but rather to an alternative interpretation of it. In section 3, in fact, we introduce the necessary toolbox to take up this possible interpretation of (ICT), whereby the 'additional resources' we are committed to when accepting a system S do not entail statements that are unprovable in S.
Our aim is to isolate the semantic core of an arithmetical theory, amounting to the xed, invariable component of our commitment to it. In section 4 we defend the thesis that the distinction between semantic core and variable components of implicit commitment resolves the tension between Tait's and Isaacson's theses and (ICT); in addition, we show that the resulting picture of implicit commitment is also compatible with the traditional reading of (ICT) associated with positions such as Feferman's. Section 5 contains some concluding remarks.

I
As we mentioned in the previous section, Dean in [6] examines the way in which Tait and Isaacson respectively justify the acceptance of PRA and PA from a ' nitist' and a ' rst-orderist' perspective. In both cases he concludes that (ICT) is incompatible with the justi cation that they give for the acceptance of the theories. Let us now brie y recall Tait's and Isaacson's theses and how Dean employs them to criticize this strong reading of (ICT).
On the view articulated by Tait, the formal system of PRA captures precisely the nitist portion of mathematics. 5 Inasmuch as the ' nitist portion of mathematics' is not itself a mathematical notion, Tait draws an analogy between his thesis and Church's thesis, suggesting that 'any plausible attempt to construct a nitist function that is not primitive recursive either fails to be nitist [. . . ] or else turns out to be primitive recursive after all' [48, p. 533, p. 537]. In accordance with the spirit of Hilbert's program, Tait then investigates what counts as nitistic 5 In the literature one can nd many formal systems that fall under the label PRA. In the following we refer to PRA as the extension of propositional logic with the de ning equations of all primitive recursive functions and the schema of quanti er-free induction -for a precise de nition, see for instance [50,Ch. 4,§5].
proof of an open formula ϕ( x) of the language of PRA, and concludes that its proofs are precisely the formal proofs of ϕ( x) in PRA. Tait's proof principles involve only a limited form of induction for nitistically acceptable types (cf. [48, p. 537]), much closer to primitive recursion than to the rst-order schema of induction. 6 In fact, already the schema of induction for Σ 2formulas, let alone the full schema of induction, would enable one to de ne recursive but not primitive recursive operations such as the Ackermann function. Such instances of induction are therefore not available to the nitist.
To discuss the consequences of accepting (ICT) for the nitist à la Tait, we move from the quanti er-free language to a more comfortable base theory formulated in the rst-order language of arithmetic. As Dean emphasizes, Kalmar Elementary Arithmetic (EA) (cf. [2]) is a good choice because (i) it enjoys a smooth arithmetization of the syntax and (ii) it is a proper subtheory of the conservative extension of PRA in which we allow for full-predicate logic but still quanti er-free induction. 7 Let us then consider the so-called uniform re ection principle for an elementary theory S, namely the claim for φ(v) a formula of L S with only v free, where Pr S ( φ(ẋ) ) canonically expresses that the result of formally substituting the variable v with the numeral for x in φ(v) -formally the L S -term sub( ϕ , v , num(x)) -is provable in S. The following is well-known: ). Over EA, full induction is equivalent to RFN(EA).
Therefore, if one understands (ICT) as including RFN(EA), then the nitist should also be committed to the very induction principle of PA, which clearly isn't available in the nitist's preferred theory PRA -nor obviously in its rst-order variant QF-IA. Dean thus concludes that the nitist à la Tait cannot include RFN(EA) (and a fortiori RFN(PRA)) into the set of principles 6 The nitistic justi cation process for PRA sketched by Tait is rooted in the fundamental operation of manipulating nite sequences of objects. All operations and notions obtained by bootstrapping this operation are nistically kosher. In particular, this process of justi cation is not itself legitimate for the nitist because it assumes the general notion of function, which is not nitistically de nable (cf., e.g., [48, pp. 531-533]). 7 This theory is called QF-IA in [47].
she is implicitly committed to when embracing PRA. If principles such as RFN(PRA) are taken to be, as in Feferman's own reading, necessary for (ICT), then (ICT) is simply an inadequate account of implicit commitment across reasonable arithmetical systems. This is basically Dean's conclusion.
Such conclusion is based on the presuppositions that (ICT) can only be interpreted along the lines of Feferman's own account of it and, as a consequence, the reference to 'resources not available in S' in the formulation of (ICT) could only be read in terms of assertions that imply, or even that are equivalent, to sentences in the language of S that are not provable in it. As we shall see later on, however, our proposal will rest indeed on refuting such presupposition; there are in fact many senses in which a resource not available in S may fail to entail unprovable sentences in S. Indeed, the lesson that we draw from Dean's point is not that (ICT) has to be rejected given the incompatibility of the nitist's justi cation of PRA and RFN(PRA). On the contrary, Dean's objection points at the possibility of embracing a plausible version of (ICT) that -relative to certain restrictive standpoints such as the nitist's -does not invoke principles equivalent to or stronger than RFN(EA). Such a version of (ICT) will be articulated in the following sections.
