Systems for Non-Reflexive Consequence

Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.


Introduction
Interest in substructural logics and in their application to logical and semantic paradoxes has grown considerably in recent years.Many recent works focus on non-transitive approaches to paradox [3][4][5]8,9,[44][45][46][47], and noncontractive approaches have also received considerable attention [7,15,23,24,27,36,40,41,48,53].Non-reflexive theories have been investigated less. 1  Nevertheless, non-reflexive theories are especially promising to model the interplay between naïve truth and consequence [38].However, a systematic 1 Some works touching upon non-reflexive logics and their relationship to paradox include [22,25,26,34,38,43].A brief comparison of our paper with those work might be helpful: [26], [43], and [25] motivate the non-reflexive approach without providing a modeltheory or a proof-theory; our work integrates those with a more comprehensive model-and proof-theoretic analysis.As explained in [38], a model-theory for a non-reflexive "validity" predicate is provided in [34], but the construction differs form our in that it does not validate many desirable principles (e.g.contraction and cut).[22] studies a pure logic of disquotational truth based on a three-sided logical system: our two-sided approach combines well with a semantics, and can be neatly extended to a fully fledged compositional theory of truth over a non-logical syntactic base.
Presented by Francesco Paoli; Received June 30, 2022 study of the logic, the semantics, and the proof-theory of non-reflexive theories of naïve truth and consequence is currently lacking, and so does a thorough philosophical analysis (and defence). 2 The purpose of this paper is to fill this lacuna, at least in part.First, we introduce the basics of non-reflexive logic(s) and semantics, and their extensions with naïve consequence (and truth) rules ( §2-3).The two main sections of the paper are §4 and §5.In the former, we first build on the work carried out in [37] on logics of truth to investigate the proof-theory of non-reflexive logics of consequence, with a special focus on cut-elimination proofs.In §5 we study the interaction between truth and consequence in non-reflexive systems: this is achieved by providing a compositional theory of truth and consequence, by establishing the adequacy of such a theory with respect to the semantics provided in §3, and by investigating its prooftheoretic properties.
The present work is mainly a technical study, which aims to consolidate non-reflexive logics as a viable basis to address semantic paradoxes, and to develop satisfactory theories of naïve semantic notions.Its main findings are that (i) naive consequence rules can be added to a non-reflexive logic while preserving an intuitive semantics and remarkable proof-theoretic properties (above all, the eliminability of cut); (ii) it is possible to provide a fully compositional theory of truth over a non-reflexive logic that admits naïve rules for truth and consequence and that axiomatizes a generalization of a standard fixed-point construction for truth.

Logics of Transparent Truth and Consequence
Let L be a first-order language with logical constants ¬, ∧, ∀, and L C := L ∪ {C} its expansion with a binary predicate C(x, y) intended to express object-linguistic consequence. 3Variables are denoted with x, y, z, . .., and terms with r, s, t, . . . .We assume that L contains constants ϕ for any formula ϕ of the language L C .The nature of the names ϕ is not fully fixed by the theory: as customary practice when dealing with logics of semantic concepts, one can assume that the denotation of ϕ in all models of the theory is ϕ itself [30,44].Such extra-theoretic assumptions will become redundant once a proper theory of syntax will be assumed in the final sections of the paper.Definition 1. (LPC) The system LPC in L C contains the following initial sequents and rules, where Γ, Δ -possibly with subscripts-are finite multisets of formulae of L C .
(i) AtFml L denotes the set of atomic formulae of L, i.e. the language without the consequence predicate, and FV(Γ) denotes the set of free variables of Γ.
(ii) We can define a theory of full disquotational truth LPT -studied in [37] -as a sub-theory of a definitional extension of LPC obtained by defining Tr(x) as C( , x). 4 In fact, in the theory LPT the rules Cl and Cr are easily shown to be admissible in LPT.
