Principles for ObjectLinguistic Consequence: from Logical to Irreflexive
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Abstract
We discuss the principles for a primitive, objectlinguistic notion of consequence proposed by (Beall and Murzi, Journal of Philosophy, 3 pp. 143–65 (2013)) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the objectlinguistic consequence predicate internalizes whichever metalinguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity (and some of its variants) by showing that it is conservative over a wide range of base theories: this result is achieved via a wellbehaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of objectlinguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naïve, selfapplicable notion of consequence.
Keywords
Objectlinguistic consequence VCurry paradox Logical consequence Irreflexive consequence1 Introduction
Objectlinguistic treatments of consequence have been extensively investigated in the recent literature: on these approaches, consequence is formalized as a predicate in some firstorder language, and principles governing its behaviour are given. These studies are motivated by diverse philosophical aims, ranging from criticisms to paraconsistent theories [39], to deflationism about consequence [34], to new versions of truththeoretical paradoxes such as Curry’s paradox [1, 18, 27]. Some of these authors, Beall and Murzi [1] and Murzi and Shapiro [23] in particular, also stress the analogy between objectlinguistic treatments of consequence, truth, and comprehension, and call for a unified solution of the resulting paradoxes, arguing that substructural approaches are preferable to fully structural ones.^{1} In order to conform with the terminology adopted in the literature, we will treat ‘consequence’ and ‘validity’ as synonymous where, crucially, consequence or validity do not necessarily coincide with logical consequence or logical validity.
In this paper, we explore different strategies to block the VCurry and related paradoxes. These strategies fall under a common intuition: starting with some metatheoretic consequence relations, we internalize them in the object language in ways that capture their fundamental traits. That is, each such strategy corresponds to the acceptance of different principles for Val and to different restrictions of the structural rules. Different solutions to the paradox, then, correspond to different ways of cashing out the idea that the acceptance of a sentence of the form \(\mathsf {Val}(\ulcorner \varphi \urcorner ,\ulcorner \psi \urcorner )\) is ultimately to be explained and justified by the acceptance of some metatheoretical validity statements, where the acceptance of the latter does not involve objectlinguistic validity principles.
A natural option to develop this strategy, and block the VCurry (and related paradoxes), is to apply (VP)only to logical derivations. Under this reading, Val becomes a primitive predicate for logical validity. This is the strategy followed by Ketland [15]: he axiomatizes Val over Peano Arithmetic (henceforth PA) and proves the consistency of the theory resulting from this restriction of (VP). This option is supported by the fact that (VP) does not preserve logical validity. The very possibility of formulating (VP) requires a wellbehaved machinery to handle the nameforming device \(\ulcorner \cdot \urcorner \) and this machinery does not satisfy uniform substitutivity, violating a basic requirement for logically valid principles.^{2}
However, Ketland’s consistency proof only applies to a restricted category of theories. These theories will be called later reflexive theories. Moreover, Ketland’s strategy is based on the possibility of reducing the primitive logical validity predicate to a provability predicate definable in PA. Besides establishing consistency, such prooftheoretic reductions (such as conservativity and variants of interpretability) also help us in characterizing the notion associated with the logical validity predicate. For instance, the reducibility of the truth predicate to the base theory may be used to assess general conceptions of truth such as truththeoretical deflationism (see [14, Ch. 7]): in the same way, prooftheoretical reductions might be employed to assess deflationary and other general conceptions of logical validity (see [34]). Therefore, in Section 2, we provide more general reduction techniques that, besides yielding a consistency proof, will give us a finergrained analysis of logical validity axiomatized over a wide range of base theories.^{3}
In Section 2.4 we will see that this stratified notion of validity may be understood in terms of a hierarchy of reflection principles over the starting theory: therefore, not only stratified objectlinguistic validity is classically consistent, but it has a natural conceptual analysis in terms of a hierarchy of soundness extensions of the starting theory.
As for truth, these cases pose problems for hierarchical and nonselfapplicable accounts of consequence.Speaker Asays: ‘the negation of everything Isay follows from what Speaker Bsays’.
Speaker Bsays: ‘everything Isay follows from what Speaker Asays’.
In Sections 3–4, we develop an approach to objectlinguistic validity that overcomes this problem: it blocks paradoxical arguments and, at the same time, avoids restrictions on (VP), recovers a natural version of (VD), and delivers a single and genuinely selfapplicable notion of validity. This will be accomplished by an inductive construction that generalizes the one in [16] (strong Kleene version). The fundamental feature of the construction is that the models (for languages with selfapplicable validity) it generates do not satisfy the structural rule of reflexivity.^{5} The smallest fixed point of our construction yields a notion that we might call grounded validity, in that it extends Kripke’s notion of grounded truth (see [16], pp. 694 and 706–707). This is because the metatheoretical notion of validity that holds in the base language determines the extension of the objectlinguistic validity predicate. As we argue in Section 5, grounded validity affords us a natural reading of the naïve validitytheoretical principles.
2 ObjectLinguistic Validity and Classical Logic
One may be tempted to read the rules (VP) and (VD) as characterizing a notion of logical validity or logical consequence. However, it became soon clear that this temptation should be resisted: objectlinguistic treatments of logical consequence simply do not give rise to paradox. This is the conclusion reached by Cook [7] and Ketland [15], and echoed by Field [9]. In particular, the former analyze the Currylike derivation sketched in the introduction and come to the main conclusion that paradox arises when the principles governing this notion (whatever it may be) of primitive consequence are themselves considered to be logically valid.^{6}
 1.
Improving on [7, 15], we give a uniform method for the conservativity of the theories of logical validity over an arbitrary theory extending Elementary Arithmetic (EA), a very weak arithmetical theory.
 2.
This method will also yield the reducibility of the theory of logical validity to reflexive base theories (e.g. PA as in [7, 15]) and its local reducibility in finitely axiomatized based theories (e.g. EA itself), in which ‘reduction’ is intended as a wellbehaved version of relative interpretability that preserves arithmetical vocabulary.
 3.
Even if logical validity is extended to purely arithmetical consequence, classical logic can consistently be kept by interpreting Val as provability in the base theory. Starting form this observation, we show that the hierarchy of validity predicates hinted at by Field [9] can be naturally interpreted as a hierarchy of local reflection principles for the starting theory.
2.1 Arithmetical Theories and Reductions
We now fix some formal details. We work in the language \(\mathcal {L}=\{0,\mathsf {S},+,\times ,\mathsf {exp},\) ≤} of arithmetic. Occurrences of the quantifiers in expressions of the form (∀x ≤ t) φ(x) and (∃x ≤ t) φ(x) where t does not contain x are called bounded. Formulas containing only bounded occurrences of quantifiers are called elementary formulas or Δ_{0}formulas.
All theories considered below will extend Elementary Arithmetic EA (or, equivalently, IΔ_{0} plus the totality of exponentiation).^{7}
The formalization of the syntax of firstorder theories as it is standardly done in, e.g., [33], is carried out without difficulties in EA. In particular, once we show that the standard arithmetization of the syntax can be captured by elementary functions, the fact that EA can Σ_{1}define precisely the elementary functions ensures us that syntactic predicates and notions can be intensionally captured in it.
A relative interpretation preserves the structure of a proof. On many occasions we will employ a more regimented notion of relative interpretation. An interpretation is direct if quantifiers are unrelativized and identity is mapped into identity. Let U and W be such that \(\mathcal {L}\subseteq \mathcal {L}_{U}\cap \mathcal {L}_{W}\). We say that U is \(\mathcal {L}\) embeddable in W if there is a direct interpretation of U into W that leaves the \(\mathcal {L}\)vocabulary unchanged. \(\mathcal {L}\)embedding is a properly stricter notion than relative interpretability.^{9} Finally, U is locally interpretable in W if any finite subtheory of U is relatively interpretable in W. The notion of local \(\mathcal {L}\) embedding is defined analogously.
