Naïve validity
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Abstract
Beall and Murzi (J Philos 110(3):143–165, 2013) introduce an objectlinguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules (over sufficiently expressive base theories). As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field (Notre Dame J Form Log 58(1):1–19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a nonclassical solution to the semantic paradoxes need not be substructural. The aim of this paper is to respond to Field’s objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi’s principles: grounded validity. The notion, first introduced by Nicolai and Rossi (J Philos Log. doi: 10.1007/s109920179438x, 2017), is a generalisation of Kripke’s notion of grounded truth (J Philos 72:690–716, 1975), and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic (nor, more generally, of a revisionary approach to semantic paradox), we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches.
Keywords
Curry’s paradox Naïve validity Substructural logics Grounded validity
Validity Proof (\(\mathsf {VP}\)) If \(\psi \) follows from \(\varphi \), then the argument \(\langle \varphi \therefore \psi \rangle \) is valid.

Validity Detachment (\(\mathsf {VD}\)) \(\psi \) follows from \(\varphi \) and from the validity of the argument \(\langle \varphi \therefore \psi \rangle \).
We should stress at the outset that the notion of validity that gives rise to paradox is not logical validity. Purely logical validity does not unrestrictedly satisfy \(\mathsf {VP}\) (if Val is to express logical validity, the rule must be restricted to purely logical subproofs) and is certainly a consistent notion.^{2}
While we do not advocate a nonclassical approach to semantic notions,^{3} in order to investigate the vCurry paradox and its philosophical implications, we’ll assume for the sake of argument that semantic paradoxes are to be solved via a revision of classical logic. Beall and Murzi (2013) point out that, on this assumption, if \(\mathsf {Val}\) satisfies both \(\mathsf {VP}\) and \(\mathsf {VD}\) (or closely related principles), one of the classical structural rules must go. More generally, Beall and Murzi argue that the vCurry paradox is a genuine semantic paradox and that, for this reason, if semantic paradoxes are to be solved via logical revision, such a revision should be substructural.^{4} Hartry Field (2017) has objected that ‘taken together, there is no reading of [\(\mathsf {VP}\) and \(\mathsf {VD}\)] that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the Second Incompleteness Theorem’ (Field 2017, p. 1). For this reason, he concludes that the vCurry paradox doesn’t call for a substructural revision of logic. Elia Zardini (2013, pp. 634–637) argues along similar lines that \(\mathsf {VD}\) is incompatible with Löb’s Theorem and Gödel’s Second Incompleteness Theorem.
Our response to Field and Zardini is twofold. We first review their specific objections, and argue that they fall short of offering conclusive reasons to question the coherence of Beall and Murzi’s naïve principles for validity. In our next step, we introduce a semantic construction for naïve validity, recently developed in Nicolai and Rossi (2017), which generalises Kripke (1975)’s fixedpoint construction for truth. Just like Kripke’s construction yields a theory of grounded truth, the construction for validity yields a theory of grounded consequence or validity—one that validates versions of Beall and Murzi’s principles. In keeping with our rejection of nonclassical approaches to semantic notions, we do not endorse the notion of grounded validity. However, we argue that this notion provides a coherent reading of the naïve validity principles, that can be used to respond to Field’s and Zardini’s criticisms.
The discussion is organised as follows. Section 1 introduces the vCurry paradox and suggests that it is a generalisation of the Knower paradox. Section 2 critically reviews Field’s and Zardini’s specific objections to the coherence of naïve validity. Section 3 introduces the notion of grounded validity and argues that it provides a coherent reading of (versions of) Beall and Murzi’s principles. Section 4 concludes.
1 Introduction
This section briefly sets the scene. After some technical preliminaries (Sect. 1.1), we introduce the Knower and Curry’s paradoxes (Sect. 1.2). We then present the vCurry paradox, and briefly introduce Beall and Murzi’s argument for \(\mathsf {VP}\) and \(\mathsf {VD}\) (Sect. 1.3) and Field’s preliminary discussion thereof (Sect. 1.4).
1.1 Technical premilinaries

There is a function Open image in new window such that for every sentence \(\varphi \), Open image in new window is a closed term. Informally, Open image in new window can be understood as a quotename forming device, so that Open image in new window is a name of \(\varphi \).