(iii) The combination of the rules (Cl) and (Cr) with unrestricted initial sequents, even in the absence of (¬r), results in inconsistency.This is essentially a version of Curry's paradox that has recently received some attention [6,38]. 5(Cr) is in fact a version of the rule VP of Beall and Murzi, generalized to include arbitrary context, while (Cl) is a metainferential version of their rule VD.The intended reading of C(x, y) is "grounded consequence" in the sense of [38].As explained in [38, §2.3], C(x, y) differs also from a formal provability predicate, in that it does not generate a hierarchical concept of consequence -by Löb's theorem, formal provability in arithmetic can in fact only validate naive principles for C(x, y) if one ascends to a stronger system.
(iv) As it happens in the standard G3 systems on which it is based, the formulation of LPC with context-sharing rules is justified by the admissibility of weakening and contraction in the system, established below.

Fixed-Point Semantics
The semantics for the logical rules of LPC is provided by a substructural (non-reflexive) logical consequence relation defined over strong Kleene semantics (K3), i.e. the tolerant-strict consequence relation (TS) defined in [8] with the assumption that atomic, truth-free formulae always receive a classical semantic value.This semantics can then be incorporated into a simple fixed-point construction (introduced in [38], and to be recalled), in order to interpret the consequence predicate of LPC.Let us start from the former.In the following, we take for granted the notion of a strong Kleene (K3) evaluation, with values 1 for true, 0 for false, and n for neither -see for instance [13].
Definition 3. Let v be a K3 evaluation function.The argument from Γ to Δ is TS-valid (for Tolerant-Strict), in symbols Γ ts Δ if: for any K3 evaluation function v, if for every ϕ ∈ Γ, v(ϕ) = 1 or n, then there is at A few basic features of TS are easily stated.Just like strong Kleene logic K3, TS does not have any classical laws not involving and ⊥.In other words, no such sequent of the form ⇒ ϕ, for ϕ classically valid, is TS-valid.In addition, and unlike K3, TS does not validate any classical inferences -with the same restriction -, i.e. no classically valid sequent of the form Γ ⇒ Δ is TS-valid.This includes, of course, reflexivity: to see that the inference from ϕ to ϕ is not unrestrictedly TS-valid, just consider a K3-evaluation which assigns value n to ϕ.However, and again unlike K3, TS is closed under all the meta-inferences (with no non-empty premises) of the classical sequent calculus G3 [52].
As announced above, the consequence relation defined by TS can be easily combined with a Kripke-style, fixed-point interpretation of the consequence predicate C(x, y) [31].That this is generally possible is guaranteed by the fact that the K3 evaluation scheme is monotone in the evaluation ordering. 6or simplicity and definiteness, we develop the model-theoretic construction in an arithmetical setting, thus identifying and ⊥ with some arithmetical truth and falsity, respectively. 7Let then L N be the language of arithmetic and L C N := L N ∪ {C}.We assume that the language of arithmetic includes the signature {0, S, +, ×} plus finitely many symbols for primitive recursive functions, which facilitates the development of formal syntax.For instance, it will contain symbols for the syntactic operations: The use of L N presupposes a more comprehensive formalization of the syntax of L C .The meaning of the Gödel quotes is now fixed by a canonical Gödel numbering and a standard formalisation of syntactic notions and operations.
In what follows, we keep assuming a canonical coding of finite sets.A sequent is thus simply a pair of finite sets.We write (Γ; Δ) for (the code of) the sequent Γ ⇒ Δ.For simplicity, we identify syntactic objects and their codes.The semantic clauses of the jump given in the next definition correspond to standard, classically valid sequent rules -i.e.classical rules to introduce complex formulae to the left and to the right of the sequent arrow -plus rules for the consequence predicate that internalize them.As remarked above, the semantics for LPC is closed under a version of all classical sequent rules, so it's no surprise that a semantics for LPC follows the same patterns to interpret the logical vocabulary.Iterations of Ψ can be defined as usual, by letting Ψ 0 (S) = S and putting:8 The operator Ψ is both increasing -i.e. S ⊆ Ψ(S) for any S -, and monotonic: S 0 ⊆ S 1 entails Ψ(S 0 ) ⊆ Ψ(S 1 ).The latter property entails the existence of fixed points of Ψ, i.e. sets T s.t.Ψ(T ) = T .A fixed point T is said to be inconsistent if, for some sentence ϕ, both (; ϕ) and (; ¬ϕ) are in T , and consistent otherwise.We are mainly interested in the minimal of these fixed points I Ψ := α∈Ord Ψ α (∅).We also write I α Ψ for Ψ α (∅).It can be shown that the minimal fixed point is indeed consistent [38].