2.2 ObjectLinguistic Logical Validity
As anticipated, in this subsection we deal with objectlinguistic treatments of logical validity: that is we focus on theories that will be obtained by restricting (VP) only to purely logical derivations. It is worth remarking here that, since we are employing classical logic, the deduction theorem holds: this makes the presentation of the theories of logical validity smoother.
Definition 1
T^{ V }_{0} results from restricting (VP) to purely logical derivations. However, since conditional introduction will be assumed throughout this section, it is convenient to work with a unary rather than a binary validity predicate.
Definition 2 (Primitive logical validity)
That T^{ V } is no essential modification of T^{ V }_{0} is guaranteed by the following:
Proposition 1
T^{ V }and T^{ V }_{0}are mutually \(\mathcal {L}\) embeddable, and so are \(T^{\mathsf {V}}\!\!\upharpoonright \) and \(T^{\mathsf {V_{0}}}\mathpunct \upharpoonright \).
Proof
□
However, consistency may be seen as a necessary but not sufficient condition for a full characterization of the concept of validity captured by T^{ V }. As we mentioned above, a primitive validity predicate is usually motivated – for instance in [1, 23, 34] – along similar lines as the truth predicate: in both cases we aim at expressing metatheoretic facts in the objectlanguage. For instance, one might want to prove in T^{ V } that all tautologies are logically valid, or that so are all implications from a finite subsets of the axioms of T^{ V }. A natural question concerns therefore the costs of the extra expressive power given by V with respect to the inferential resources of the base theory. Moreover, it is mathematically interesting to weigh these costs across a wide range of possible syntactic base theories by abstracting away from specific conditions related to a particular choice of the base theory. From this point of view, a general study of the properties of the theory of objectlinguistic validity such as its \(\mathcal {L}\)embedding in the base theory T, conservativity over T, finite axiomatizability over T, become integral part of the study of this notion of validity.
Next we show that the primitive notion of consequence given by T^{ V } cannot serve an expressive role of finite reaxiomatization.^{12}
Lemma 1
T ^{ V } is not finitely axiomatizable.
Proof
Let I_{ V }^{ n } be the least fixed point of this inductive definition. \((\mathbb {N},\mathsf {I}_{V^{n}})\) is a model of A but it cannot be a model of T^{ V }. □
Lemma 2
 (i)
If T is reflexive, then \(\mathfrak {a}\) is an \(\mathcal {L}\) embeddingof T^{ V }in T.
 (ii)
If T is finitely axiomatizable, \(\mathfrak {a}\) cannot be an interpretation of T^{ V }in T.
Proof
Both proofs are immediate. For (i), one simply notices that T, being reflexive, proves \(\mathsf {Pr}_{\varnothing }(\ulcorner \varphi \urcorner )\rightarrow \varphi \) for all \(\varphi \in \mathcal {L}\) by Rfn(S).
Part (i) of the previous lemma obviously entails the interpretability of T^{ V } in T for reflexive T, being \(\mathcal {L}\)embeddings stricter than intepretability. Two further remarks: the interpretation \(\mathfrak {a}\) is a variant of the one contained in [15], which only takes care of external occurrences of V without applications of the recursion theorem. Moreover, in Lemma 2(i), \(\mathfrak {a}\) is indeed an \(\mathcal {L}\)embedding of T^{ V } in T when the latter is reflexive. This clearly indicates that, in the case of reflexive theories, the notion of validity governed by (VP_{1}) and (VD_{1}) can be unequivocally understood as a definable notion of logical validity. Lemma 2(i) also immediately yields the conservativity of T^{ V } over T for reflexive T.
However reflexive theories are in many senses very special and they have a peculiar behaviour with respect to interpretability and related notions. For instance, by Orey’s compactness theorem,^{13} reflexive theories collapse the distinction between local and global interpretability and they have the very convenient feature of proving the reflection principle for pure logic that is, as we have seen, closely related to (VD_{1}). It is therefore natural to generalize the picture given by Lemma 2 and ask ourselves whether the notion of consequence captured by (VP_{1}) and (VD_{1}) can be uniformly characterized also in the case of nonreflexive theories. As we anticipated, we will focus on finitely axiomatized theories, which are provably distinct from reflexive theories due to Gödel’s second incompleteness theorem.
We recall that U is locally interpretable in V if every finite subtheory of U is interpretable in V. Similarly, U is locally \(\mathcal {L}\) embeddable in V if every finite \(U_{0}\subseteq U\) is \(\mathcal {L}\)embeddable in V. We also recall that T^{ V } is formulated in a Hilbertstyle calculus in which modus ponens is the only rule of inference. Proofs in T^{ V } of φ are therefore objects of the form \( \mathcal {D}=\langle \varphi _{0},\ldots ,\varphi _{n1},\varphi \rangle \) where each element of the sequence is either an axiom of T^{ V } or it has been obtained from previous elements by modus ponens. Also, the code \(\mathcal {D}\) of \(\mathcal {D}\) is the code of \(\langle \ulcorner \varphi _{0}\urcorner ,\ldots ,\ulcorner \varphi _{n1}\urcorner \rangle \).
To prove the local \(\mathcal {L}\)embeddability of T^{ V } in T, we need the following, well known fact:
Lemma 3 (Σ_{1}completeness)
For every Σ_{1}formula φof \(\mathcal {L}\) and every T extending EA, if \(\mathbb {N}\vDash \varphi \), then \(\, T\vdash \varphi \).
The informal idea for the proof of the local \(\mathcal {L}\)embeddability of T^{ V } in T is straightforward: we translate only the outermost occurrences of V because only one ‘layer’ of the logical validity predicate matters in logical proofs. This enables us to dispense with uses of more sophisticated devices, such as the recursion theorem, to translate within Gödel corners.
Proposition 2
T^{ V }is locally \(\mathcal {L}\) embeddablein T.
Proof
Let B be a finite subsystem of T^{ V }. In B, we can safely assume that there are at most m applications of (VP_{1}) to logical proofs \(\mathcal {D}_{i}\), i ≤ m.
Here \(\mathsf {Prf}_{\varnothing }(\cdot ,\cdot )\)is elementary and expresses Hilbertstyle provability in\(\mathsf { PL}(\mathcal {L}^{+})\), predicate logicin the language \(\mathcal {L}^{+}\). This n is fixed and will be kept so throughout the proof.
If the latter, then we are done by Lemma 3 because V_{ n }iselementary. If the former disjunct obtains, there is a purely logical derivation\(\mathcal {D}_{j}\)of ψ such that\(\mathcal {D}_{j}\leq n\). Then there is a purelylogical derivation \(\mathcal {D}^{\mathfrak {b}}_{j}\)of \(\psi ^{\mathfrak {b}}\)obtained bytranslating its elements.^{14}
For the induction step, we only need to worry about (VP_{1}). Now if φ is obtained by anapplication of (VP_{1}),it has the form \(\mathsf {V} (\ulcorner \psi \urcorner )\)for ψ an\(\mathcal {L}^{+}\)sentence and there isa purely logical proof \(\mathcal {D}_{i}\)of ψ. Byassumption, \(\,\mathcal {D}_{i}\,\leq n\),therefore \(T\vdash \mathsf {V}_{\!n}(\ulcorner \psi \urcorner )\)byLemma 3. □
Now Proposition 2 immediately yields, besides the consistency of T^{ V } that wasn’t seriously doubted, the conservativity of T^{ V } over T for any T extending EA.
Corollary 1
T^{ V }is a conservative extension of T.