For every open formula \(\varphi (x)\) there is a term \(t_{\varphi }\) such that the term Open image in new window is \(t_{\varphi }\), where ‘\(\varphi (t_{\varphi }/x)\)’ is the result of replacing every occurrence of x with \(t_{\varphi }\) in \(\varphi \).
1.2 The Knower and Curry’s paradox
1.3 The vCurry paradox
1.4 Field on the VSchema
In a nutshell, Field (2017) argues that there is no coherent reading of \(\vdash \) and Val for which both \(\mathsf {VP}\) and \(\mathsf {VDm}\) (or \(\mathsf {VD}\)) hold.^{12} According to Field, validity is standardly defined in one of three ways: as necessary truthpreservation, as preservation of truthinamodel (for suitably chosen models), or as derivabilityinS (for a suitably chosen formal system S). However, Field argues that none of these notions makes both of \(\mathsf {VP}\) and \(\mathsf {VDm}\) coherent. We discuss Field’s argument in detail in Sect. 2 below. We first focus on what he has to say about the VSchema, a naïve validity principle that Beall and Murzi take to justify \(\mathsf {VP}\) and \(\mathsf {VD}\).
Field then mentions a possible strengthening of \(\mathsf {V}\)\(\mathsf {Schema}\)—one that, given SRef, actually delivers both \(\mathsf {VP}\) and \(\mathsf {VD}\): However, Field also dismisses the \(\mathsf {V}\)\(\mathsf {Schema^{+}}\), on the grounds of cases such as the following:Beall and Murzi’s likening of the \((\mathsf {V}\)\(\mathsf {Schema}\)) to the truth schema [\(\ldots \)] seems incorrect: even on the assumption that ‘\(\vdash \)’ represents a kind of validity and ‘\(\mathsf {Val}\)’ the same kind of validity, their schema has a ‘double occurrence of validity’ (‘\(\vdash \mathsf {Val}\)’) on the left side and a ‘single occurrence’ (‘\(\vdash \)’) on the right, making the argument from right to left [\(\ldots \)] problematic. And without the assumption that ‘\(\vdash \)’ represents a kind of validity and ‘\(\mathsf {Val}\)’ the same kind of validity, there seems even less reason to accept \(\mathsf {VP}\). (Field 2017, p. 7)
Field’s argument fails to convince, however. To be sure, both the \(\mathsf {V}\)\(\mathsf {Schema}\) and the \(\mathsf {V}\)\(\mathsf {Schema}^{+}\) fail if \(\mathsf {Val}\) is interpreted as expressing logical validity. However, such a reading is already known to be unsuitable for \(\mathsf {VP}\) (Ketland 2012; Cook 2014; Nicolai and Rossi 2017, §2). Hence, a fortiori, it does not fit stronger principles such as the \(\mathsf{V}\hbox {}\mathsf{Schema}\) and the \(\mathsf {V}\)\(\mathsf {Schema}^{+}\). In any event, absent a precise characterisation of \(\vdash \) and \(\mathsf {Val}\), it is unclear whether one should accept or reject (1) and (2), and the \(\mathsf{V}\hbox {}\mathsf{Schema}\) and the \(\mathsf {V}\)\(\mathsf {Schema}^{+}\) more generally. Field contends that no coherent notion of validity simultaneously satisfies Beall and Murzi’s principles. We aim to show otherwise.
2 The case against VP and VD
We now turn to Field’s positive case for claiming that there is a fundamental asymmetry between truththeoretical and naïve validitytheoretical principles. We first discuss two classicality constraints for \(\mathsf {Val}\), which Field expresses sympathy for but doesn’t endorse (Sect. 2.1). We then turn to Field’s argument from definability, that the standard ways of defining validity are incompatible with at least one between VP and VD (Sect. 2.2).
2.1 Classicality constraints
In Field’s view, both principles are incompatible with a naïve conception of validity. As he writes, the weaker principle ‘would immediately rule out substructural solutions to the validity paradoxes in otherwise classical languages’ Field (2017, p. 8). What is more, Field maintains that WCC also rules out nonclassical solutions to Knowerlike paradoxes generated using \(\mathsf {NEC}\) and \(\mathsf {FACT}\). But why should validity be classically constrained? Field mentions two possible arguments.Weak Classicality Constraint (WCC) If the \(\mathsf {Val}\)free fragment of \(\mathcal {L}_{V}\) is classical, then sentences containing \(\mathsf {Val}\) (restricted to inferences in \(\mathcal {L}\)) should also be classical, in the sense of obeying classical laws like excluded middle and explosion.