The following lemma, proved in [38], shows that I Ψ is a model of a naïve, self-applicable consequence predicate.
Via the definition Tr(x) :↔ C( 0 = 0 , x) the minimal fixed point for self-referential truth from [31] essentially "lives" inside I Ψ .In particular, one can restrict the construction above to empty contexts, and the clauses for C(x, y) then obviously can be restricted to: This restriction of the monotone operator Ψ above reaches then fixed points X in which Properties ( 3) and ( 4) correspond to the so-called "intersubstitutivity" of truth.For definiteness, let's call J the set of truths of the minimal fixed point X 0 so obtained, that is: It's clear that we can express Ψ(•) as a formula of the language L 2 of secondorder arithmetic in such a way that for F(x, X) arithmetical and X occurring only positively -i.e.not in the scope of an odd number of negation symbols -in it.Therefore 1 .Moreover, by the relationships between J and I Ψ outlined above, and by Π 1  1 -hardness of J, 9 we have: 1 -sets have a natural presentation in terms of cut-free infinitary derivability [1,42].The case we are considering is not an exception, and a suitable infinitary calculus LPC ∞ can be developed along the lines of the infinitary system for non-reflexive truth developed in [37].LPC ∞ is obtained from LPC by (essentially): replacing the axioms for ⊥ and with corresponding rules for arithmetical truth and falsity, and replacing (∀r) with an ω-rule. 10By adapting the analysis in [37], it can be shown that LPC ∞ has nice proof-theoretical properties: weakening and contraction can be proved to be admissible in a way that preserves the (possibly infinite) length of the derivation, its rules are invertible, and (crucially) cut is eliminable in it.
In addition, it is possible to show that (possibly infinitary) proofs in LPC ∞ closely "match" the construction of I Ψ .More precisely: the ordinal stage of the inductive definition in which a sequent Γ ⇒ Δ enters in I Ψi.e. its ordinal norm -can be associated to the lengths of cut-free proofs of Γ ⇒ Δ in LPC ∞ .By a well-known result, this ordinal norm cannot exceed the first non-recursive (countable) ordinal ω CK 1 : If one restricts their attention to pairs of sentences, the above result entails the existence of a tight correspondence between the extension of the consequence predicate in I Ψ , and the consequence ascriptions derivable in LPC ∞ .More specifically, for all ϕ, ψ ∈ L C , the following are equivalent

Proof Theory of LPC
In this section we focus on the proof-theoretic properties of LPC.Our analysis culminates in the full eliminability of cut in it.The key technical insight that makes cut fully eliminable-and that is extensively investigated in [37]-is a strong form of invertibility of the C-rules (Lemma 10).This section is essentially an adaptation to the setting with primitive consequence of the proof-theoretic analysis of [37].
The notions of length of a derivation is standardly defined [49,52]. 12Given a calculus with rules that are at most α-branching, the length of a derivation D is the supremum of the lengths of its direct sub-derivations D γ increased by one: 11 See for instance, [42,Thm. 6.6.4]. 12Our notion of length amounts to what is called depth in [52].
Clearly, for LPC, α = 2 and the length of derivations is finite.We will write LPC Γ ⇒ Δ to indicate that there is a derivation of the sequent Γ ⇒ Δ in LPC, and D LPC ϕ to indicate that D is a proof of ϕ in LPC.