Proof
If \({T^{\mathsf {V}}} \vdash \varphi \) and \(\varphi \in \mathcal {L}\), then already a finite subsystem \(B\subset {T^{\mathsf {V}}}\) proves φ. By Proposition 2, \(T\vdash \varphi ^{\mathfrak {b}}\). But \(\varphi ^{\mathfrak {b}}\) is nothing more than φ itself by definition of \(\mathfrak {b}\). □
The conservativity of T^{ V } over T immediately yields the consistency of T^{ V }, relative to the consistency of T, that was assumed in Definition 2. It should be noted that the conservativity of \(T^{\mathsf {V}}\!\!\upharpoonright \) can be obtained in a straightforward way since any model \(\mathcal {M}\) of \(T\supseteq \mathsf {EA}\) can be expanded to a model \((\mathcal {M},S)\) of \(T^{\mathsf {V}}\!\!\upharpoonright \) where S is the set specified in the inductive definition (6). It is not clear to us whether this strategy can be adapted to the full T^{ V }. The strategy employed in Proposition 2, however, has the additional advantage of being formalizable with only weak arithmetical assumptions.
Moreover, we obtain another proof of the interpretability of T^{ V } in T, for T reflexive, by Orey’s compactness theorem:
Corollary 2
For \(T\supseteq \mathsf {EA}\) and reflexive, T^{ V }is interpretable in T.
As far as the authors know, the question of the global interpretability of T^{ V } for arbitrary \(T\supseteq \mathsf {EA}\) is still open.
2.3 Extending Logical Consequence
Observation 1
The theory T^{ V } + K+9is \(\mathcal {L}\) embeddablein and conservative over T for reflexive \(T\supseteq \mathsf {EA}\).
Let’s abbreviate T^{ V } + K by writing T^{ V }^{+}. Can we obtain analogues of Proposition 2 and Corollary 1 for T^{ V }^{+} over arbitrary \(T\supseteq \mathsf {EA}\)? It turns out that we can, by suitably tweaking the proofs given above.^{15} The fundamental idea is to modify the bound given in the definition of V _{ n } in Proposition 2 to allow for the concatenation of the logical proofs of formulas φ and \(\varphi \rightarrow \psi \) of \(\mathcal {L}^+\) when the translations of the antecedent of K are assumed.
Proposition 3
T^{ V }^{+}islocally \(\mathcal {L}\) embeddablein T.
Proof
As before, let B be a finite subsystem of T^{ V }^{+}. Again, we fix a standard n as bound for the codes of the finitely many logical proofs \(\mathcal {D}_{i}\) preceding an application of (VP_{1}).
As before, we define the translation\(\mathfrak {c}\)that, like\(\mathfrak {b}\), only replaces outeroccurrences of V in proofs,clearly this time with C_{ n }and not V_{ n }.
Therefore \(\mathbb {N}\vDash \mathsf {C}_{n}(\ulcorner \psi \urcorner )\)and \(T\vdash \mathsf {C}_{n}(\ulcorner \psi \urcorner )\)byLemma 3. □
By the same argument given in Corollary 1:
Corollary 3
T^{ V }^{+}is a conservative extension of T.
Again, this guarantees the consistency of T^{ V }^{+} relative to the consistency of T. As above, Orey’s compactness theorem gives us:
Corollary 4
For T reflexive, T^{ V }^{+}is interpretable in T.
The results just presented improve the picture discussed in [7, 15] and tell us that in many respects – especially if one focuses on conservativity – primitive logical validity is uniformly reducible to the resources of the base theory for a much wider class of theories than the one considered before. However, we were not able to show the interpretability, let alone the \(\mathcal {L}\)embedding, of T^{ V } and T^{ V }^{+} in T. This is, however, not an unexpected difficulty: by Orey’s and analogous results, there is no gap between local and global interpretability in the context of reflexive theories such as PA. In the case of finitely axiomatizable theories, by contrast, the relationships between these two notions vary considerably and are usually hard to characterize.^{17}
From the point of view of the theory of logical validity the lesson to learn is apparent: the combination of (VP) and (VD)cannot be taken to characterize logical validity, which is unparadoxical and uniformly conservative over base theories that contain just a minimum amount of syntactic reasoning.
2.4 Arithmetical Consequence and Hierarchies
In the previous two subsections we analyzed primitive logical consequence based on an introduction rule (VP_{1}) for the primitive validity predicate restricted to purely logical proofs. It turns out that no incisions on classical reasoning are needed even if one liberalizes (VP_{1}) to arithmetical consequence. We now show that by iterating this idea to the transfinite we obtain a symmetry between the hierarchy of validity predicates suggested in [1, 9] and the hierarchy of local reflection principles for a starting theory T. For the sake of determinateness, we assume our starting theory to be EA, although the arguments would proceed in an analogous way for any \(T\supseteq \mathsf {EA}\).
To define a hierarchy of primitive notions of validity, we assume a notation \((\mathsf {OR},\prec )\) for ordinals up to Γ_{0}, available in EA,^{18} and a countable stock of predicates V_{ a }(x) – where a ranges over codes of ordinals α < Γ_{0}, that is we take a Latin alphabet letter to code the corresponding ordinal in the Greek alphabet. We let \(\mathcal {L}_{0}\) to be \(\mathcal {L}\) itself and \(\mathcal {L}_{\alpha +1}\) is \(\mathcal {L}_{\alpha } \cup \{\mathsf {V}_{\!a}\}\); \(\mathcal {L}_{\lambda }\), for λ limit, contains all V_{ b } for β < λ.
Definition 3 (Hierarchical validity)
Proposition 4
For α < Γ_{0}, S^{ α }is \(\mathcal {L}\) embeddablein R^{ α }.
Proof
Proposition 4 suggests at least the following two remarks: for the reader interested in the mathematical strength of the theories S^{ α }, by a result of Beklemishev [2], these theories will prove no more Π_{1}arithmetical sentences than ω^{ α } iterated consistency progressions over EA. At the philosophical level, the theories S^{ α } embody a notion of arithmetical validity corresponding to a proper extension of arithmetical provability in the starting theory EA stratified along ordinal paths that are meaningful in the starting theory. No incision on classical logic is needed at this stage. However, as pointed out also in [9], the formulation of the theories S^{ α } relies on how many ordinals we can code in the starting theory. In order to read off a notion of validity from this stratified picture there seem to be only two options: either validity is inherently stratified, or there is a specific countable ordinal α such that S^{ α } fulfills the requirements we are willing to ascribe to the notion of validity. Neither of these alternatives, however, is completely satisfactory; for one thing, there is no reason to think that validity should be stratified, unless one is happy to concede that also truth is a stratified notion. Moreover, the countable ordinals that are provably wellfounded in arithmetical theories vary considerably, and it is highly implausible that the notion of validity should be tied to these implementation details.
This is not to say, however, that stratified validity lacks importance. Even if it doesn’t afford a viable notion of validity, it gives us a tool to generate validities starting from valid inferences in the base language. This picture will be improved in the in the next two sections, where we will turn the hierarchical increase of validities into a positive inductive definition. This technical shift will yield a truly selfapplicable notion of validity whose extension and properties are independent of how we represent ordinals. As in the case of truth, this will require restricting classical reasoning.
3 A New Construction for Naïve Validity
In this section we propose a way of transcending the stratified picture of validity that generalizes Kripke’s method to provide models for languages with a selfapplicable truth predicate (see [16]). We will be mainly concerned with providing a class of models that makes the naïve principles for validity consistent, and not so much with formulating effectively presented theories of validity (as in the previous section).^{19} The models are obtained via fixed points of an inductive construction. Similarly to what happens in Kripke’s theory of truth, we have only one validity predicate that can be introduced without restrictions, via (VP). At a fixed point, (VP) and all the principles that are accepted in the construction can be iterated arbitrarily, thus internalizing all the inferences deriving from validities of the base language.