Dean draws a similar conclusion in relation to Isaacson's thesis, according to which PA captures "an intrinsic, conceptually well-de ned region of arithmetical truth" [24, p. 203]. Indeed, Isaacson suggests that PA may be regarded as sound and complete with respect to a conception of arithmetical truths as "directly perceivable" by articulating "our grasp of the structure of the natural numbers" [24, p.217], [25]. Unprovable truths in PA such as Goodstein theorem and the Paris-Harrington sentence are ones that involve hidden higher-order (or in nitary) concepts. 8 If these claims have a clear mathematical meaning, however, it is also well-known that they are equivalent, over PA, to claims of apparent meta-mathematical meaning such as the Gödel sentence for PA or a canonical consistency statement Con(PA). 8 Note that Isaacson characterization of arithmetical truth seems to entail that sentences like the Goldbach conjecture are un-arithmetical, being neither directly perceivable by grasping the structure of natural number, nor perceivable from some arithmetical truth [1]. Against the claim that a proof of any true PA sentence which is independent of PA needs ideas that go beyond those that are required in understanding PA, see [42].
A similar correspondence between the mathematical and the meta-mathematical can be found at the level of the principles which are usually involved in strong readings of (ICT) such as Feferman's. Let's consider again RFN(PA). It is a classical result by Gentzen that PA proves trans nite induction up to any ordinal smaller than ε 0 (henceforth TI ωn ) -i.e. up to the limit of all ordinals of the form ω ω . . . ω for towers of order n [18]. Hence, by the properties of formal provability, PA proves the formalization of this fact for all n. By RFN(PA), therefore, one can conclude, within PA+RFN(PA), the claim that for all n, TI ωn , that is the schema of transnite induction up to ε 0 (TI ε 0 ). Also the other direction -that is the claim that PA + RFN(PA) proves TI ε 0 -is well-known, although the proof, which can be found in [31], is de nitely more involved.
As a consequence, a principle that is naturally justi ed by appealing to semantic or metamathematical considerations such as RFN(PA) on the one hand, and a principle concerning how many countable trans nite ordinals can be well-ordered on the other, are equivalent over PA. Therefore, if Isaacson's thesis on PA is to be understood in a radical way as to entail that anything that is unprovable in PA should not be part of the principles allowed by (ICT), both RFN(PA) and TI ε 0 should be ruled out. In a less categorical reading of Isaacson's thesis, one may still think that principles that are not provable in PA may be allowed in the set speci ed by (ICT); however, as stressed by Dean himself, these truths should now assume the instrumental role of con rming the theorems of PA as clear boundaries for nite mathematics (see [24, §3]). 9 However, it is not clear to us in which sense this more liberal reading of (ICT) should di er from the radical one, since the inclusion of these additional arithmetical truths in the set speci ed by (ICT) only rea rms and does not characterize PA as a self-standing portion of mathematical truth. The message that Dean extracts from Isaacson's thesis looks, again, 9 It should be noticed that we haven't made any reference to the notions of 'higher-order' or 'in nitary' in this description, and this is not by accident: it is not completely clear to us, indeed, where the boundary between nitary and in nitary should lie in the case of PA. Isaacson seems to think that such a boundary coincides with the distinction between what can be proved or not in PA: but can there be a sense in which 'higher-order' or 'in nitary' notions are not at odds with PA? To cite one simple example, consider well-orderings of order type α < ε0, that can be proved in PA by a well-known theorem of Gentzen; other examples that come to mind are versions of semantical re ection that, unlike RFN(PA), are conservative over PA and therefore do not lead us outside of the realm of what is acceptable by the ' rst-orderist'. The next section will present and discuss examples of semantical re ection of this sort. uncontroversial: if one endorses it, she is also committed to a reading of (ICT) that eschews claims that are unprovable in PA, casting serious doubts on the plausibility of (ICT) itself. And again, we will see in the next section that there are senses in which the 'resources not available in PA' we might be implicitly committed to when accepting PA may fail to imply statements that are unprovable in PA itself.