Strong Classicality Constraint (SCC) Even for nonclassical [\(\mathsf {Val}\)free] languages \(\mathcal {L}\), \(\mathsf {Val}\) (applied to \(\mathcal {L}\)) should be a classical predicate, in the sense that classical laws like excluded middle and explosion apply to sentences containing it.
First, given that ‘the notion of validity should serve as a regulator of reasoning’, Field argues that it ‘would seem as it would hamper that role if there were inferences for which we had to reject that they were either valid or not valid (or accept that they were both) \([\ldots ]\)’ (Field 2017, p. 9). Second, Field mentions what he calls the hypocrisy problem. He argues that if validity were nonclassical, one would have to formulate a theory of validity within a nonclassical metatheory. But because it is very hard to give a nonclassical metatheory, one might as well endorse one of WCC and SCC, thus avoiding the hypocrisy problem.
To be sure, WCC and SCC might be construed as requiring that sentences containing \(\mathsf {Val}\) behave fully classically, where this includes the satisfaction of the structural rules. This is where WCC and SCC part ways, however. If one interprets WCC in this more stringent way, the criterion is still satisfied by several substructural theories of naïve validity, including the approach of Ripley (2012) and the theory developed in Nicolai and Rossi (2017), which will be also described in Sect. 3. Just as in the case of many nonclassical theories of truth, in such theories the \(\mathsf {Val}\)free sentences (and also some sentences featuring \(\mathsf {Val}\)) satisfy all classical rules, operational and structural alike. By contrast, SCC is incompatible with substructural approaches that validate \(\mathsf {VP}\) and \(\mathsf {VDm}\). However, in absence of a plausible independent reason to accept SCC (in its stricter reading), this requirement simply begs the question against substructural logicians who are such because of the vCurry and related paradoxes.
2.2 Field’s argument from definability
Field merely expresses sympathy towards WCC and SCC: his main argument against the coherence of naïve validitytheoretical principles is independent of either principle. In a nutshell, the argument is that none of the three main accounts of validity (validity as necessary truthpreservation, validity as preservation of truthin\(\mathfrak {M}\), and validity as provabilityinS) is naïve. Hence, pending an alternative reading of Val, there seems to be no good reason to accept both of \(\mathsf {VP}\) and \(\mathsf {VDm}\).
2.2.1 Validity as necessary truthpreservation
On this view, Field argues, one between \(\mathsf {VP}\) and \(\mathsf {VD}\) must fail. For ‘any paradoxes of validity will simply be paradoxes of truth in the modal language. Standard resolutions of the paradoxes of truth \(\ldots \) [will] carry over’ (Field 2017, p. 10). Thus, Field concludes, ‘Beall and Murzi’s idea that there are new paradoxes of validity \(\ldots \) requires rejecting this reduction of validity to truth and \(\ldots \) modality’ (ibid.).
 (\(\mathsf {VTP}\))
The argument \(\Gamma \therefore \varphi \) is valid if and only if necessarily, if all the \(\psi \in \Gamma \) are true, then \(\varphi \) is also true.
One first difficulty with the argument is that, on a natural reading of it, it seems premised on a standard revisionary approach, i.e. one validating the structural rules of \(\mathsf {SRef}\), \(\mathsf {SContr}\), and \(\mathsf {Cut}\). But such rules are incompatible with naïve validity. Presumably, then, Field intends the argument to establish that standard paracomplete and paraconsistent approaches can already cope with the vCurry paradox, if \(\mathsf {VTP}\) holds. But there are difficulties with this suggestion, too. As Field (2008, pp. 42–43, pp. 284–286, and pp. 377–378) has long pointed out, \(\mathsf {VTP}\) cannot be consistently asserted in a fully structural setting, on pain of Currydriven triviality.^{13} But then, \(\mathsf {VTP}\) cannot be used to show that fully structural revisionary theorists have a reason to invalidate one between \(\mathsf {VP}\) and \(\mathsf {VD}\): such theorists reject \(\mathsf {VTP}\).
Field’s argument may be recast as the contention that fully structural solutions that invalidate \(\supset \)I can reject \(\mathsf {VP}\), and that fully structural solutions that invalidate \(\supset \)E can reject \(\mathsf {VDm}\) and \(\mathsf {VD}\). However, this observation by itself does not tell against proponents of naïve validity. Substructural theorists who are such because of the vCurry paradox can retort that they can offer a more compelling package: they can not only retain each of \(\supset \)I, \(\supset \)E, \(\mathsf {VP}\) and either \(\mathsf {VDm}\) or \(\mathsf {VD}\); they can also consistently assert (suitable versions of) \(\mathsf {VTP}\) (see Murzi and Shapiro 2015).