3.1 Initial Sequents and Rules of Inferences
Since, in the perspective of an unstratified picture of validity, (VP) is not in question, what are we do to with the VCurry Paradox? We have seen in the introduction that the paradox forces a restriction of contraction, cut, or (VD). The idea of transcending the hierarchical conception of validity via an inductive definition is at odds with (VD), even though it calls for an unrestricted acceptance of contraction and cut.^{20} For sure, (VD) looks perfectly fine if we read Val as an unspecified ‘following from’ and clearly also as the notion of logical validity studied in Sections 2.2 and 2.3, but things are different if we accept (VP) unrestrictedly: in so doing, we take Valstatements to represent a naïve notion of consequence, namely metainferences that hold in virtue of logical, basetheoretic, and validitytheoretic principles. However, in the presence of full (VD), this idea translates into the acceptance of sentences that we might not want to accept, such as ν.
How can we avoid this new path to triviality? Our proposed solution consists in the development of a Kripkestyle positive inductive definition that, while restricting (VD) and (Ref), consistently satisfies (VP), (VDm), contraction, cut, and indeed every other classically valid rule of inference (with nonzero premises).^{21} More generally, our construction will operate uniform restrictions on initial sequents: this harmonizes with the motivation to restrict (VD) outlined above, since arbitrary initial sequents might contain Valsentences codifing problematic inferences. Rules of inference, by contrast, are safe: if we can control the sequents that we accept, we can adopt all such rules. A Kripkestyle construction along the lines of the one developed here has been hinted at by Field in [9]. Meadows [20] also develops an inductive construction that recovers all Beall and Murzi’s principles. His construction also rejects reflexivity but, unlike ours, it is not closed under contraction and cut.^{22}
3.2 The KVConstruction
The generalization of Kripke’s construction we propose here consists in dealing with sequents rather than single sentences. By ‘sequent’, from now on, we will mean an object of the form \({\Gamma } \Rightarrow {\Delta }\), where Γ (the antecedent) and Δ (the consequent) are finite sets of \(\mathcal {L}_{V}\)sentences.

start with a set of sequents containing at least the ones the form \({\Gamma } \Rightarrow {\Delta }, s_{0}=t_{0}\) and \(s_{1}=t_{1}, {\Gamma } \Rightarrow {\Delta }\), for s_{0} = t_{0} an atomic arithmetical truth, and s_{1} = t_{1} an atomic arithmetical falsity,

apply the operational and structural rules of inference to them,

internalize the sequents so obtained within the validity predicate, interpreting Val via principles that are modelled after the rules of inference for the classical material conditional \(\supset \).
Definition 4
 (i)
n ∈ S; or
 (ii)
n is \({\Gamma } \Rightarrow s=t, {\Delta }\), and \(\mathbb {N} \models s=t\); or
 (iii)
n is \({\Gamma }, s=t \Rightarrow {\Delta }\), and \(\mathbb {N} \models s\neq t\); or
 (iv)
n is \({\Gamma } \Rightarrow \varphi \wedge \psi , {\Delta }\), and \({\Gamma } \Rightarrow \varphi , {\Delta } \in S\), and \({\Gamma } \Rightarrow \psi , {\Delta } \in S\); or
 (v)
n is \({\Gamma }, \varphi \wedge \psi \Rightarrow {\Delta }\), and \({\Gamma }, \varphi , \psi \Rightarrow {\Delta } \in S\); or
 (vi)
n is \({\Gamma } \Rightarrow \varphi \vee \psi , {\Delta }\), and \({\Gamma } \Rightarrow \varphi , \psi , {\Delta } \in S\); or
 (vii)
n is \({\Gamma }, \varphi \vee \psi \Rightarrow {\Delta }\), and \({\Gamma }, \varphi \Rightarrow {\Delta } \in S\), and \({\Gamma }, \psi \Rightarrow {\Delta } \in S\); or
 (viii)
n is \({\Gamma } \Rightarrow \forall x \varphi (x), {\Delta }\), and for all \(t \in \mathsf { Cter_{\mathcal {L}_{V}}}\), \({\Gamma } \Rightarrow \varphi (t), {\Delta } \in S\); or
 (ix)
n is \({\Gamma }, \forall x \varphi (x) \Rightarrow {\Delta }\), and for some \(t \in \mathsf { Cter_{\mathcal {L}_{V}}}\), \({\Gamma }, \varphi (t) \Rightarrow {\Delta } \in S\); or
 (x)
n is \({\Gamma } \Rightarrow \mathsf {Val}(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner ), {\Delta }\), and \({\Gamma }, \varphi \Rightarrow \psi , {\Delta } \in S\); or
 (xi)
n is \({\Gamma }, \mathsf {Val}(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner ) \Rightarrow {\Delta }\), and \({\Gamma } \Rightarrow \varphi , {\Delta } \in S\), and \({\Gamma }, \psi \Rightarrow {\Delta } \in S\).

for every \(S \subseteq \omega \), we have that \(S \subseteq {\Psi }(S)\);

for every \(S_{1}, S_{2} \subseteq \omega \), if \(S_{1} \subseteq S_{2}\), then \({\Psi }(S_{1}) \subseteq {\Psi }(S_{2})\).
In the next section, we prove that I_{Ψ} can be used to interpret \(\mathcal {L}_{V}\)sequents in a nontrivial way. We will then investigate the behaviour of the structural rules and of the validitytheoretical principles in fixed points of Ψ.
4 Main Properties of the KVConstruction
We start by showing that there are consistent fixed points of Ψ. A fixed point S_{Ψ} is consistent if it does not contain the empty sequent \(\varnothing \Rightarrow \varnothing \). Consistency typically avoids triviality: if \(\varnothing \Rightarrow \varnothing \) is in a fixed point closed under weakening, then every sequent is in that fixed point.^{23}
Proposition 5
I _{Ψ} is consistent.
Proof
The proof is by induction on the stages \(\mathsf {I}^{\alpha }_{\Psi }\) of the construction of I_{Ψ}. The claim is trivial for \(\mathsf {I}^{0}_{\Psi }\) and \(\mathsf {I}^{1}_{\Psi }\). Assuming the claim for the stage \(\mathsf { I}^{\alpha }_{\Psi }\), one simply notices that the stage \(\mathsf {I}^{\alpha +1}_{\Psi }\) is obtained from \(\mathsf {I}^{\alpha }_{\Psi }\) by adding to it all sequents resulting from an application of the clauses of Definition 4. It is then clear that, if \(\varnothing \Rightarrow \varnothing \) is not in \(\mathsf {I}^{\alpha }_{\Psi }\), no such clause can introduce it in \(\mathsf {I}^{\alpha +1}_{\Psi }\). The limit case follows straightforwardly from the successor cases. □
We first notice that every stage \(\mathsf {I}_{\Psi }^{\alpha }\) of I_{Ψ} is closed under left and right weakening. By construction, any sequent \({\Gamma } \Rightarrow {\Delta }\) in \(\mathsf {I}_{\Psi }^{\alpha }\) is obtained by applying a series of Ψclauses to sequents containing arithmetical truths or falsities, with arbitrary side sentences. Therefore, in order to have \({\Gamma }, {\Gamma }^{\prime } \Rightarrow {\Delta },{\Delta }^{\prime }\) in \(\mathsf { I}_{\Psi }^{\alpha }\), we simply consider the same succession of Ψclauses applied to starting sequents with \({\Gamma }^{\prime }\) and \({\Delta }^{\prime }\) as extra side sentences.