In addition to the cases examined by Dean, similar questions arise in analogous foundational theses that rely on a restriction of the full induction schema of PA. For instance, one might look at the ultra nist thesis advocated by Edward Nelson in [37], and echoed in several commentator's works, according to which one should mistrust the assumption of the totality of exponentiation. 10 A theory that fully meets Nelson's standards is the theory S 1 2 from [3,4]. S 1 2 has several further advantages: besides being consistent with the negation of exponentiation, S 1 2 is also remarkable from a purely proof-theoretic point of view: it can be seen as improving on EA as a theory for formalizing in a natural way the syntax of rst-order theories as it is commonly done for the incompleteness theorems and as it is required for formulating re ection principles and semantic extensions of our starting theories. These notions are in fact all p-time and the functions Σ 1 -de nable in S 1 2 coincide with the p-time computable functions. S 1 2 is formulated in L * = L ∪ {0, S, +, ×, | · |, #, 1 2 · }, where | · | is the length function that gives the number of symbols in the binary representation of the input, # is such that x#y = 2 |x|×|y| and 1 2 · gives the lower integer part of x 2 . Its axioms are the de ning equation of these symbols and the schema for ϕ in the class Σ b 1 , which is similar to the usual class Σ 1 formulas with the additional assumption that quanti ers in the formula have to be bounded by a term of the form |t|, except 10  . . . the basic informal argument says, roughly, that the number of steps needed to terminate a recursion de ning exponentiation is of the order of magnitude of exponentiation itself -a perceived circularity. [14, p. 2] the outermost string of existential quanti ers that can be bounded by an arbitrary term. Crucially, S 1 2 is interpretable in Robinson arithmetic Q, witnessing its minimality, and is nitely axiomatizable [21, Ch. V].
Assuming therefore that S 1 2 is ultra nitistically-acceptable, let us consider a re ection principle of the form where Pr ∅ ( ϕ(ẋ) ) expresses the fact that an arbitrary numeral instance of the formula ϕ is provable in rst-order predicate logic. It's important to notice that now, for the formalization of provability in S 1 2 , instead of non standard numerals one has to consider but dyadic numerals whose formalization is polynomially bounded. 11 However, even under these minimal assumptions, we obtain a result similar to Proposition 1.
is provable in rst-order logic by a series of modus ponens and universal instantiations starting from ϕ(0). This proof, however, may not be captured in general by S 1 2 . Therefore we argue as follows: assuming that ϕ is provably progressive in S 1 2 -that is, S 1 2 proves that it holds for 0 and that, if it holds for x, it holds for x + 1 as well -, by employing Solovay's shortening of cuts technique (cf. again [21, Ch. V]), we downwards close ϕ under ≤ so that the resulting formula de nes an initial segment of the S 1 2 -numbers J . We can safely assume J to be closed under multiplication and the function #. 11 Dyadic numerals are de ned as The codes of the numeral n, in this way, is of order n c for a xed c -therefore can be handled with # -and not 2 cn for xed c, which would require exponentiation.
Then we can prove that (2) by crucially considering dyadic numerals. Now reasoning in S 1 2 and starting with the proof of J (0), we can reason as usual to obtain a proof of J (n). Therefore, in S 1 2 , which can be expressed as a single sentence A, plus RFN(∅), Proposition 2 strengthens the conclusion that, if one reads (ICT) as referring to 'resources not available in S' that entail claims that are not provable in S, then S cannot be taken to capture a self-standing, self-justifying portion of mathematical reality. The ultra nitst embracing S 1 2 , in fact, cannot even be committed to a re ection principle for logic, on the pain of the acceptance of the full induction schema of PA that, obviously, also entails the claim that the exponentiation function is total.
To summarize, the discussion of the theses of Tait, Isaacson, and Nelson, coupled with a strong reading of (ICT) à la Feferman that seems to be taken for granted by Dean, leads to at least two options: either we reject (ICT) across the board, deeming it as inadequate, or we provide a di erent interpretation of (ICT) equipped with an alternative reading of what the 'resources not available' in the chosen system could amount to. In the next section we set the basis for such an alternative interpretation: we will introduce in particular a wide array of semantical extensions of an arithmetical system S that, although crucially resorting to notions that are not immediately available in S -such as a truth predicate -do not entail sentences in the language of S that are not provable in S itself.

S
As noticed by several authors, 12 resorting to schemata such as RFN(S) above may be plausibly seen as a surrogate for single sentences of the form where T is unary truth predicate. These surrogates only become necessary when a notion of truth is not part of the signature of the theory. Any soundness claim seems in fact to be intrinsically related to the notion of truth. If one wants to express in the object language that all non-logical axioms of S are true, for instance, one can of course resort to a schema of the form is the representation of all non-logical axioms of S. Yet, this option merely highlights the fact that we are relegating the notion of truth in the meta-theory.