2.2.2 Validity as preservation of truthin\(\mathfrak {M}\)
Field’s general point is that in each of these cases, validity cannot be paradoxical on the grounds that ‘the notion of validity is to be literally defined in set theory’ (Field 2008, p. 298).[f]ocusing on onepremise inferences, the general form [of these definitions] is either (i) that the inference from \(\varphi \) to \(\psi \) is valid if and only if in all models \(\mathfrak {M}\) of type \(\Psi \), if \(\varphi \) has a designated value in \(\mathfrak {M}\) then so does \(\psi \); or (ii) that it is valid if and only if in all models \(\mathfrak {M}\) of type \(\Psi \), the value of \(\varphi \) is less or equal to that of \(\psi \). (Field 2017, p. 17; Field’s notation has been adapted to ours)
The argument fails to convince, however. If it were legitimate to assume that validity is modeltheoretically definable in order to show that there are no paradoxes of naïve validity, then it would also be legitimate to assume that truth is modeltheoretically definable in order to show that there are no paradoxes of naïve truth. But this seems unacceptable (Murzi 2014, pp. 77–8). As proponents of naïve theories of truth point out, what holds for modelrelative notions need not hold for the corresponding modelindependent notions (see e.g. Field 2007, p. 107). To be sure, Field might object that there is no coherent modelindependent notion of naïve validity. However, his argument from modeltheoretic definability does not establish this stronger conclusion.
2.2.3 Validity as provabilityinS
Let S be a consistent, recursively axiomatisable theory (formulated in \(\mathcal {L}_{V}\), or in a language that extends it) that is strong enough to simulate selfreference. For simplicity, we could require that S interprets \(\mathsf {PA}\) or \(\mathsf {ZF}\). Either way, the notion of derivability in S, in symbols \(\vdash _{S}\), is also a recursively enumerable relation. Field (2017, p. 12) suggests that S might be taken to be a ‘mathematical theory \(\ldots \) identical to that we use in our informal reasoning’ , whose consequence relation \(\vdash _{S}\) plausibly models the notion of validity associated with S, or at least one such notion. If the validity predicate \(\mathsf {Val}(x, y)\) is to express \(\vdash _S\) in the objectlanguage, then it is natural to interpret \(\mathsf {Val}(x,y)\) as derivability in S. To indicate this specific reading, and in this subsection only, we will write \(\mathsf {Val}_{S}(x,y)\). But here lies the problem.
Likewise, Zardini argues that[g]iven that \(\mathsf {PA}\) and \(\mathsf {ZF}\) are presumably consistent, we must reject \(\mathsf {VD}\) [...]. That, I assume, is a fact that we have come to terms with long ago. (Field 2017, p. 12)
If validity is derivability in a recursively enumerable system, \(\mathsf {VDm}\) and \(\mathsf {VD}\) must fail.derivabilty in \(\mathsf {PA}\) actually coincides with validity relative to \(\mathsf {PA}\). It then becomes utterly unclear why, in view of these facts, one should still expect \(\mathsf {VD}\) to be correct for \(\mathsf {Val}\). (Zardini 2013, p. 636)
There are some difficulties with the foregoing argument, however. Even conceding Field’s and Zardini’s assumption that naïve validity can be equated with validity relative to S, it is not at all clear that the latter notion can be identified with derivability in S. A wellknown argument from the First Incompleteness Theorem, first given (as far as we know) by John Myhill (1960, pp. 466–7), suggests that validity outstrips derivability in any recursively axiomatisable theory that interprets a small amount of arithmetic, and whose axioms and rules we can at least implicitly accept as correct.