In developing the proof theory of LPC, it is convenient to work in a system that is extensionally equivalent to LPC, but that features an explicit labelling of formulae in sequents in a proof.Extensional equivalence means in this context that labelling does not allow one to obtain new proofs, but only to keep track of existing ones.This machinery is left implicit in work on the restriction of identity sequents [21,50], but it's required for a formally precise cut-elimination argument, and in particular to define the main measure of complexity -called C-complexity -for applications of the C-rules to formulae in derivations.For each proof D, we assume a labelling function l D : Form L C → ω \ {0, 1} applying to formulae in initial sequents.Labels then expand in a uniform way depending on the rule employed.For instance, for different rules configurations, we have: (1,l) , δ (1,m)  , (m,q)  .
All other rules conform to one of these patterns, and are labelled in an analogous way.Full details of the labelling machinery, including its extension to infinitary rules, can be found in [37].Once we know in principle that we can always employ labels to uniquely refer to formulae and their "history" throughout a proof, we can choose to omit labels for the sake of readability.We will often choose to do so.In a nutshell, the C-complexity of a formula keeps track of the applications of the C-rules: initial sequents and L-formulae have complexity 0, and the only way to increase the C-complexity of a formula is by introducing the consequence predicate.
and the C-complexity of the formulae in Γ, Δ is unchanged.Similarly for (¬r) and (∀r).
) and the complexity of occurrences in side formulae is the maximum of the corresponding occurrences of side formulae in premisses.κ(ψ)) + 1 and the complexity of occurrences in side formulae is the maximum of the corresponding occurrences of side formulae in premisses.
) and the C-complexity of the formulae in Γ, Δ is unchanged.
(ix) If D ends with Γ ⇒ Δ, ϕ ϕ, Γ ⇒ Δ Γ ⇒ Δ then the complexity of occurrences in side formulae is the maximum of the corresponding occurrences of side formulae in premisses.In this case the complexity of the cut formula is the maximum of its two active occurrences.
We start by observing that, in proofs of sequents containing on the left, and ⊥ on the right, the occurrences of such constants can be omitted.Both claims follow by a straightforward induction on the length of the proof that preserves the C-complexity of the formulae in the contexts.In both claims, the length of the derivation is preserved.
Next, we turn to subsitution and weakening lemmata.Again, a proof by induction on the length of derivations is required.In the proof of weakening, the formulation of (ref), ( ), (⊥) with arbitrary contexts is of course essential.
Lemma 9. (Substitution, Weakening) Γ * , Δ * are obtained by uniformly replacing in Γ, Δ, a variable x by a term t which is free for x and does not contain variables employed in applications of (∀r) in the proof of Γ ⇒ Δ.Moreover, the C-complexity of the formulae involved in the substitution and in the contexts does not change. (ii In both claims, the length of the derivation is preserved.
The next lemma marks out the key property of LPC which makes it possible to generalize the standard G3-strategy for the admissiblity of cut to the present setting.All rules of LPC, including the rules for the consequence predicate, are invertible in a strong sense that preserves, and in the appropriate cases reduces, the C-complexity of formulae.

Lemma 10. (κ-invertibility of LPC-rules)
and in which the C-complexity of the side formulae does not increase.
and in which the C-complexity of the side formulae is no greater than their κ-maximal occurrence in the premisses.Crucially, the invertibility of the rules preserves the length of the proof.
If Γ, C( ϕ , ψ ) ⇒ Δ is not an axiom, there are two cases to consider.The first in which C( ϕ , ψ ) is principal in the last inference of D, the second in which it is not.In the former case, κ(C( ϕ , ψ )) > 0, and the claim follows immediately by definition of C-complexity in the case of an application of (Cl).In the latter case, suppose that D ends with (we treat the case of an arbitrary binary rule, the case of unary rules is simpler).The induction hypothesis applied to D 0 and D 1 yields derivations Remark 11.In the presence of unrestricted initial sequents (ref), the inversion strategy considered above will not go through.For instance, the derivability of a sequent of the form Γ, C( ϕ , ψ ) ⇒ C( ϕ , ψ ), Δ does not guarantee, for instance, the derivability of a sequent Γ, ϕ ⇒ ψ, Δ with κ(ϕ) ≤ κ(C( ϕ , ψ )).This fact is crucial for the next lemma, in which contraction is shown to be κ-admissible.