Clearly someone might have independent motivations to stick with the expressive limitations of the arithmetical language in asserting the soundness of a theory. Tennant in [49], for example, has made use of the well-known fact that schematic versions of re ection, such as RFN(S), enable us to go beyond what's provable in S to defend the possibility of a de ationary account of the notion of truth employed in these soundness claims. However, Tennant does not fully articulate a justi cation for these principles, although he hints at the schematic version of re ection as su cient for xing the norms for assertion of these soundness claims [49, p. 574]. More generally, while it is uncontroversial a soundness extension of S will contain forms of re ection such as RFN(S), it remains problematic whether the presence of RFN(S) is sucient for de ning a soundness extension, in the sense that its principles amount to a coherent articulation of the concepts needed to state soundness claims for S. A good illustration of how soundness claims can be derived within an adequate framework for provability and truth is 12 Cf. for instance, [31], [20, p.  shows that already weak truth axioms seem to collapse the ne structure of the subsystems of PA.
Proposition 3. The result of extending S 1 2 -whose language is expanded with a fresh predicate T -with the schema 14 for all L-formulas ϕ(v) derives the full induction schema of L.
Proof. Since S 1 2 in L T := L ∪ {T} contains I∆ 0 in L T , the following is derivable in the former 14 Recall the slight shift in meaning of the numerals (cf. footnote 11). 15 A similar argument would even hold in the case of set theories formulated by syntactically restricting schemata.
Nevertheless, concluding this would simply be trading on a confusion on the meaning of 'truth axiom'. The theory of truth employed in Proposition 3 is obtained by extending the mathematical induction schemata of the base theory to the truth predicate. If the axioms (utb) are unequivocally truth-theoretic in character, it is natural to think of the extended induction as a mathematical and not as a truth-theoretic axiom. There seems to be in fact a substantial di erence between metalinguistic principles declaring the truth conditions for a sentence of L, as (utb) seems to be (partially) doing, and the extension to the truth predicate of a schema whose justi cation is apparently non-metalinguistic. As observed by Hartry Field, such a justi cation essentially depends on a 'fact about natural numbers, namely, that they are linearly ordered with each element having nitely many predecessors' [15, p. 538]. from the mathematical or, more generally, the object theoretic universe (see [20,22,38]). It's not our intention here to consider the details of this alternative framework: we will keep implicit the distinction between metalinguistic and object-linguistic instances of the induction schema.
However, in what follows we will not extend the induction schema of S to the truth predicate to avoid any con ation between the two levels.
This is not to say, however, that we will not be able to state the truth of the induction schemata of S: if in fact the extended induction schema in combination with natural truth axioms would lead us to very strong theories, the assumption of the truth of all its instances is fairly innocent. As we shall see shortly, indeed, the result of adding to a wide class of base theories S the claim 'all instances of the induction schema of S are true' is still compatible with the alternative reading of (ICT) that we suggested in the previous section and that is aimed at harmonizing (ICT) with foundational standpoints such as Tait's, Isaacson's and Nelson's.
3.1. The semantic core. We have seen that a strong reading of (ICT) may con ict with foundational standpoints based on a form of 'arithmetical completeness' or 'epistemic stability' of some arithmetical system S. In fact, if (ICT) entails re ection principles for S and therefore claims in the arithmetical language that are not provable in S alone, then in accepting S one is also bound to accept arithmetical consequences that go beyond S, thus contradicting its alleged completeness.
In concluding §2, we envisaged the possibility of an alternative reading of (ICT) that could be immune from this problem. But how could this alternative reading look like? A hasty thought may be to let (ICT) depend exclusively on one's foundational standpoint. This is highly problematic. Let's consider, for example, someone who embraces only what's derivable or interpretable in PA: by a well-known result of Feferman, she will also accept ¬Con(PA). 16 By contrast, we have seen that there are several authors disposed to accept Con(PA) after accepting PA. Under this relativistic view of (ICT), therefore, di erent readings of it would not only lead to alternative sets of principles, but rather to sets of principles inconsistent with each other. In the speci c case of ¬Con(PA) just mentioned, moreover, there is a clear departure from what we previously defended as a necessary condition for any plausible reading of (ICT), namely the truth of the principles at play. The interpretation of (ICT) that we now introduce 16 See [9]. will keep a strong link with the notion of truth, while rejecting the sort of rigidity detected in Feferman's reading of (ICT). Our approach substantiates a dynamic reading of (ICT) as displaying a xed, semantic component -called the semantic core of implicit commitment -and a variable component that is relative to one's foundational standpoint.