From this perspective, the notion of validity that arises from \(\mathsf {PA}\), \(\mathsf {ZF}\), or indeed any sufficiently expressive, recursively axiomatisable theory S is not identifiable with the corresponding notion of derivability. While the latter is classically expressible in the target theory and fails to respect \(\mathsf {VDm}\) and \(\mathsf {VD}\), the former requires methods and tools that extend the target theory, such as local reflection principles.[i]t is possible to prove [\(\kappa \)] by methods which we must admit to be correct if we admit that the methods available in [S] are correct. (Myhill 1960, pp. 466–7)
 (i)
The progression reaches a halting point, namely a theory \(S^{\mathsf {H}}\) such that the progression technique that was adopted at the outset cannot be applied to \(S^{\mathsf {H}}\) to yield a stronger theory that is (computationally) simple enough for Löb’s Theorem to apply.^{15}
 (ii)
The progression reaches a stage (which may or may not be its halting point) such that the theories beyond that stage are too complex for Löb’s Theorem to apply.
However, not even highly complex iterations to which Löb’s Theorem doesn’t apply offer positive reasons for accepting \(\mathsf {VDm}\) or \(\mathsf {VD}\). The problem is that even in the case of theories that are too complex to have a workable provability predicate, it is unclear that anything like \(\mathsf {VDm}\) or \(\mathsf {VD}\) is fully justified. In the construal of validity we are considering, namely validity relative to a theory S, there is no point, in any progression of theories along the lines sketched above, at which a theory \(S^{\star }\) is closed under the local reflection principle for \(S^{\star }\). \(\mathsf {VP}\) and \(\mathsf {VDm}\) or \(\mathsf {VD}\) are a sort of unattainable ‘limit’ of the notions of validity relative to a theory that the acceptance of Myhill’s argument suggests—a limit that fuels the progression of theories but that remains always one step beyond the reach of every theory so generated.
2.2.4 Hierarchical validity
But it can. Nicolai and Rossi (2017, §§3–4) develop a construction that is in effect a naïve validitytheoretical generalisation of Kripke ’s (1975) construction for truth. Their construction, called ‘KVconstruction’ (for ‘Kripke’ and ‘validity’), delivers nontrivial models of \(\mathcal {L}_{V}\) (or languages extending it) where \(\mathsf {VP}\) and \(\mathsf {VDm}\) (together with the \(\mathsf {V}\)\(\mathsf {Schema}^{+}\)) hold unrestrictedly. The significance of this result is not only technical: the construction can also be used to meet Field’s challenge of finding a coherent reading of the naïve validitytheoretical principles.^{17}The thought might be that just as Kripke (1975) showed how to transcend the Tarski hierarchy in a nonclassical setting (introducing a single unstratified nonclassical truth predicate [...]), we should do the same for validity in a nonclassical setting. Extending the analogy, the idea might be to argue in a nonclassical setting that by starting from a hierarchy of validity predicates and allowing sentences to ‘seek their own level’, an unstratified predicate that satisfied \(\mathsf {VP}\) and \(\mathsf {VD}\) would emerge at some fixed point. [\(\dots \)] Obviously there’s no way that anything like this could happen if the nonclassical setting were merely paracomplete or paraconsistent, with standard structural rules—[\(\ldots \)] the whole point of the vCurry argument was that mere paracompleteness or paraconsistency don’t suffice to allow for \(\mathsf {VP}\) and \(\mathsf {VD}\) together. But perhaps if we did a construction modeled after Kripke’s in a substructural setting, \(\mathsf {VP}\) and \(\mathsf {VD}\) together would emerge? That would certainly be interesting if it could be done, but Beall and Murzi don’t claim it can, and nothing in their paper gives any reason to think that it can. (Field 2017, pp. 15–6)
3 A Kripkean construction for naïve validity
We begin by offering a (largely informal) presentation of the KVconstruction in Sect. 3.1.^{18} We then argue in Sect. 3.2 that one of the models that results from the KVconstruction suggests a coherent interpretation of naïve validity: grounded validity.
3.1 The KVconstruction
The KVconstruction generalises Kripke’s treatment of truth (strong Kleene version) to naïve validity. Rather than constructing successions of sets of sentences (leading to a fixed point), it builds successions of sets of inferences or sequents. We work with the language of arithmetic, enriched with a primitive binary predicate \(\mathsf {Val}(x,y)\), for validity; we call this language \(\mathcal {L}^{a}_{V}\). More precisely, the KVconstruction generalises inferences to multipleconclusion \(\mathcal {L}^{a}_{V}\)sequents, i.e. objects of the form \(\Gamma \vdash \Delta \), where both \(\Gamma \) and \(\Delta \) are finite sets of \(\mathcal {L}^{a}_{V}\)sentences. From now on, we will work with finite sets rather than multisets. We will continue using capital Greek letters (such as \(\Gamma \) and \(\Delta \)) to denote finite sets.