Lemma 4 (Weakening)
For every ordinal α, if \({\Gamma } \Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then for every \({\Gamma }^{\prime }, {\Delta }^{\prime } \subseteq \mathsf {Sent_{\mathcal {L}_{V}}}\), the sequent \({\Gamma }, {\Gamma }^{\prime } \Rightarrow {\Delta },{\Delta }^{\prime }\) is in \(\mathsf { I}_{\Psi }^{\alpha }\).
Also, since we are dealing with finite sets, left and right contraction hold for every sequent in every stage of the construction of I_{Ψ}.
Lemma 5 (Contraction)
For every ordinal α, if \({\Gamma },\varphi ,\varphi \Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then \({\Gamma },\varphi \Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\). Similarly, if \({\Gamma }\Rightarrow \psi ,\psi ,{\Delta }\) is in \(\mathsf { I}_{\Psi }^{\alpha }\), then \({\Gamma }\Rightarrow \psi ,{\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\).
A crucial feature of our construction is that, at any stage, sequents are grounded in at least one sentence in their antecedent or consequent.
Lemma 6 (Groundedness)
For every ordinal αand every sequent \({\Gamma } \Rightarrow {\Delta }\), if \({\Gamma } \Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then there is at least one sentence φin Γsuch that \(\varphi \Rightarrow \varnothing \) is in \(\mathsf {I}_{\Psi }^{\alpha }\), or at least one sentence ψin Δsuch that \(\varnothing \Rightarrow \psi \) is in \(\mathsf {I}_{\Psi }^{\alpha }\).
Proof
We reason by induction on the construction of I_{Ψ}. The claim is trivial for \(\mathsf {I}_{\Psi }^{0}\). For \(\mathsf {I}_{\Psi }^{1}\), the claim is also immediate since this set contains only sequents with atomic arithmetical truths in the consequent or atomic arithmetical falsities in the antecedent. We now assume the claim up to α, and prove it for α + 1. Let \({\Gamma } \Rightarrow {\Delta } \in \mathsf {I}_{\Psi }^{\alpha +1}\) be obtained by applying one Ψ clause to sequents in \(\mathsf {I}_{\Psi }^{\alpha }\). We consider two cases.
If \({\Gamma } \Rightarrow {\Delta }\)is obtained via the Ψclause (xi) of Definition 4,then it has the form \({\Gamma }^{\prime }, \mathsf {Val}(\ulcorner \varphi _{0}\urcorner ,\ulcorner \varphi _{1}\urcorner )\Rightarrow {\Delta }\) and \(\mathsf {I}_{\Psi }^{\alpha }\) contains \({\Gamma }^{\prime } \Rightarrow \varphi _{0},{\Delta }\) and \({\Gamma }^{\prime },\varphi _{1}\Rightarrow {\Delta }\). If the induction hypothesis gives us sequents \(\psi \Rightarrow \varnothing \)or \(\varnothing \Rightarrow \chi \) where ψ or χ are in \({\Gamma }^{\prime }\)or Δ respectively, we are done. In the only other case, the induction hypothesis gives us that\(\varphi _{1}\Rightarrow \varnothing \) and \(\varnothing \Rightarrow \varphi _{0}\) are in \(\mathsf {I}_{\Psi }^{\alpha }\). By the Ψclause (xi) of Definition 4, then, we get \(\mathsf {Val}(\ulcorner \varphi _{0}\urcorner ,\ulcorner \varphi _{1}\urcorner )\Rightarrow \varnothing \)in \(\mathsf {I}_{\Psi }^{\alpha +1}\). □
To prove the closure of the stages of the construction of I_{Ψ} under cut, we need the following inversion lemma.
Lemma 7 (Inversion)
 (i)
If \({\Gamma } \Rightarrow \varphi \wedge \psi , {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then \({\Gamma } \Rightarrow \varphi , {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\) and \({\Gamma } \Rightarrow \psi , {\Delta } \) is in \(\mathsf { I}_{\Psi }^{\alpha }\).
 (ii)
If \({\Gamma }, \varphi \wedge \psi \Rightarrow {\Delta }\) is in \( \mathsf {I}_{\Psi }^{\alpha }\), then \({\Gamma }, \varphi , \psi \Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\).
 (iii)
If \({\Gamma } \Rightarrow \varphi \vee \psi , {\Delta }\) is in \( \mathsf {I}_{\Psi }^{\alpha }\), then \({\Gamma } \Rightarrow \varphi , \psi , {\Delta }\) is in \( \mathsf {I}_{\Psi }^{\alpha }\).
 (iv)
If \({\Gamma }, \varphi \vee \psi \Rightarrow {\Delta }\) is in \( \mathsf {I}_{\Psi }^{\alpha }\), then \({\Gamma }, \varphi \Rightarrow {\Delta }\) is in \( \mathsf {I}_{\Psi }^{\alpha }\) and \({\Gamma }, \psi \Rightarrow {\Delta } \) is in \(\mathsf { I}_{\Psi }^{\alpha }\).
 (v)
If \({\Gamma } \Rightarrow \forall x \varphi (x), {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then for all \(t \in \mathsf {Cter_{\mathcal {L}_{V}}}\): \({\Gamma } \Rightarrow \varphi (t), {\Delta }\) is in \( \mathsf { I}_{\Psi }^{\alpha }\).
 (vi)
If \({\Gamma }, \forall x \varphi (x) \Rightarrow {\Delta } \) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then for some \(t \in \mathsf {Cter_{\mathcal {L}_{V}}}\): \({\Gamma }, \varphi (t) \Rightarrow {\Delta }\) is in \(\mathsf { I}_{\Psi }^{\alpha }\).
 (vii)
If \({\Gamma } \Rightarrow \mathsf {Val}(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner ), {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then \({\Gamma }, \varphi \Rightarrow \psi , {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\).
 (viii)
If \({\Gamma }, \mathsf {Val}(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner ) \Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\), then \({\Gamma } \Rightarrow \varphi , {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\) and \({\Gamma }, \psi \Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\).
Proof
Finally, we show that every stage of the construction of I_{Ψ} is closed under cut.
Proposition 6 (Closure under cut)
For every α, if \({\Gamma }\Rightarrow {\Delta },\varphi \) and \(\varphi ,{\Gamma }\Rightarrow {\Delta }\) are in \(\mathsf {I}_{\Psi }^{\alpha }\), then also \({\Gamma }\Rightarrow {\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\).
Proof
The proof is by induction. The case for \(\mathsf {I}_{\Psi }^{0}\) is trivial. The case for \(\mathsf {I}_{\Psi }^{1}\) is also immediately obtained since, for \({\Gamma }\Rightarrow {\Delta },\varphi \) and \(\varphi ,{\Gamma }\Rightarrow {\Delta }\) to be in \(\mathsf {I}_{\Psi }^{1}\), Γ or Δ have to contain at least one atomic arithmetical falsity or truth respectively.