The semantic core amounts to a set of principles of meta-theoretic nature that enable us to re ect in a natural and uniform way on our acceptance of di erent arithmetical theories. To introduce it, we argue in three stages. In the rst step, we need to expand the language of S with semantic resources, a truth predicate T in particular, and characterize it with a minimal set of principles capturing its disquotational nature. More precisely, given a suitable S, the theory TB[S] is obtained by expanding L S with the predicate T and extending its axioms with the schema (tb) T ϕ ↔ ϕ for all L S -sentences ϕ. An immediate consequence of (tb) is the truth of each axiom of S; it is clear therefore that if S has nitely many non-logical axioms, (tb) su ces to conclude The sentences above clearly do not su ce to count as axioms for the truth predicate T: in the rst case the the resulting theory is clearly interpretable in S by taking the truth predicate in question to be de ned by Ax S (x) itself; in the second case, the full schema (tb) is not necessary to derive (4), as the 'modal' axiom T φ → φ su ces. This suggests that, in these extensions of S, concepts other than truth could be employed as natural readings for the predicate T.
where Sent n L (x) expresses that x is a sentence of L of complexity ≤ n for any given n but not for arbitrary sentences of L and the expression → . (and f . more generally) represents in S the syntactic operation of entailment (resp. f ). 17 Therefore pure disquotation is not su cient for our purposes. Therefore, as second step, one might think of extending TB[S] with further truth-theoretic principles so as to derive the nonrestricted versions of (5). Obvious candidates are the so-called compositional truth axioms such as '¬ϕ is true if and only if ϕ is not true', which govern the interaction of the truth predicate and the logical constants. For instance, since we might safely assume that S is formulated in a calculus in which modus ponens is the only logical rule of inference (see, for instance, [8]), we would only need to add to S the sentence 17 Here the complexity of a formula can simply be taken as the number of logical symbols in it.
to derive the truth-preserving character of modus ponens.
If S is nitely axiomatized, therefore, TB[S]+(6) enables us to prove that all non-logical axioms of S are true and that -if the logic is rightly chosen -that all rules of inferences of S preserve truth. However, there are at least two problems with this theory: in the rst place, it does not articulate a coherent semantic notion as we usually demand that the truth of a compound sentence depends on the truth of its compounds, and this theory has no such feature.
In short, the theory is not (fully) compositional. Secondly, if S is not nitely axiomatizable, it cannot prove that all non-logical axioms of S are true. In fact, as the next lemma shows, it cannot do so even if we add to S a fully compositional theory of truth: Lemma 1. Let S be formulated in L T and assume it satis es full induction for L S -that is the truth predicate is not allowed into instance of induction. This theory extended with the sentences cannot prove that all axioms of S are true.
In (7), R ranges over the relation symbols of L S .
Proof. Assume that S+(7)-(10) proves We can then show that the formula is progressive in it. In (12), Prv S is a ∆ b 1 predicate expressing the notion of being a proof in S and end(·) is a Σ b 1 -function that outputs the last element of of a S-proof. Therefore, still by Solovay's result on subcuts (see Proposition 2), we nd an initial segment of the S-numbers satisfying the property expressed by K(x) in which all logical axioms of S are true and then prove the consistency of S relative to this initial segment. 18 By a strengthening of Gödel's second incompleteness theorem due to Pudlák ([44,Cor. 3.5]), therefore, this is su cient to show that S cannot interpret S+(7)- (10). However, S+(7)-(10) is known to be interpretable in S (see [7, §16.5]).
The full compositional clauses (7) and then proceeds via an attempt to eliminate cuts on formulas of the form Ts from derivations in this theory. Leigh in [33] shows that this strategy can only remove cuts of a provably xed complexity (cf. [33, §3.7]). He then shows how to x Halbach's strategy by nding suitable bounds to the complexity c(·) of truth-cut-formulas in CT[S]-derivations -for S interpreting EA -so that CT[S] can be embedded in the system resulting from replacing the full cut rule for formulas of the form Ts with a weaker set of rules Truth-free proof . . .
for each n and a suitably bounded version of (11). Crucially, this system enjoys a standard version of cut-elimination for cuts on truth ascriptions. Derivations of truth-free sequents of the form Γ ⇒ ∆ are then regimented via the notion of approximation of a sequent, rst considered by Kotlatski, Krajewski, and Lachlan in [28], that enables one to control such proofs Proposition 4 tells us that the semantic principles of the theory CT[S] can safely be included into the semantic core of the implicit commitment for S. Our main thesis is now taking shape: in accepting an arithmetical theory S, we are always implicitly committed to the theory CT[S], which amounts to the xed, invariable component of our commitment. Crucially, whether or not CT[S] exhausts our commitments depends on the particular foundational standpoint that led us to accept a given theory S in the rst place. 19 19 If one grants a claim repeatedly reported in print (for instance by [16,Thm 3.4]) one might think that the theory CT[S] could not be extended with the assertion of the truth of non-logical axioms of S because CT[S]+'all logical axioms of S are true' proves the consistency of S. Unfortunately, the argument for that claim contains a gap as shown in detail in the Appendix to [51]. We strongly conjecture that CT[S]+'all logical axioms of S are true' is conservative over S, but no full proof has been found yet.
