Notes on Models of (Partial) Kripke–Feferman Truth

This article investigates models of axiomatizations related to the semantic conception of truth presented by Kripke (J Philos 72(19):690–716, 1975), the so-called fixed-point semantics. Among the various proof systems devised as a proof-theoretic characterization of the fixed-point semantics, in recent years two alternatives have received particular attention: classical systems (i.e., systems based on classical logic) and nonclassical systems (i.e., systems based on some nonclassical logic). The present article, building on Halbach and Nicolai (J Philos Log 47(2):227–257, 2018), shows that there is a sense in which classical and nonclassical theories (in suitable variants) have the same models.


Introduction
Formal theories of truth can be divided into two main categories: semantic theories and axiomatic theories. These two types of approaches to truth are intimately intertwined. In fact, many axiomatic theories have been obtained in the attempt to characterize proof-theoretically certain semantic conceptions of truth, and-from the opposite direction-investigating models of axiomatic theories sheds light on their conceptual aspects. 1 This article investigates models of axiomatic theories related to [19]. The semantic conception of truth presented by Kripke is still the most popular semantic theory of type-free truth.
As it is well known, Kripke describes an inductive procedure for obtaining a class of models for a language L T containing a self-applicable, monadic truth predicate T. These models are called fixed-point models. Since the 1980s, a number of proof systems have been suggested as axiomatizations of fixed-point models (e.g., [1,7,12,16,18,20,30]). In recent years, two alternatives have received particular attention. On the one hand, there are classical systems (i.e., systems based on classical logic); on the other hand, there are nonclassical systems (i.e., systems based on some nonclassical logic). The standard classical systems are variants of the theory known as Kripke-Feferman (KF), devised by Feferman [12], 2 and they will be referred to as KF-like systems (or just KF-systems, or KF-theories). The standard nonclassical systems are variants of the theory known as Partial-Kripke-Feferman (PKF), developed by Halbach and Horsten [16], 3 and they will be referred to as PKF-like systems (or just PKF-systems, or PKF-theories).
There is an ongoing debate on whether classical and nonclassical systems do an equally good job in axiomatizing the fixed-point semantics, and especially in the last few years, researchers have been interested not only in KF-and PKF-systems per se. Rather, they have been interested in how they are related to each other, and in how they are related to the partial conception of truth developed by [19]. In this respect, a number of interesting facts have been discovered, and it is now known that there are several senses in which classical and nonclassical theories can be taken to be on a par (e.g., [8,9,17,22,25]). 4 These results also bear an important contribution to questions concerning the so-called classical recapture, that is, the idea that within nonclassical systems we can restore classical reasoning in most circumstances (see e.g., [4,21,31]).
The present article is a technical contribution to the just mentioned debates, as it focuses on the relationship between classical and nonclassical systems. In particular, we are interested in comparing their models. More specifically, since KF and PKF (and variants thereof) have both been devised as a proof-theoretic characterization of the fixed-point semantics, it seems natural to ask whether there is a sense in which they can be taken to have the "same" models. Of course, since they are formulated over different base logics, there is an obvious sense in which KF and PKF do not have the same models: models of classical theories satisfy the laws of classical logic, but models of nonclassical theories do not satisfy every such law. However, the disparity between the underlying logics of KF and PKF does not prevent one from investigating and comparing their models in a sensible way. In fact, one can still ask, for instance, whether models of classical and nonclassical theories are such that they satisfy the same sentences to be true. That is, one can still ask whether a sentence of the form T ϕ -where the expression ϕ is a name of the sentence ϕ-is satisfied in a model of KF iff the same sentence is satisfied in a model of PKF. More in general, one can still ask whether a given structure M can serve as a model for both KF and PKF.
Some facts concerning models of KF-and PKF-systems are already known. In particular, we know that KF and PKF have the "same standard models". What this means exactly will be clarified below. Roughly, though, letting a standard model be a structure (N, E) expanding the standard model of arithmetic N by an interpretation E for T, it can be shown that (N, E) is a model of KF iff (N, E) is a model of PKF. Additionally, Halbach and Nicolai [17] have recently moved the first steps towards investigating nonstandard models of KF and PKF, i.e., structures whose arithmetical reduct is a possibly nonstandard model of Peano arithmetic. In particular, in [17] it is shown that, if one considers the variants of KF and PKF restricting the schema of induction to formulae not containing T, call them KF − and PKF − , then any model of PKF − is also a model of KF − .
In what follows, building up on [17], we study further the relationship between nonstandard models of KF-and PKF-theories. The main result to come extends the observation by Halbach and Nicolai in several ways, thereby showing that the connection between models of classical and nonclassical systems can be fairly strong. Specifically, it will be shown that: 5 (1) the KF-variants and the PKF-variants with restricted induction have the same models, (2) the KF-variants with internal induction and the PKF-variants with full induction have the same models, Whether the converse of (4) holds, i.e., whether models of PKF-variants with additional transfinite induction up to < ε 0 are models of KF with full induction, is a question that will remain open.
In the next section we fix language and notation, introducing the base logics underlying the various KF-and PKF-systems. Section 3 then defines the truth-systems, specifying their truth-theoretic principles, and the different induction schemata. Section 4 contains the main results of the present paper: it starts by recalling some known facts about standard models of KF and PKF, and it then proceeds with the analysis of their nonstandard models.