 (a) We consider the case in which φ is of the form \(\mathsf {Val}(\ulcorner \varphi _{0}\urcorner ,\ulcorner \varphi _{1}\urcorner )\).Therefore the sequentsare in \(\mathsf {I}_{\Psi }^{\alpha }\). By the weakening lemma applied to (II), also \({\Gamma }\Rightarrow \varphi _{0},\varphi _{1},{\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\). By induction hypotesis, since (I) is in \(\mathsf {I}_{\Psi }^{\alpha }\), also \({\Gamma }\Rightarrow \varphi _{1},{\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\). Therefore, since (III) is also in \(\mathsf {I}_{\Psi }^{\alpha }\), \({\Gamma }\Rightarrow {\Delta }\) will be in \(\mathsf {I}_{\Psi }^{\alpha }\subseteq \mathsf {I}_{\Psi }^{\alpha +1}\) as well, as desired.$$\begin{array}{@{}rcl@{}} \text{(I)}\hspace{10pt}{\Gamma},\varphi_{0}\Rightarrow\varphi_{1}, {\Delta} \qquad\qquad \text{(II)}\hspace{10pt} {\Gamma}\Rightarrow \varphi_{0},{\Delta} \qquad\qquad \text{(III)}\hspace{10pt} {\Gamma},\varphi_{1}\Rightarrow {\Delta} \end{array} $$
 (b) We only consider the case in which φ is \(\mathsf {Val}(\ulcorner \varphi _{0}\urcorner ,\ulcorner \varphi _{1}\urcorner )\). We assume, moreover, that \({\Gamma }\Rightarrow {\Delta }, \mathsf {Val}(\ulcorner \varphi _{0}\urcorner ,\ulcorner \varphi _{1}\urcorner )\) is obtained via the Ψclause (viii) from sequents in \(\mathsf {I}_{\Psi }^{\alpha }\) of the formfor all i ∈ ω, and that\(\mathsf {Val}(\ulcorner \varphi _{0}\urcorner ,\ulcorner \varphi _{1}\urcorner ),{\Gamma }\Rightarrow {\Delta }\) is obtainedvia the Ψclause(xi) from \({\Gamma }\Rightarrow \varphi _{0},{\Delta }\) and \({\Gamma },\varphi _{1}\Rightarrow {\Delta }\) in \(\mathsf {I}_{\Psi }^{\alpha }\). On the one hand, by the inversion lemma applied to all sequentsof the form (IV), we obtain that all sequents of the form \({\Gamma },\varphi _{0}\Rightarrow {\Delta }^{\prime }, \varphi (t_{i}),\varphi _{1}\) are in \(\mathsf {I}_{\Psi }^{\alpha }\). By the weakening lemma, since \({\Delta }={\Delta }^{\prime },\forall x\varphi (x)\),$$\text{(IV)}\hspace{10pt}{\Gamma}\Rightarrow {\Delta}^{\prime}, \varphi(t_{i}), \mathsf{Val}(\ulcorner \varphi_{0}\urcorner,\ulcorner\varphi_{1}\urcorner) $$is in \(\mathsf {I}_{\Psi }^{\alpha }\) for all i ∈ ω. On the other, from the fact that \({\Gamma }\Rightarrow \varphi _{0},{\Delta }\) and \({\Gamma },\varphi _{1}\Rightarrow {\Delta }\) are in \(\mathsf {I}_{\Psi }^{\alpha }\), the weakening lemma gives us$$\text{(V)}\hspace{10pt}{\Gamma},\varphi_{0}\Rightarrow {\Delta}, \varphi(t_{i}),\varphi_{1} $$in \(\mathsf {I}_{\Psi }^{\alpha }\). By induction hypotesis, since for all i ∈ ω(V) and (VI) are in \(\mathsf {I}_{\Psi }^{\alpha }\), also \({\Gamma }\Rightarrow \varphi (t_{i}), \varphi _{1},{\Delta }\) is in \(\mathsf {I}_{\Psi }^{\alpha }\) for all i ∈ ω. Therefore, since for all i ∈ ω (VII) is also in \(\mathsf {I}_{\Psi }^{\alpha }\), \({\Gamma }\Rightarrow \varphi (t_{i}), {\Delta }\) will be in \(\mathsf {I}_{\Psi }^{\alpha }\) for all i ∈ ω. An application of the Ψclause (viii) gives us the desired result.$$\begin{array}{@{}rcl@{}} \text{(VI)}\hspace{10pt}{\Gamma}\Rightarrow\varphi(t_{i}),\varphi_{0}, \varphi_{1},{\Delta} \qquad\qquad \text{(VII)}\hspace{10pt} {\Gamma},\varphi_{1}\Rightarrow\varphi(t_{i}), {\Delta} \end{array} $$
 (c) We consider the case in which \({\Gamma }\Rightarrow {\Delta },\varphi \) and \(\varphi ,{\Gamma }\Rightarrow {\Delta }\)are obtained by application of the Ψclause (viii) to sequents in \(\mathsf {I}_{\Psi }^{\alpha }\) of the formfor all i ∈ ω. Bythe groundedness lemma applied to all sequents of the form (VIII), we obtain, for each i ∈ ω, that either thereis a sequent \(\psi _{i}\Rightarrow \varnothing \) is in \(\mathsf {I}_{\Psi }^{\alpha }\) for ψ_{ i } ∈ Γ, or there is a sequent \(\varnothing \Rightarrow \chi _{i}\) in \(\mathsf {I}_{\Psi }^{\alpha }\) with \(\chi _{i}\in {\Delta }^{\prime },\varphi _{0}(t_{i}),\varphi \). In the former case, we are done by the weakening lemma; in the latter case, if\(\chi _{i}\in {\Delta }^{\prime }\)we are also done by weakening, otherwise we reason as follows. If χ_{ i }is φ_{0}(t_{ i }) for all i, an application of the Ψclause (viii) gives us that \(\varnothing \Rightarrow \forall x\varphi _{0}(x)\) is in \(\mathsf {I}_{\Psi }^{\alpha +1}\), therefore the claim follows by the weakening lemma. If χ_{ i }is φ for some i, we apply the groundedness lemma to all sequents of the form (IX). By the consistency of I_{Ψ}, the induction hypothesis cannot give us \(\varphi \Rightarrow \varnothing \) in \(\mathsf {I}_{\Psi }^{\alpha }\). In all other possible outcomes of the groundedness lemma applied to all sequents of the form (IX), we reason as we did in the corresponding cases of (VIII).$$\begin{array}{@{}rcl@{}} \text{(VIII)}\hspace{10pt}{\Gamma}\Rightarrow{\Delta}^{\prime},\varphi_{0}(t_{i}),\varphi \qquad\qquad \text{(IX)}\hspace{10pt}\varphi,{\Gamma}\Rightarrow{\Delta}^{\prime\prime},\varphi_{1}(t_{i}) \end{array} $$
Reflexivity is the only structural rule that does not hold unrestrictedly in I_{Ψ} (the proof is a slight variant of the VCurry derivation sketched in the introduction).
Lemma 8
Lemma 8 shows also that dropping reflexivity is ‘best possible’: we have a single structural rule that cannot be consistently accepted.^{24} I_{Ψ}, however, features a weaker form of reflexivity, which follows immediately from the weakening lemma. For every \(\varphi \in \mathcal {L}_{V}\), we can always find Γ and \( {\Delta } \subseteq \mathsf {Sent}_{\mathcal {L}_{V}}\) such that \({\Gamma }, \varphi \Rightarrow \varphi , {\Delta } \in \mathsf {I}_{\Psi }\), where Γ and Δ can be taken to be disjoint.
4.1 Naïve Validity in I _{Ψ}
Several principles for naïve validity (including the (ValSchema)^{+} formulated and discussed in [9]) are recovered in I_{Ψ}, in the sense made precise by the following result.
Lemma 9
Since (VAlSchema)^{+} holds in I_{Ψ}, it is clear that (VP) and (VAlSchema) are recovered in I_{Ψ} as well.
(VDm) is not to be understood as a ‘weaker version’ of (VD), since there are theories that validate (VD) but for which (VDm) is too strong and yields triviality. The nontransitive approach of Ripley [29] and Cobreros et al. [6] is a case in point: adapting the theory developed there to the validity predicate, we see that (VD) holds, while an unrestricted acceptance of (VDm) would trivialize the theory. Essentially, this is because (VDm) incorporates a form of cut, which is clearly inadmissible in a nontransitive approach.^{25}
In Section 3.1 we suggested that a uniform way to avoid VCurrydriven triviality consists in restricting our acceptance of initial sequents, avoiding the acceptance of Valsentences that express inferences that we cannot control. Rules of inference, on the other hand, are safe, since the construction of I_{Ψ} operates a selection over the sequents that are accepted in the first place. This view sits naturally with a restriction of (Ref) and (VD) (the only schematic inferences amongst the structural rules and the validitytheoretical principles) and an unrestricted acceptance of weakening, contraction, cut, (VP), (VDm), and (ValSchema)^{+}. The results 4, 5, 6, 8, and 9, establish that I_{Ψ} realizes the solution to the paradoxes described in Section 3.