Lemma 12. (κ-admissibility of contraction) ) and in which the C-complexity of the side formulae does not increase.
) and in which the C-complexity of the side formulae does not increase.
Crucially, in both claims the length of the original derivation is preserved.
Proof.The proof is by induction on the length of D. One proves (i) and (ii) simultaneously.
The case of initial sequents follows immediately by the definition of Ccomplexity.The sub-case of the induction step in which neither ϕ k 0 nor ϕ k 1 is principal in the last inference is immediate by induction hypothesis.
What remains is the case in which Γ, ϕ k 0 , ϕ k 1 ⇒ Δ or Γ ⇒ ϕ k 0 , ϕ k 1 , Δ are not initial sequents, and one of ϕ k 0 or ϕ k 1 is principal in the last inference.We treat the crucial cases in which ϕ is C( ϕ , ψ ).
We can then apply the inversion Lemma to D 0 to obtain a Similarly, inversion applied to D 1 yields By induction hypothesis, we obtain: An application of (Cl) yields the desired By two applications of the induction hypothesis, we obtain a proof Therefore, by (Cr), one obtains a derivation of Γ ⇒ C( ϕ , ψ ), Δ with It is worth noticing that the formulation of (∀l) and its associated Ccomplexity renders the case of (i) in which one of the ϕ's is principal in the last inference and of the form ∀xϕ straightforward.
We can finally state and prove the cut-elimination lemma for LPC.We start with the reduction lemma.
Lemma 13. (Reduction) If D 0 is a cut-free proof of Γ ⇒ Δ, ϕ k in LPC, and D 1 is a cut-free LPC-proof of ϕ l , Γ ⇒ Δ, then there is a cut-free proof D of Γ ⇒ Δ in which the C-complexity of the side formulae is no greater than their κ-maximal occurrence in the premisses.
Proof.The proof is by a main induction on κ(ϕ) = max(κ(ϕ l ), κ(ϕ k )), with side inductions on the logical complexity of ϕ and on the sum d 0 + d 1 of the lengths of D 0 and D 1 .We consider the main cases.
Case 1.One of D 0 , D 1 is an initial sequent, say D 0 .If ϕ is not principal, then Γ ⇒ Δ is already an initial sequent.If ϕ is principal in it, then we can distinguish two cases.If D 0 Γ 0 , ϕ ⇒ ϕ k , Δ, then we can apply Lemma 12 to D 1 to obtain a derivation of Γ ⇒ Δ whose formulae have the required C-complexity.If D 0 Γ ⇒ k , Δ, then D 1 l , Γ ⇒ Δ.By Lemma 8(i), Γ ⇒ Δ is derivable with the expected C-complexity.
Case 2. The cut formula is not principal in one of the premises, say D 1 .For instance the last inference of D 1 is an application of (Cl).Then, with Γ := C( ϕ , ψ ), Γ 0 , the derivation D ends with: By the weakening lemma, D can be transformed into a derivation D whose last inference is an application of (Cl), whose premises are The upper cuts in D can be eliminated by side induction hypothesis, since d 0 +d 11 , d 0 +d 10 < d 0 +d 1 .Moreover, since the weakened formulae have lowest possible C-complexity, an application of the contraction lemma to the transformed derivation yields the claim.The other cases in which the cut formula is not principal are easier.
Case 3. The cut formula is principal in the last inference of D 0 and D 1 .The case in which the cut formula is C( ϕ , ψ ) is particularly easy, by main induction hypothesis, because the cut can be pushed upwards and applied to the ancestors of the cut formula, which have strictly smaller Ccomplexity.The case in which the cut formula is principal and of the form ∀xϕ is treated standardly as well but one has first to get rid of the universal quantifier in the premise of (∀l).This involves an essential application of the substitution lemma [52, § 4.1].
Remark 14.Although our proof of lemma 13 above relies heavily on lemma 12, the role of κ-admissibility of contraction can be circumscribed to the role it plays in Case 1 -that is, the case in which one of the premisses is an axiom and the cut formula is principal.