This completes the presentation of the semantic core for implicit commitment: it amounts to an extension of S with compositional truth axioms and the claim that all the (non-logical) axioms of S are true. In our account, it is a necessary condition for implicit commitment but possibly not a su cient one: this will depend on the foundational standpoint that one is adopting in justifying a speci c formal system S.
Before giving concrete examples of how our reading of (ICT) in the light of the semantic core applies to the positions considered above, we anticipate two possible objections to the structure of the semantic core. 20 The semantic core may be accused of being too arti cial given (i) the absence of natural soundness principles such as Con(S), and (ii) the absence of the closure under rst-order reasoning. We consider the two objections separately.
To the rst objection a natural reply is that it is not the task that we are assigning to the semantic core to decide which soundness extension of the base theory S is natural or not.
The question we are addressing is in fact whether someone who considers a base theory S as epistemically stable in the sense of §1 can consistently accept (ICT): with the semantic core we aim at providing a framework to answer this question a rmatively. In other words, we do not claim that, say, Con(S) is not a natural principle to endorse once that one has endorsed S, but we share with Dean the view that if the justi cation of Con(S) is equivalent to principles that are incompatible with the alleged epistemic stability of S, as we have seen is the case in the case of nitism and rst-orderism, then such a justi cation cannot be implicit in the mere acceptance of S but should stem from more general considerations. For instance, as we shall see in a moment, although Isaacson considers Con(PA) a principle of in nitary nature, this does not mean that its acceptance should be denied: it simply follows from the acceptance of a suitable portion of in nitary mathematics although it is not implicit in the acceptance of PA that -according to rst-orderism -delimits the boundaries of nite mathematics. This is all compatible with the framework provided by the semantic core.
To address the second objection a similar line of reasoning can be employed: closure of truth under logical reasoning is not a principle that we deem incorrect nor undesirable. However, 20 We thank the anonymous referees for allowing us to clarify this point.
under the condition of the epistemic stability of a theory S, the re ection principle for logic entails principles that are incompatible with this epistemic status such as Con(S). Still, we have shown that there are weaker forms of soundness -such as the truth of all axioms of Sthat are on the one hand not available in S but on the other deductively innocent with respect to S: this makes, we will argue, these weaker soundness claims t the demands of the implicit commitment thesis (ICT) without falling prey to Dean's objections. Once again, the point is that someone may be implicitly committed to the semantic core even if she believes that in accepting S she is not implicitly committed to accept principles unprovable in S: in turn, this does not rule out that she might also have an independent justi cation for these unprovable claims such as Con(S) or the re ection principle for logic.

S
Several foundational standpoints, including the ones considered above, can be compared and distinguished by taking into account the role of the schemata of induction of the arithmetical systems associated to them. In this section we employ these di erent understandings of schematic reasoning to assess the e ectiveness of our dynamic analysis of implicit commitment based on the distinction between the constant semantic core and its variable components. is also committed to the rich ontology of second-order logic (see [40]).
Feferman's notions of re ective closure of a theory S (see [12]) represent a less radical alternative. It comes in two versions: the re ective closure of S and the schematic re ective closure of S. In both cases, the interaction of semantic resources and the power of the induction of PA enable one to reach strong subsystems of second-order arithmetic. In the case of the re ective closure of PA one reaches the strength of rami ed analysis up to ε 0 via the theory of self-applicable truth KF, whereas the schematic re ective closure of PA takes the form of a type-free theory of truth as strong as rami ed analysis up to the Feferman-Schütte ordinal Γ 0 (i.e, roughly speaking, the theory resulting form iterating predicative comprehension αtimes for α < Γ 0 ) [11,45]. 22 Feferman's approach therefore, although clearly committed to schematic reasoning, is clearly weaker than McGee's, since it only delivers a proper subsystem of second-order arithmetic. 21 As McGee writes: Our understanding of the language of arithmetic is such that we anticipate that the Induction Axiom Schema, like the laws of logic, will persist through all such changes. There is no single set of rst-order axioms that fully expresses what we learn about the meaning of arithmetical notation when we learn the Induction Axiom Schema, since we are always capable of generating new Induction Axioms by expanding the language [35, p. 58]. 22 More precisely, such a theory amounts to an extension of the type-free theory of truth KF in L T ∪ {P } equipped with a schematic rule of substitution ψ(P )/ψ(χ), with ϕ(P ) not containing truth, that replaces every subformula P of ψ(P ) with by χ. The axioms of KF are the axioms of PA formulated in L ∪ {T} and the sentences Among the authors that hold an intermediate position between the ones just considered we nd Isaacson himself. He does not seem to impose any restriction to the class of formulas allowed to appear into instances of induction; however, he also clearly states that any further instance of induction involving extra-vocabulary would be intrinsically higher-order, inasmuch as the axioms of full PA su ce to characterize what he calls ' nite mathematics' ([24, p. 204]).