It is easy to turn I_{Ψ} into a proper model of the language \(\mathcal {L}_{V}\), as it is standardly done with Kripke fixed points. Let the extension of the validity predicate generated by I_{Ψ}, in symbols E_{Ψ}, be the set of those pairs of \(\mathcal {L}_{V}\)sentences 〈φ, ψ〉 such that \(\varnothing \Rightarrow \mathsf {Val}(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner )\) is in I_{Ψ}, and the antiextension of the validity predicate generated by I_{Ψ}, in symbols A_{Ψ}, be the set of pairs of \(\mathcal {L}_{V}\)sentences 〈φ, ψ〉 such that \(\mathsf {Val}(\ulcorner \varphi \urcorner , \ulcorner \psi \urcorner ) \Rightarrow \varnothing \) is in I_{Ψ}. The model of \(\mathcal {L}_{V}\) naturally associated with I_{Ψ}, thus, is \((\mathbb {N}, \mathsf {E}_{\Psi }, \mathsf {A}_{\Psi })\), and its evaluation clauses can be read off Definition 4. In particular, \((\mathbb {N}, \mathsf {E}_{\Psi }, \mathsf {A}_{\Psi })\) can be associated with a threevalued semantics, say with values {0,1/2,1}, where the logical vocabulary is interpreted as in strong Kleene semantics (with Val being treated as the strong Kleene conditional), and where \({\Gamma } \Rightarrow {\Delta }\) has a tolerantstrict reading, that is whenever every φ ∈ Γ has value 1/2 or 1, then there is a ψ ∈ Δ with value 1.^{26}
4.2 NonMinimal Fixed Points and Extensions

(xii) n is \(({\Gamma }, {\Gamma }^{\prime } \Rightarrow {\Delta }^{\prime }, {\Delta })\), and \(({\Gamma } \Rightarrow {\Delta }) \in S\).
Let’s adapt the notation adopted for Ψ to Ψ^{+}. The following result is immediate (the first item follows from Lemma 4 and the proof of Proposition 6, and the second by an induction on the buildup of \(S_{{\Psi }^{+}}\)).
Lemma 10

\(\mathsf {I}_{\Psi } = \mathsf {I}_{{\Psi }^{+}}\).

For every \(S \subseteq \omega \), S_{Ψ} is consistent if and only if \(S_{{\Psi }^{+}}\) is consistent.
The properties of I_{Ψ} transfer to \(\mathsf {I}_{{\Psi }^{+}}\), and the consistency of a fixed point S_{Ψ} transfers to the fixed point of Ψ^{+} generated by the same set S.^{27} The operator Ψ^{+}, however, guarantees closure under weakening, contraction, and cut.^{28}
Observation 2
5 From Logical to Grounded Consequence
Key Findings
Consequence  Key Finding 

Logical  (i) Conservativity for all \(B \supseteq \mathsf {EA}\) 
(ii) Local \(\mathcal {L}\)embeddability for all \(B \supseteq \mathsf {EA}\)  
Arithmetical  Analysis of Field’s hierarchy via local reflection principles 
Grounded  (i) Consistency of (VP), (VDm), (ValSchema)^{+} 
(ii) Kripkestyle theory for Val: development of Ψ  
(iii) Make sense of Valprinciples via I_{Ψ} and grounded validity 
The rules (VP) and (VD) support a strict reading of ⤙ as logical derivability, and therefore of Val as the class of the logically valid inferences. On this approach, (VP), namely (VP_{1}) in the formalism of Section 2, is restricted to purely logical inferences, while the full (VD) – (VD_{1}) in Section 2 – can be consistently kept: from this point of view, we can naturally read (VD) as preservation of truth in logically valid arguments. A subtheory of our theory of logical validity is the theory given by (VD) and a suitable version of (VP): this theory meets the criterion (suggested by Field in [9]) of giving the same reading to both the metatheoretic notion expressed by \(\vdash \) and to the predicate Val. The theory of logical validity is therefore simple and wellbehaved: it’s then natural to study in more depth the corresponding notion of objectlinguistic validity by comparing it to the inferential resources of the underlying base theory. Corollary 1 tells us that logical validity does not have any impact on the underlying syntactic structure: for any theory extending a very weak arithmetical system, (VD) and the restricted version of (VP) do not enable us to prove new syntactic or arithmetical facts. This extends to further principles for logical consequence, like internalized modus ponens, as shown in Corollary 3. These results seem to suggest that ‘deflationary’ approaches to logical consequence, such as Shapiro’s [34], may endorse conservativity requirements for logical validity. This is not to say, however, that the predicate of logical consequence does not play an indispensable expressive role: although the theory of logical validity is uniformly locally interpretable in the base theory, our results do not show that it is relatively interpretable in the object theory and therefore, arguably, expressively reducible to it.
As we have seen, the consistency of the theory of logical validity follows from a restriction of (VP) to purely logical derivations: consistency, however, is preserved even if we internalize purely arithmetical derivations. This led us, following [9], to investigate a hierarchy of arithmetical consequence predicates. We have shown that every stage in this hierarchy corresponds to a stage of a parallel hierarchy of local reflection principles; as a consequence, the hierarchical notion of validity can be read as iterated arithmetical provability.
The hierarchical approach to consequence suffers from several wellknown problems: in particular, there seem to be notions of ‘following from’ that cannot be accounted for in the stratified picture, such as implication. Transcending the hierarchy calls for an unrestricted (VP). We have achieved this via a Kripkestyle construction, the KVconstruction, in which the unrestricted (VP) is balanced by a rule form of (VD). The least fixed point of the KVconstruction, I_{Ψ}, embodies a notion that, following Kripke’s theory of truth, we might call grounded validity, i. e. validity grounded in truths and falsities of the base language, in our case arithmetical truths and falsities.^{29}
5.1 Future Work
We conclude by sketching some directions for further work. In the context of logical validity, the main open problem is the question of the global interpretability and, most importantly, of the \(\mathcal {L}\)embedding of the theory of logical validity in the base theory. As for grounded consequence, the construction described by I_{Ψ} can be turned into an axiomatic theory. It would then be natural to study the relationships between this theory and the class of models extending \((\mathbb {N}, \mathsf {E}_{\Psi }, \mathsf {A}_{\Psi })\). Finally, it would be interesting to relate irreflexive validity with paracomplete theories of truth and validity.^{32}
Footnotes
 1.
To emphasize even more the analogy with truth, we note that an objectlinguistic predicate for consequence is needed for the same purposes that motivate a truth predicate, such as blind ascriptions (‘all the derivations made this morning in the Logic lecture are valid’) or generalizations.
 2.
Suppose in fact that (VP) is logically valid, that there is a logical derivation of φ ⤙ ψ, and that f is a function in the language from names of sentences to terms that do not name sentences. Then applying (VP) and uniform substitutivity we conclude that \(\mathsf {Val}(f(\ulcorner \varphi \urcorner ), f(\ulcorner \psi \urcorner ))\) comes out as logically derivable, which is absurd. This is clearly remarked in [7].
 3.
Ketland’s results will turn out to be special cases of our findings.
 4.
See [28].
 5.
For a quirk of terminology, ‘reflexive’ will be employed in two different senses: (i) as referred to arithmetical systems that prove the consistency of all their finite subsystems, and (ii) as referred to theories that satisfy the structural rule of reflexivity. We apologize for the possible, but unavoidable, confusion.