In Case 2, and in the specific sub-case treated above, one can apply the inversion lemma to D 0 to obtain LPC-proofs D 00 Γ 0 ⇒ Δ, ϕ, χ and D 01 ψ, Γ 0 ⇒ Δ, χ.These can then be combined with D 10 and D 11 respectively, and then (Cl) applied to the results of the shorter cuts.Such a template, with inversion playing the fundamental role, can be applied to all other sub-cases of Case 2 except of course (∀l).In such case, D has the form: In such case, one can therefore weaken D 1 , apply cut to such weakened derivation and D 00 , and then apply (∀l).
By repeated applications of the Reduction Lemma, we can then obtain: Since the cut-elimination proof above displays standard bounds for the reduction, Corollary 15 can be formalized in IΔ 0 + superexp, where IΔ 0 is the subsystem of PA featuring only bounded induction, and The strategy leading to the cut-elimination theorem above clearly generalizes to the case of the theory obtained by replacing the C-rules with the rules (Tr-l) and (Tr-r).One simply has to replace the C-complexity with a truth complexity measure (cf.Sect. 5 below).Similarly, one can apply the strategy to a theory of naïve abstraction (or property predication) based on rules of the form where instead of a naming device one assumes a term-forming abstraction operator {• | •} -e.g.along the lines of the one employed for a contraction-free set theory in [7].∈-complexity is then defined in the obvious way: given a derivation D ending with ∈l, the ∈-complexity of t ∈ {x | ϕ} is defined as the ∈-complexity of ϕ(t) plus one.On can then follow the template of Definition 7. All results above then carry over with only minimal modifications.

A Compositional Theory of Non-Reflexive Truth and Consequence
In their [29], Halbach and Horsten develop a formal system, called PKF (for Partial Kripke-Feferman), which axiomatizes Kripke's fixed point models over Peano Arihtmetic (PA) in strong Kleene logic.PKF constitutes the basis of any theory of truth that extends Kripke's theory with extra-resources -e.g. a new conditional [19,20,32].In this section, we develop a twin theory of PKF -or better, a version of PKF based on the logic K3 -, which we call RKF, whose logic is based -somewhat unsurprisingly -on a restriction of (ref).PKF and RKF are twins in the sense that for X a fixed point model for the language L Tr obtained in the manner suggested in Sect. 3 (that serves as the extension of the truth predicate), (N, X) k3 PKF iff (N, X) ts RKF This obviously entails that RKF is also a theory of naïve truth, and in fact an axiomatisation of Kripke's theory of truth in partial (substructural) logic.Actually, RKF is still richer: it is also a theory of naïve consequence, whereas PKF cannot be.In fact, just as a naïve truth predicate can be defined from the naïve consequence predicate of LPC, so can a predicate for naïve consequence (obeying the rules (Cl) and (Cr)) be defined from the naïve truth predicate of RKF (the naïveté of the latter, in turn, follows from the compositional rules of RKF).Definition 16, Lemma 20, and Corollary 21 will establish this claim more precisely.By contrast, since PKF is a fully structural theory, the presence of naïve consequence rules would immediately entail triviality by an internalized version of Curry's paradox-the V-Curry paradox by [6].
In addition to the vicinity of RKF to well-known theories with restricted operational rules, RKF displays some important theoretical virtues.First, it admits a nice semantics (via the simple, inductive construction reviewed in Sect.3) which is matched by the axiomatic theory.More specifically, RKF enjoys an adequacy result with respect to the fixed points of the inductive construction (Proposition 22).Adequacy results have been defended as a theoretical virtue for theories of truth, e.g. by [18].Moreover, non-reflexive approaches of the kind we discuss here admit a full inter-definability of naïve validity and naïve truth (via the conditional), and both notions enjoy fully symmetric rules (Lemma 20 and Corollary 21).Here, we leave open the question of which approach is ultimately preferable as an environment to formalize naïve semantic notions.This work is aimed at producing new results concerning non-reflexive theories, in order to better assess their prospects as formal approaches to naïve semantic notions.