We now need to explain how the acceptance of a collection of sequents can lead to the acceptance of other sequents. Since we are dealing with sequents, and not with sentences, this cannot happen (as in Kripke’s case) via some evaluation scheme. However, we can resort to metainferences, namely principles that determine which sequents are to be accepted given the acceptance of one or more other sequents. In the KVconstruction, we can consistently use inductive clauses modelled after all the classical metainferences. Of course, we need to devise clauses for the validity predicate too, namely clauses that tell us when a sentence of the form Open image in new window can be introduced in a sequent, given some previously accepted sequents. An inspection of the naïve principles for validity suggests an obvious option: these principles are classical implication principles formulated using a predicate, namely \(\mathsf {Val}\), rather than a connective. It is then natural to use metainferences for \(\mathsf {Val}\) modelled after the classical metainferences adopted to introduce conditionals in sequents.
We conclude this section by noticing that the computational complexity of \(\mathsf {I}_{\Psi }\) is identical to the computational complexity of the least Kripke fixed point for truth—a relatively low complexity in the context of semantic theories of truth. We also observe that, just as in the case of Kripke’s theory, the clauses of the definition of \(\Psi \) can be turned into a recursively enumerable theory that axiomatizes adequately, in the sense of Fischer et al. (2015), the set of the fixed points extending \(\mathsf {I}_{\Psi }\). Naïve validity need not be too complicated to reason with.
3.2 Grounded validity
\(\mathsf {I}_{\Psi }\) provides a coherent reading of the notion of validity—one that makes sense of many of the naïve principles discussed in Beall and Murzi (2013). Following Kripke’s construction, we call this reading grounded validity, i.e. validity as grounded in truths and falsities of the base language.^{20} The idea of grounded validity is simple: a sequent \(\Gamma \vdash \Delta \) is to be accepted if and only if it results from iterated applications of the clauses of \(\Psi \) to sequents having atomic arithmetical truths in their consequent, or atomic arithmetical falsities in their antecedent. This option is naturally associated with \(\mathsf {I}_{\Psi }\), since it follows the idea of grounded truth, associated to the least Kripkean fixed point for truth. In what follows, we argue that the notion of grounded validity, as articulated by \(\mathsf {I}_{\Psi }\), addresses Field’s challenge of finding a coherent reading for Beall and Murzi’s principles for naïve validity. We should stress, however, that we are not endorsing naïve validity. Our claim is simply that it can be made sense of, via grounded validity, especially if one can already make sense of the Kripkean notion of grounded truth.
3.2.1 The naïve principles for validity
We now review the case for \(\mathsf {VP}\), \(\mathsf {VDm}\), \(\mathsf {V}\)\(\mathsf {Schema}\), and \(\mathsf {V}\)\(\mathsf {Schema}^{+}\), construing naïve validity as grounded validity. In doing so, we also address some of Field’s more specific objections.
\(\mathsf {VP}\) states that it is possible to internalise the metatheoretical notion of naïve validity represented by \(\vdash \), and express it via \(\mathsf {Val}\). In the reading offered by \(\mathsf {I}_{\Psi }\), \(\mathsf {VP}\) says that if \(\psi \) follows from \(\varphi \) on the basis of arithmetical truths and falsities via the \(\Psi \)clauses, then it follows on the basis of arithmetical truths and falsities via the \(\Psi \)clauses that \(\psi \) follows from \(\varphi \) on the basis of arithmetical truths and falsities via the \(\Psi \)clauses. This much is obvious, since the \(\Psi \)clauses themselves include a version of \(\mathsf {VP}\), that lets one express via \(\mathsf {Val}\) at level \(\alpha +1\) the \(\vdash \)inferences accepted at level \(\alpha \). This arguably answers Field’s worry that there might be no reasons to accept a ‘double occurrence’ of the notion of naïve validity on the right of \(\mathsf {VP}\). Field also asks why couldn’t there be true validity claims that are not valid. While \(\mathsf {I}_{\Psi }\) does not exclude this possibility, it nevertheless shows that there is a uniform construal of \(\vdash \) and \(\mathsf {Val}\) under which this is admissible. True groundedvalidity claims are themselves groundedly valid, since grounded validity just consists in the iterative generation of all the validities that derive from our acceptance of arithmetical truths and falsities.