Language and Notation
Let L T be the language L PA ∪ {T}, where L PA is the language of arithmetic specified by the signature {0, S, +, ×}, and where T is a unary truthpredicate. Terms and formulae are generated in the usual way via ¬, ∧, ∀. We use u, v, x, y, . . . for variables, . . . r, s, t for arbitrary terms-all possibly with indices. Greek letters ϕ, ψ, ξ, . . . range over formulae of L T . By an L T -expression we mean a term or a formula of L − T . The numeral S . . . S n 0 corresponding to the number n ∈ ω is denoted by n. We fix a canonical Gödel numbering of L T -expressions and a primitive recursive (p.r.) formalization of syntactic notions and operations. If e is an L T -expression, the Gödel number of e is denoted by #e and e is the term representing #e in L T . The sets of terms, closed terms, variables, formulae, and sentences of L T are p.r., and can be represented in L T . In practice, we take the following L T -predicates to abbreviate the equations for the (p.r.) characteristic function for such sets: Term(x) (Ct(x)) := x is the gn of a (closed) term; Var(x) := x is the gn of variable; For each of the above syntactic predicates P (x), we let P denote the set of codes of P 's. That is, for instance, St T is the set of codes of L T -sentences. For simplicity, we take L T additionally to contain function symbols for the following primitive recursive operations on Gödel numbers: The expression e[t/v k ] is the result of replacing, in the expression e, each free occurrence of v k by the term t. We also assume to have a valuation function val(x) (which is PA-definable) such that val( t ) = t for closed terms t. We will make use of the following abbreviations: We often write sb(t, num(z)) for sb(t, num(z), s). The expression ϕ(v) denotes a formula ϕ with at most the variable v free. Similarly, the expression t(x) denotes a term t with at most the variable x free. We adopt the abbreviation e( v) := e(v 1 , . . . , v k ), where e(v 1 , . . . , v k ) denotes an expression e whose free variables are among v 1 , . . . , v k . Sometimes we write ϕ(t) for ϕ[t/v], when it is clear from the context that t is a term with which a designated free variable of ϕ has been replaced.

Sequent Calculi
We present KF-and PKF-system as Gentzen calculi for sets of formulae (i.e., we let a sequent be an expression of the form Γ ⇒ Δ, for Γ and Δ finite sets of L T -formulae), making use of standard notions to be found, e.g., in [34]. Given a set of formulae Θ, we let ¬Θ := {¬θ | θ ∈ Θ} and T Θ := {T θ | θ ∈ Θ}. Moreover, we let Θ ( Θ) denote the iterated conjunction θ 1 ∧ . . . ∧ θ n (disjunction θ 1 ∨ . . . ∨ θ n ) of elements of Θ := {θ 1 , . . . , θ n }. We begin by introducing the various logics underlying the systems of truth employed in the paper. The basic nonclassical logic we will be interested in is the fourvalued Belnap-Dunn Logic ( [5,11]), also known as First Degree Entailment (FDE). In the next definition we introduce a two-sided sequent calculus for FDE with classical equality, i.e., with the identity relation = behaving classically. 6 Definition 2.1. (FDE = ) The system FDE = consists of the following axioms and rules. Ax Conditions of application: u eigenvariable. • Strong Kleene, SK = , is the system obtained from FDE = by replacing the rule =¬L with the stronger rule • Logic of Paradox, LP = , is the system obtained from FDE = by replacing the rule =¬R with the stronger rule tableaux system [28] and a system of natural deduction [27] (on natural deduction see also [24]). For a survey of various semantics and proof systems for FDE and some expansion thereof, see [23].
• Kleene's Symmetric Logic, KS = , is the system obtained by adding to FDE the initial sequents 7 Semantically, the rule ¬L (¬R) restricts the class of models to those in which there is no glut (gap), while GG (which stands for "gaps or gluts") excludes the simultaneous occurrence of gaps and gluts.