4.1.
Restricted schemata and (ICT). In §2, we have defended the claim that the notion of truth is integral to any reasonable articulation of what we are implicitly committed to when accepting a given arithmetical theory. Of course this comes as no surprise and, as we have seen, our view is shared by many authors. Our intention, however, is not to reformulate a widespread position on the role of truth in foundations, but to suggest something further. What concern us, indeed, is to examine how the notion of truth, as a device to unravel our commitments, can coexist with narrow readings of the implicit commitment thesis (ICT), namely readings which do not allow for claims that are underivable in the accepted arithmetical theory, above all uniform re ection principles.
The case studies of narrow readings of (ICT) stem Dean's analysis of Tait's and Isaacsons' theses. For instance, in the case of Tait's nitism, the uniform re ection principle for the subtheory EA of PRA was su cient to deliver the full schema of induction of PA (see Proposition 1). If the nitist's reading of (ICT) involved principles such as RFN(PRA), she would also be committed to PA, which clearly outstrips primitive recursive reasoning. There is, therefore, a strong temptation for concluding that (ICT) is incompatible with nitism or, even more drastically, that it is false. 23 This temptation, we argue, should be resisted. The semantic core for implicit commitment introduced in §3 gives us a way to accommodate the strong intuition that, even for the nitist's defence of PRA, (ICT) is best spelled out in terms of truth; the semantic core also tells us, however, that these additional resources, being clearly of meta-theoretic and not of object-theoretic nature, do not interfere with the arithmetical content of PRA that the nitist wants to preserve. 23 Dean seems to support something along the lines of the rst claim.
Over PRA -or better its conservative extension in rst-order logic QF-IA -which is known to be not nitely axiomatizable, the semantic core does not only involve compositional truth axioms of the form (7) for instance, in the case of the law of excluded middle, one starts with Tx ∨ ¬Tx for Sent L S (x) and concludes, by (8) and (9), ∀x(Sent L S (x) → T(x∨ . ¬ . x)).
The bearing of this fact should now be clear: we have already argued that truth provides a powerful and natural tool to express one's commitment to a base theory, PRA in the case at hand. Dean cast doubts on the possibility of harmonizing a satisfactory notion of truth and the exclusive commitment to theorems of PRA that appears to be essential to Tait's standpoint. The semantic core o ers a minimal sense in which this balancing process can actually succeed; we do have a notion of truth satisfying some adequacy requirements, such as the partial metalinguistic re ection available in CT[PRA] just considered, and yet we cannot go beyond what's provable in

PRA.
Moving to what we called ultra nitism, in order to draw conclusions along the lines of the ones just obtaineed for PRA, we would need an analogue of Proposition 4 for all theories containing S 1 2 . This claim is, unfortunately, still only a likely conjecture. At any rate, this more general version of Proposition 4 would then establish that the semantic core for implicit commitment relative to a theory S gives us a theory that does not give us new theorems in L S , and in particular Π 1 -sentences such as the consistency of Robinson arithmetic, Con(Q), that are not available in ultra nitistically acceptable theories. Arguably, Isaacson would regard the semantic components of CT[PA] as intrinsically in nitary, but this is not a problem for our reading of (ICT). The implicit commitment to PA, if one regards it as 'arithmetically complete', would be delimited by the semantic core, and its nonarithmetical, in nitary components do not interfere in any way with its mathematical ones in CT[PA]-proofs. This is once more an example of how the semantic core can combine the idea of a privileged access to a de nite portion of mathematical reality given by a speci c theory with the natural act of re ection on the metalinguistic aspects of this theory via semantic notions. 24 Isaacson's position clearly contrasts with views such as Feferman's, who considers the extension of the induction schema of PA as essential to unravel the class of arithmetical assertions we are implicitly committed to when accepting PA in the rst-place. In such positions, schemata are open-ended, and there is no need to stop the truth predicate to interact with the arithmetical content of PA. The semantic core CT[PA], in such view, counts only as a class of necessary conditions that our notion of truth has to satisfy. The theory of truth Feferman is putting forward to fully articulate our commitment to PA, namely KF, contains CT[PA] and is spectacularly stronger than it: it corresponds in fact to ε 0 -many iterations of ACA. In terms of classical ordinal analysis, KF will prove the same arithmetical theorems as PA plus trans nite induction up to ϕ ε 0 (0). 25 According to our proposed reading of (ICT), therefore, Feferman's 24 This separation between object-linguistic and meta-linguistic aspects of theories can be even made more drastic.