The justification of the \(\mathsf {V}\)\(\mathsf {Schema}\) follows similar lines. We have already seen how \(\mathsf {I}_{\Psi }\) makes it coherent to accept its direction corresponding to \(\mathsf {VP}\). As for the other direction, it follows immediately from the fixedpoint property of \(\mathsf {I}_{\Psi }\), i.e. from the fact that the \(\Psi \)clauses are to be read as an ‘if and only if’ once we reach a fixed point. We can thus reverse the claim that closes the previous paragraph: groundedly valid validity claims are also just true groundedvalidity claims. The extra iteration of the notion of grounded validity on the righthand side of the \(\mathsf {V}\)\(\mathsf {Schema}\) does not add anything substantial to the metatheoretical grounded validity claim on its lefthand side: the \(\mathsf {V}\)\(\mathsf {Schema}\) just guarantees that the two expressions (metatheoretical and objectlinguistic) of the same notion (grounded validity) are equivalent.
Finally, the acceptance of \(\mathsf {VDm}\) in \(\mathsf {I}_{\Psi }\) follows from the fact that \(\mathsf {I}_\Psi \) is closed under clauses which essentially express all the classical metainferences. In the case of \(\mathsf {I}_{\Psi }\), it is hard to see why some classical metainference should fail. Groundedly valid inferences, expressed metatheoretically or via Val, are determined by perfectly classical claims (about arithmetical truths and falsities), so we see no plausible reason why one should not accept all the inferences that follow from applying classical patterns of reasoning to them. \(\mathsf {I}_{\Psi }\) delivers all the sequents that follow from closing the initial arithmetical sequents under all the classical metainferences.
3.2.2 What’s rejected: reflexivity and the full \(\mathsf {VD}\)
A grounded conception of validity makes it coherent to restrict \(\mathsf {Ref}\) and the full \(\mathsf {VD}\). \(\mathsf {Ref}\) and \(\mathsf {VD}\) have ungrounded instances, namely instances that cannot be obtained from inferences having arithmetical atomic truths in their consequents, or arithmetical atomic falsities in their antecedents. In \(\mathcal {L}_{V}^{a}\), or superlanguages of it, such inferences crucially feature sentences which themselves encode ungrounded inferences, via the naïve validity predicate. Inferences formed with the vCurry sentence \(\pi \) are a typical example, and indeed a grounded conception of validity rejects the instance of reflexivity that involves \(\pi \), i.e. \(\pi \vdash \pi \).

From the fact that the inference from this very sentence to \(\bot \) is naïvely valid, it follows that the inference from this very sentence to \(\bot \) is naïvely valid.
3.2.3 Grounded validity and Löb’s theorem
The notion of naïve validity encoded by \(\mathsf {I}_{\Psi }\) would appear to avoid Field’s and Zardini’s objection from Löb’s Theorem: that \(\mathsf {VD}\) and \(\mathsf {VDm}\) are in conflict with Löb’s Theorem and Gödel’s Second Incompleteness Theorem. Call this the LGobjection. Running it against \(\mathsf {I}_{\Psi }\) does not make much sense, since the LGobjection targets some recursively enumerable theory. However, as was mentioned at the end of Sect. 3.1, an axiomatic theory can be associated with the KVconstruction, and shown to contain only sequents grounded in arithmetical axioms, thus fleshing out a weaker form of grounded validity.
The notion of grounded validity provides a possible way of expressing the material conditional as an implication predicate in the objectlanguage. Because of the vCurry and related paradoxes, some principles that hold for the classical material conditional must be abandoned—in the case of grounded validity, reflexivity. At the same time, however, grounded validity is characterised by some principles that are constitutive of the material conditional but not of provability, such as VDm, which is a version of modus ponens. For this reason, grounded validity and provability overlap, but are not even extensionally identical.
In the case of nonclassical, naïve theories of truth, a standard reply is that such theories employ a nonclassical logic, and hence do not violate classical limitative results. But the same holds for grounded validity: it might be argued that just like the conditional of (5) has to be nonclassical, so too must the sequent arrow (\(\vdash \)) in (4). Therefore, either the LGobjection fails to apply to irreflexive, grounded validity, or structurally similar objections apply to naïve truth, thus allowing one to conclude that we should ‘have come to terms with’ the rejection of naïve truth ‘long ago’.