Semantics
Next we present a semantics for the calculi just introduced.
• symmetric, if it is either consistent or complete.
We now define a relation fde σ between four-valued models and L Tformulae. We let the assignment σ[v i : a] be just like σ except that it maps v i to a ∈ |N |. Variable assignments can be recursively extended, in the usual way, to provide a value for all terms t. For notational simplicity, we do not distinguish between a variable assignment σ and its recursive extension. In particular, for any term t (including variables), we let t N σ denote the value of t in M := N , (E, A) under σ.
¬ϕ Remark 2.5. Note that (i) since identity statements behave classically, by induction on the positive complexity of formulae, one can show that every formula ϕ ∈ L PA behaves classically as well, 8 We now extend the relation fde σ to sequents as follows: It may be worth noticing that our definition of M fde σ Γ ⇒ Δ differs from that of [16,17,22], where the following clause was added: if M fde σ ¬δ for all δ ∈ Δ, then M fde σ ¬γ for some γ ∈ Γ.
However, the two versions give rise to the same notion of fde-logical consequence, as shown in the next proposition. The reason for working with Df. 2.6-besides it being simpler-is that we will be considering classes of consistent, respectively complete four-valued models, in which contraposition fails.
To obtain M + , convert the gluts (gaps) of M into gaps (glut) as follows: first define M − from M by: That is, E − and A − contain, respectively, everything which is only true and only false in M. Then define: That is, E + and A + contain, respectively, everything which is only true and only false in M, and additionally both E + and A + contain all gaps of M. Using ( †), verify that M + satisfies (*). 9 Lemma 2.8. (Soundness and completeness) There are several completeness proofs for calculi equivalent to ours. We refer the reader to [6] and [17], whose arguments can be adapted to the present setting.

KF-Like and PKF-Like Theories
This section introduces the truth-theoretic principles (defined in Table 1) and the various induction schemata (defined in Table 2) of the KF-like and PKF-like theories whose models we are going to investigate. In order to introduce schemata of transfinite induction, we fix a standard notation system of ordinals up to Γ 0 . 10 We use a, b, c . . . to denote the code of our notation system whose value is α, β, γ · · · ∈ On (with the exception of ωand ε-numbers, for which we use the symbols 'ω' and 'ε' themselves), and we use ≺ to denote a standard primitive recursive ordering defined on codes of ordinals. Moreover, we let Here D(t) means that t is determinate, that is either true or false, and not both. This predicate was defined by Feferman [12]. 11 Definition 3.1. (KF) KF is the system obtained from classical logic (with equality) by adding: initial sequents Γ ⇒ Δ for Γ ⇒ Δ an initial sequent of PA (see, e.g., [33]); defining axioms for additional function symbols; truththeoretic initial sequents of Table 1 except the initial sequents T¬, Cons, Comp, Sym; the rule IND.
Variants of KF are obtained (i) by adding one of the axioms of consistency, completeness, symmetry, and/or (ii) by modifying the induction schema. More precisely: (i) KF cs is obtained from KF by adding the initial sequent Cons.
(ii) KF cp is obtained from KF by adding the initial sequent Comp.
(iii) KF S is obtained from KF by adding the initial sequent Sym.
(PKF) PKF is the system obtained from FDE = by adding: initial sequents Γ ⇒ Δ for Γ ⇒ Δ an initial sequent of PA; defining axioms for additional function symbols; truth-theoretic initial sequents of Table 1 except Cons, Comp, Sym; the rule IND.
Variants of PKF are obtained (i) by modifying the base logic and/or (ii) by modifying the induction schema. More precisely: (i) PKF cs is obtained from PKF by adding the rule ¬L.

Induction for
for ϕ ∈ L and u eigenvariable.