As anticipated above, in order to formulate RKF, it is more convenient to take the truth predicate as primitive.Let L Tr be the language given by adding a fresh unary predicate Tr to the language of arithmetic, i.e.L N ∪ {Tr}.In this language, we can define the consequence predicate via a combination of truth and conditional, putting C(x, y) :↔ (Tr(x) → Tr(y)), i.e. ¬(Tr(x) ∧ ¬Tr(y)).Due to the fact that the logic TS has all the classical meta-inferences, and thus the conditional can be introduced and eliminated just as the consequence predicate, one can easily define truth in terms of consequence and the other way around.
One last piece of notation, following [17].Let num(x) be the function symbol representing the primitive recursive function that sends each number to its numeral.Given a formula ϕ(v), we write ϕ( ẋ) for the result of formally substituting the variable v for the numeral of x in ϕ (see e.g.[51]).Moreover, x(y/v) stands for the result of formally substituting y for the (code of) the variable v in x (we follow the conventions in [28]) Definition 16. (RKF) The theory RKF in L Tr has the following components: (i) The logical component of LPC, that is the initial sequents and rules of LPC except (Cl) and (Cr) (ii) The initial sequents Γ ⇒ Δ, ϕ for ϕ a basic axiom of PA, including identity axioms: for P an atom of L Tr (iii) All instances of the induction schema for all formulae ϕ(v) of L Tr : Γ ⇒ ϕ(0), Δ Γ, ϕ(x) ⇒ ϕ(x + 1), Δ Γ ⇒ ∀xϕ, Δ with x not free in Γ, Δ, ∀xϕ.
(i) Syntactic functions operating on codes of L Tr -expressions will be presented in simplified form for the sake of readability.
(ii) The Tr-rules for connectives and quantifiers are presented in simplified forms, with variables intended to range over sentences and terms, according to the form of the rules.For instance, (Tr∧1) is short for: Finally, the non-abbreviated form of (Tr∀1) reads: The consistency of RKF will be a corollary of Proposition 22, whose proof requires a few preliminary results that have also independent interest.The following lemma, which follows from a simple external induction on the length of ϕ, indicates a form of recapture: RKF (and extensions thereof) features full initial sequents for the language L N .
Next, observe that weakening is length-preserving admissible in RKF in the sense specified in the following lemma.We can now show that the naïve rules to introduce the truth predicate to the left and to the right of the sequent arrow are derivable in RKF. 13emma 20.Every instance of the naïve truth rules is admissible in RKF: Γ, Tr( ϕ( ẋ1 , . . ., ẋn ) ) ⇒ Δ Proof.We prove the admissibility of both Trr and Trl by simultaneous induction, with the main induction on the logical complexity of ϕ and secondary induction on the length of the derivations.We do only some cases, and only for the rule Trr, for the sake of brevity.Call D the derivation of Γ ⇒ ϕ, Δ in the premiss of Trr.Case 1. Suppose ϕ has logical complexity 0. Therefore, it is atomic.There are two cases.
Case 1.1.ϕ is an atomic formula of L N .In this case, the rule Trat1 provides the desired conclusion: . . .
Footnote 13 continued admissible in PA (in our non-reflexive logic) extended with the naive rules; any subsystem of the latter system will feature only a finite number of applications of naive truth rules, and one can find models of such sub-systems in which compositionality fails for sentences of a sufficiently high syntactic complexity.Thanks to an anonymous referee for raising this issue.
The above Lemma also shows that, via the definition of C(x, y) as Tr(x) → Tr(y), RKF unrestrictedly validates the naïve rules for consequence.

Corollary 21. Every instance of the naïve consequence rules is admissible in
Finally, thanks to the above Corollary, RKF can be shown to be adequate with respect to the semantics articulated in Sect.Proof.(Proof sketch) The right-to-left direction is immediate: a quick inspection shows that if S is a consistent fixed point of Ψ, then N, S 0 TSsatisfies all the axiom and rules of RKF.For the left-to-right direction, notice that if N, S 0 ts RKF, then the set of (codes of) sentences in S 0 is consistent and the corresponding sequents are closed under all the logical clauses of the operator Ψ (for otherwise N, S 0 would not TS-satisfy the logical rules of RKF) and, by Corollary 21, also under the naïve consequence-theoretic clauses of Ψ.Therefore, Ψ(S) = S.