 6.
It is also clear that the paradox can be derived by avoiding any syntactic/arithmetical assumption. To see this, one can employ a familiar trick due to Montague [21]. A finite formulation of the axioms of a finitely axiomatizable T (see below for the choice of T) can be pushed into the instance of the diagonal lemma needed in the derivation of the paradox. Starting with the instance
of the diagonal lemma, where A is a finite reaxiomatization of T, one easily obtains a version of the paradox that does not rely on the assumption of the logical validity of the underlying arithmetical theory but only of the principles for logical consequence.$$\nu \leftrightarrow \mathsf{Val}(\ulcorner \nu\land { A}\urcorner,\bot) $$  7.For more details on EA and elementary functions the reader may consult [3, 33]. The axioms of EA are the universal closures of the following:In addition, EA features the principle of induction for elementary formulas φ(x):$$\begin{array}{llllll} \mathsf{EA1}\quad& 0\neq \mathsf{S}x \\ \mathsf{EA2}\quad& \mathsf{S}x=\mathsf{S}y\rightarrow x=y\\ \mathsf{EA3}\quad& x+0=x\\ \mathsf{EA4}\quad& x+\mathsf{S}y=\mathsf{S}(x+y)\\ \mathsf{EA5}\quad& x\times 0=0\\ \end{array} \hspace{1cm} \begin{array}{llllll} \mathsf{EA6}\quad& x\times \mathsf{S}y=(x\times y)+x\\ \mathsf{EA7}\quad& \mathsf{exp}(0)=1\\ \mathsf{EA8}\quad& \mathsf{exp}(\mathsf{S}x)=\mathsf{exp}(x)+\mathsf{exp}(x)\\ \mathsf{EA9}\quad& x\leq 0 \,\leftrightarrow\,x=0\\ \mathsf{EA10}\quad& x\leq \mathsf{S}y\,\leftrightarrow\,(x\leq y\vee x=\mathsf{S} y) \end{array} $$
 8.
The notion of relative interpretation goes back to Tarski et al. [36]. Although it is a syntactic notion, it has a natural semantic counterpart: if U is interpretable in W, in any model of W we can construct an internal model of U.
 9.
Feferman’s theorem on the interpretability of inconsistency (see [38]) represents a separating example between relative interpretation and \(\mathcal {L}\)embeddings. Let T be ωconsistent: by Feferman’s theorem, T + ¬Con(T) is interpretable in T, but it cannot be \(\mathcal {L}\)embeddable in T because \(\mathcal {L}\)embeddability clearly preserves ωinconsistency. For a study of how to separate \(\mathcal {L}\)embeddings from stronger notions of equivalence, see [26].
 10.
 11.
 12.
The same proof applies to \(T^{\mathsf {V}}\!\!\upharpoonright \).
 13.
The theorem states that if V is reflexive and any finite \(U_{0}\subseteq U\) is interpretable in V, also U is interpretable in V (see [12, Secton 3]).
 14.
By employing the sequence encoding sketched above, one can estimate for the translated proof:\(\mathcal {D}^{\mathfrak {b}}_{j}\leq G(n)\), where\(G(n)=\underbrace {\langle n(n+1){2^{n}},\ldots ,n(n+1){2^{n}}\rangle }_{n\text {times}}=\left (n(n+1){2^{n}}+1\right ){2^{(n(n+1){2^{n}})}}\)
 15.
We adapt to the present setting a more general strategy suggested to us by Albert Visser.
 16.
In the definition of C_{ n }, (x)_{ y } = z is an elementary functional expression corresponding to the projection of the y^{ th }element ofx.
 17.
For a study of the asymmetry between finitely axiomatized and reflexive theories in the context of theories of truth, see [25].
 18.
In particular, OR is an elementary set of ordinal codes and \(\prec \) an elementary relation isomorphic to the usual wellordering of ordinals up to Γ_{0}.
 19.
It is nonetheless possible to develop axiomatic theories that are adequate for these models and therefore avoid any arbitrariness in choosing a natural halting point for progressions of theories of stratified validity.
 20.
Some substructural theories compatible with Beall and Murzi’s principles are available in the literature: the noncontractive theory in [41] (see also [5]) and the nontransitive approach of [29] and [6] (see also [37]) support both (VP) and (VD), and block the derivation of ⊥. Strictly speaking, these are theories of naïve truth that feature a conditional obeying conditional versions of (VP) and (VD), but they can be easily adapted to naïve validity.
 21.
 22.
For a strengthening of Meadows’ approach, see [35].
 23.
Consistency is typically defined as the absence of a contradiction, but our definition is equivalent to that. We could introduce a connective ¬, interpreting ¬φ as \(\mathsf {Val}(\ulcorner \varphi \urcorner , \ulcorner 0=1 \urcorner )\), and show that the classical rules for negation hold for the sodefined ¬ in I_{Ψ}. Then, it would be easy to show that \(\varnothing \Rightarrow \varnothing \notin \mathsf {I}_{\Psi }\) if and only if there is no \(\mathcal {L}_{V}\)sentence φ s. t. \(\varnothing \Rightarrow \varphi \wedge \neg \varphi \in \mathsf {I}_{\Psi }\). We note that the resulting negation is weaker than classical negation, and indeed even weaker than intuitionistic negation (not al the ex falso sequents \(\varphi \wedge \neg \varphi \Rightarrow \bot \) are in I_{Ψ}, although the corresponding rule of inference is admissible in I_{Ψ}).
 24.
[32] has recently remarked that a similar restriction of reflexivity in the context of rules for naïve comprehension can avoid paradoxes and, at the same time, make both cut and contraction admissible.
 25.The choice between (VD) and (VDm) reflects an ongoing debate in the truththeoretical literature (especially concerning substructural theories of truth) on the ‘correct’ form of modus ponens (see [42] and [30]). Two of the main contenders are analogous to (VD) and (VDm):$$\varphi, \varphi \rightarrow \psi \vdash \psi \quad\text{and} \quad \text{from } {\Gamma} \vdash \varphi \text{ and } {\Delta} \vdash \varphi \rightarrow \psi \text{ infer } {\Gamma}, {\Delta} \vdash \psi. $$
 26.
We are grateful to Paul Egré, Francesco Paoli, and Robert van Rooij for suggesting this point to us. Tolerantstrict consequence is also employed in [20].
 27.
We did not use Ψ^{+} in the first place in order to simplify the proofs of the properties of I_{Ψ}: this has no practical effects, however, since I_{Ψ} and \(\mathsf {I}_{{\Psi }^{+}}\) are identical.
 28.
This result improves on the structural rules recovered by Meadows in [20].
 29.
 30.
Grounded validity is clearly distinct from analytical validity, i. e. validity based on analytical truths and falsities. In fact, it is possible to start our construction from nonanalytic claims, e. g. about snow being white and grass being green. Thanks to Andreas Fjellstad for bringing this point to our attention.
 31.
The standard counterpart of of reflexivity in natural deduction is the rule of assumption (see [24], Section 1. 3). If we accept a grounded picture of consequence, we can safely assume all the sentences of the base language, and apply all rules of inferences to them: the only sentences we cannot assume are the ungrounded ones, in line with our conception of consequence. Moreover, it is easy to see that the notion of consequence captured by I_{Ψ} includes all the logical consequences grounded in the base language. Many thanks to Kit Fine, Lionel Shapiro, and Luca Tranchini for useful discussions on these points.
 32.
The latter research question has also been suggested in [10].
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). We are grateful to Paul Egré, Kit Fine, Andreas Fjellstad, Volker Halbach, Julien Murzi, Francesco Paoli, Lionel Shapiro, Luca Tranchini, Robert van Rooij, Albert Visser, and an anonymous referee for several useful comments and discussions about the material presented in this paper.
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