Perhaps in this setting the distinction between arithmetical and syntactico/semantic content may even be more convincing for authors that stress the epistemic stability of an arithmetical theory S. We refer to [38] for an overview of such options. 25 For a de nition of the Veblen functions, see [43]. acceptance of PA is tied not only to the semantic core, but to a substantial amount of mathematical principles that can be measured by the big gap separating the trans nite induction schemata for L associated to the ordinals 0 and ϕ ε 0 (0). In moving from Isaacson's to Feferman's position, the semantic core stayed the same, whereas the variable component, which was empty in the case of Isaacson, now includes a large amount of analysis.
There is, however, an unexpected bridge between Isaacson's and Feferman's positions. Once the truth predicate is not allowed into the induction schema of PA, KF becomes much closer to CT[PA]. This theory, called KF in [20], is in fact conservative over PA. Any model M of PA can be expanded to a model (M, S) of KF by tanking S to be a xed point of a suitable positive inductive de nition capturing the clauses of the construction of a Kripke truth set (see [32]).
Instead of being a mere curiosity, this point highlights how the di erence between the view of implicit commitment associated with the rst-orderist à la Isaacson and with the predicativist may be seen as not lying in their conception of truth, but in their understanding of schematic reasoning. If in fact our distinction between object-linguistic and metalinguistic component of a truth theory is granted, then the rst-orderist can articulate a robust notion of truth and yet distinguishing between the arithmetical reality that PA is isolating and the mere metalinguistic consequences that become available once one moves to its extension CT[PA].
She might even move to a type-free notion of truth, as articulated by KF , for instance, without exceeding the arithmetical consequence of PA. Once the truth predicate is allowed to do mathematical work, however, the situation drastically changes.
This scenario reinforces the usefulness of our analysis of implicit commitment via the semantic core: the latter in fact gives us necessary conditions for soundness extensions of a mathematical theory we accept and it is compatible with both restrictive and relaxed readings of (ICT).
Of course once one has reached a satisfactory halting point, such as KF for Feferman's analysis of implicit commitment, nothing prevents one from asking herself what we are implicitly committed to when we are accepting the theory of truth. If Feferman's strategy is extended to the theory of truth, for instance, one can obtain extensions of KF via uniform re ection principles. Indeed, Horsten and Leigh in [23] have shown that extensions of KF can be obtained by starting with TB[PA] via nitely many iterations of uniform re ection. 26 However, since we are not interested in the theory of truth itself, but only in the boundary between acceptable and non-acceptable characterizations of the implicit commitment for the base theory, we do not consider further this possible extension of our analysis.

C
The implicit commitment thesis (ICT) prescribes that, in accepting a system S formalizing some portion of mathematics -arithmetic in our case studies -one is committed to resources not immediately available in S. Traditionally, these additional resources have been understood in terms of sentences in the language of S that are not provable in S already, typically re ection principles for S expressing the soundness of S.
As recently shown by Dean, however, certain foundational standpoints consider a particular arithmetical theory S as delimiting a privileged region of mathematical reality. Re ection principles for the theory S therefore, being closely related to mathematically meaningful principles that lie beyond the space of mathematics occupied by S (see §2 ), should be considered as incompatible with those foundational standpoints. Examples of such positions are Tait's justi cation of PRA, Isaacson's thesis on PA, and to some extent Nelson's strict nitism.
Starting with the observation that soundness claims of S can only be fully articulated by resorting to the notion of truth, we have proposed a dynamic and widely applicable reading of (ICT). The additional resources we are committed to when accepting S will contain principles for truth: these principles, what we called the semantic core for implicit commitment, are xed and shared by any reasonable justi cation for the acceptance of a system S. They amount to compositional truth principles and include minimal soundness claims for S such as the truth of all its non-logical axioms, the truth of all instances of each propositional tautology and, in reasonably chosen cases, the truth-preserving character of its rules of inference. Further 26 A similar strategy for a nonclassical setting in which the starting point are type-free principles of the form T ϕ ⇔ ϕ, with ⇔ a suitable non classical biconditional has been carried out by [17].
principles extending the semantic core of implicit commitment depend on the justi cation for S provided by the idealized mathematician.
This analysis that we have provided is adequate with respect to the case studies considered in the rst part of the paper: the semantic core, when added to S, prevents one from proving new consequences in the language of S besides the ones already available in S itself. Moreover, all natural articulations of soundness assertions of S in the form of stronger truth principles will contain the semantic core; whatever variable components one is willing to add to the semantic core, therefore, they will not be incompatible with it. 27