We now turn to the proof-theoretic analysis of RKF.We will establish an upper-bound for RKF-provability.We show that RKF can be embedded in the theory PKF -first proposed by [29].We assume a sequent calculus formulation of K3 with identity (see, e.g., [39,  The idea of the reduction is as follows: since the sequent arrow of RKF is modelled after the material conditional of K3, we can translate the provability of a sequent as provability of the corresponding material conditional, plus the condition that the sentences in the conclusion are fully classical in PKF.
Proof.The proof is by induction on the length of the derivation in RKF.
For the base case: if Γ ⇒ Δ is an initial sequent in RKF, then ϕ ∈ Δ may be either t = t or a basic axiom of PA.In both cases, PKF ⇒ ϕ; the claim then follows by logic.If there is a ϕ ∈ L N ∩ Γ ∩ Δ, then again ⇒ ϕ, ¬ϕ is derivable in PKF, and the claim is obtained again by logic.
For the induction step, each rule must be considered.We report the key cases of induction.The other cases, including the identity rule of RKF, are straightforward.
Since PKF proves ⇒ ϕ ∨ ¬ϕ for ϕ ∈ L N , we have the desidered corollary: The study of the proof-theoretic lower bound for RKF appears to be more involved.Assuming a standard notation for ordinals < ε 0 , one would hope to define in RKF the truth predicates of the theory of ramified truth up to the ordinal ω ω (RT <ω ω ) -see [28, §9.1] for a definition.If one succeeded, then it would follow that any arithmetical theorem of PKF is also a theorem of RKF: this is because RT <ω ω is an upper bound for the arithmetical theorems of PKF.This strategy would be realized if one could show that the rule We therefore list the claim as an open, although we conjecture a positive answer to it: Open problem 26.RKF defines the truth predicates of RT <ω ω .That is, there is a relative interpretation of RT <ω ω in RKF that leaves the arithmetical vocabulary unchanged.Therefore, all arithmetical theorems of RT <ω ω are theorems of RKF.

Further Work
Much work remains to be done on non-reflexive systems and their applications.Just to mention a few: fully compositional, non-reflexive theories of consequence should be formulated and studied (by analogy with the compositional, non-reflexive theory of truth presented in Sect.5).Moreover, the relations between non-reflexive and other non-classical systems (paracomplete, paraconsistent, non-contractive, and non-transitive) should be fully investigated.For instance, the non-reflexive logic TS is known to be dual to the non-transitive logic ST, in a precise technical sense: 14 therefore, TS-based theories could be dual, in the same sense, to ST-based theories.Another nonclassical system in the vicinity of RKF may involve logical constants interpreted by means of other truth functions such as the ones of Weak-Kleene logic.

Definition 7 .
(C-complexity) The ordinal C-complexity κ(•) of a formula ϕ of L C in a derivation D is defined inductively as follows: (i) formulae of L have C-complexity 0 in any D;

Lemma 8 .
(i) If LPC , Γ ⇒ Δ, then LPC Γ ⇒ Δ and the C-complexity of the formulae in the contexts is unchanged.(ii)If LPC Γ ⇒ Δ, ⊥, then LPC Γ ⇒ Δ and the C-complexity of the formulae in the contexts is unchanged.

Lemma 19 .
(Length-preserving admissibility of Weakening) If there is an RKF-derivation D of length n of Γ ⇒ Δ, then for any multisets Γ ⊇ Γ and Δ ⊇ Δ, there is an RKF-derivation D of length n of Γ ⇒ Δ .
Appendix A]).23. (PKF) The system PKF extends first-order K3 formulated in L Tr with the basic axioms of PA as initial sequents, the induction principle Notational abbreviations have been applied as in the definition of RKF.