4 Concluding remarks
We hope to have shown that such a construction can be done and that, pace Field, the cases of truth and naïve validity are not ‘totally different’. The naïve notion of grounded validity appears to indicate that truth and naïve validity not only give rise to similar paradoxes, but can also be understood in similar ways. Then, the resulting paradoxes can be dealt with in a similar fashion. As in the case of the paradoxes of truth, a revisionary resolution of the paradoxes of naïve validity calls for an appropriate nonclassical logic, and for a coherent reading for the naïve semantic principles involved. We hope to have provided both.we have a further respect in which the situation with the validity principles \(\mathsf {VP}\) and \(\mathsf {VD}\) seems totally different from the situation with the principles of naive truth. (Field 2017, p. 16)
Footnotes
 1.
Sentences such as \(\pi \) can be shown to exist in a number of ways, both in formal and natural languages. For present purposes, we simply assume their existence.
 2.
 3.
 4.
 5.
A multiset is a collection of objects that is just like a set, except that repetitions count. Thus, for instance, \(\{\varphi , \psi , \psi \}\) and \(\{\varphi , \psi \}\) are identical sets but different multisets.
 6.
For more details on this formalism, see e.g. Troelstra and Schwichtenberg (2000, p. 41 and ff).
 7.
Field formulates these principles by means of a unary validity predicate, and calls (the resulting versions of) \(\mathsf {NEC}\) and \(\mathsf {FACT}\), respectively, \(\mathsf {VALP}\) and \(\mathsf {VALD}\) (Field 2017, p. 7). We stick to the binary validity predicate and employ the constant \(\top \) for this reason. Moreover, we adapt Field’s principles to our framework.
 8.
The following derivations make also tacit use of the rules for intersubstitutivity of equivalents (e.g., in the passage labelled ‘Definition of \(\sigma \)’). We will always assume intersubstitutivity of equivalents without making it explicit amongst our rules for the sake of readability.
 9.
For more on how \(\mathsf {VP}\) and \(\mathsf {VDm}\) are generalisations of, respectively, \(\mathsf {NEC}\) and \(\mathsf {FACT}\) see Murzi and Shapiro (2015, §2.1).
 10.
 11.
 12.
To be precise, Field does not explicitly address \(\mathsf {VDm}\). However, since he never considers restrictions of reflexivity, and \(\mathsf {VD}\) is derivable from \(\mathsf {VDm}\) given \(\mathsf {Ref}\), we will treat \(\mathsf {VD}\) and \(\mathsf {VDm}\) as equivalent until Sect. 2 included (the difference between \(\mathsf {VD}\) and \(\mathsf {VDm}\) will only come into play in Sect. 3). Accordingly, we will interpret Field as rejecting both pairs \(\mathsf {VP}\) and \(\mathsf {VDm}\), and \(\mathsf {VP}\) and \(\mathsf {VD}\).
 13.
 14.
The study of the progressions of theories resulting from the systematic addition of schematic principles (such as consistency statements, reflection principles, and others) to a starting theory was pioneered by Turing (1939). Their systematic investigation was started by Kreisel (1960, 1970) and Feferman (1962), leading to the crucial notion of autonomous progression (see also Feferman 1964, 1968). For an accessible presentation of progressions of theories by iterated addition of reflection principles, see Franzen (2004).
 15.
A wellunderstood example is provided by the theory of ramified analysis up to the the FefermanSchütte ordinal \(\Gamma _{0}\) (see Feferman 1964, pp. 20–21).
 16.
 17.
Toby Meadows (2014) also offers a Kripkestyle construction for naïve validity. A proper assessment of Meadows’ construction would lead us too far afield. Here we limit ourselves to observe that (i) the construction is extremely weak from the structural standpoint, since it forces restrictions of each of the classical structural rules (reflexivity, contraction, and cut) and that (ii) it is not clear whether it addresses Field’s challenge. For a strengthening of Meadows’ theory, see Pailos and Tajer (2017).
 18.
 19.
The clause for introducing \(\forall \) on the right is an \(\omega \)rule. This choice was made to make the KVconstruction into a genuine generalisation of Kripke’s construction. Moreover, in order to simplify the construction, we don’t include a clause for negation. A negation connective obeying the classical metainferences is definable from \(\mathsf {Val}\), putting \(\lnot \varphi \) as Open image in new window .
 20.
 21.
See Boolos (1993). In particular, it cannot satisfy the Valtheoretic version of the second 4like Hilbert–Bernays condition, namely Open image in new window .
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF).
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