Determinate induction IND D
Determinate Induction is the ternary rule obtained by adding the following premises to IND Internal determinate induction (ii) PKF cp is obtained from PKF by adding the rule ¬R.
(iii) PKF S is obtained from PKF by adding the initial sequent GG.  [7] to define a variant of KF, called KF t . However, Cantini formulates KF with the consistency axiom, which excludes in general the possibility of inconsistent predicates. This means that Cantini's tot(t) was de facto defining t to be a classical predicate. Since here we also study variants of KF with no axiom concerning the consistency and/or the completeness of T, we use the rule IND int D to restrict inductive reasoning on classical predicates. (Of course, in presence of Cons or Comp, IND int D is partly redundant.) As for the rule IND D used in theories PKF D , it is a straightforward counterpart of IND int D . As far as I know, the theories PKF D have not been studied in the literature so far. 13

Models of KFand PKF-systems
This final section investigates models of KF-and PKF-systems, and it contains the main contribution of the present article. 12 Unlike [22], we do not have contraposition as a primitive rule, which is why we need to add the counterpositive of transfinite induction as an additional principle. For the other induction schemata, the counterpositive can be shown to be admissible. See [9] for details. 13 Thanks to an anonymous referee for suggesting us to define a PKF counterpart of Cantini's KFt.

Standard Models
As already mentioned in the introduction, KF-and PKF-systems were devised as a proof-theoretic characterization of models of [19], and we also mentioned that KF-and PKF-systems have the same standard models. Let us begin by making this claim precise. To this end, we introduce the socalled Kripke Jump, i.e., a monotone operator yielding so-called fixed-point models. 14

Let (E, A) ≤ (E , A ) be defined as E ⊆ E and A ⊆ A . The Kripke Jump is a monotone operator, in the sense that Φ(E, A) ≤ Φ(E , A ) whenever (E, A) ≤ (E , A ). By Tarski-Knaster theorem, Φ has fixed-points, that is, pairs (E, A) such that Φ(E, A) = (E, A).
• symmetric, if it is either consistent or complete.
The following lemmata give formal expression to the claim that KF-and PKF-theories are a proof-theoretic characterization of fixed-point models. In fact, it is known they are N-categorical axiomatizations of these models, in the sense of [14]. 15 Lemma 4.3. (essentially, [12,16]) One consequence of this observation is that classical and nonclassical systems have the same standard models, in the sense that Similarly for other KF-and PKF-variants.

Nonstandard Models
Having clarified in what sense KF-and PKF-theories can be taken to have the same standard models, we now move on to the analysis of their nonstandard models. As already mentioned, a contribution in this analysis has been provided by [17]: they have shown that any model of PKF − , including those whose arithmetical reduct is a nonstandard model of PA, is also model of KF − . More precisely: Lemma 4.4. ( [17]) For N arbitrary model of PA, for any variable assignment σ, the following jointly hold: 16 In the rest of the paper, Lemma 4.4 will be sharpened. Here's is an outline of what we shall see: (1) It will be proven that the converse direction of Lemma 4.4 holds. That is, given a model (N , E) of KF − , it can be shown that the structure  N , (E, A) is a model of PKF − , where A is defined via E as the set of sentences (in the sense of N ) such that their negation is in E, plus all elements which do not code a sentence.
(2) It will be shown that the same result holds for the pairs KF int -PKF and KF D -PKF D . That is, it will be shown that the structure (N , E) N , (E, A) is a model of PKF + , for A defined as above.
Before proving (1)-(4), let me introduce some notation. We begin with an auxiliary lemma, showing that T ϕ(ẋ) is classically satisfied in a model (N , E) of a KF-system iff ϕ(x) is fde-satisfied in the structure N , (E, Eˆ) . 18 17 , (N , E) and σ be such that (N , E) σ KF • , and ϕ(x) an L T -formula with at most x free. 19 Then Proof. The proof is by induction on the positive complexity of ϕ. We show the crucial cases of truth ascriptions. Let ϕ(x) ≡ T t(x), for t(x) a term with at most x free. Then, For ϕ(x) ≡ ¬T t(x), we reason as follows: As it will become clear below, this Lemma plays a crucial role, as it allows us to move in and out the scope of the truth predicate when we move between classical and nonclassical satisfaction relations. This will be particularly important for proving the equivalence between the classical satisfability of the rules of internal induction and the nonclassical satisfability of the corresponding induction schemata.
Let us now begin by proving part of the converse of Lemma 4.4.
We first show that each initial sequent of PKF − is satisfied in N , (E, Eˆ) , and then we show that every rule of inference is sound. PKF − 's initial sequents are either of the form ϕ, Γ ⇒ Δ, ϕ, or they are truth-theoretic initial sequents. The former kind of sequent are satisfied simply by definition of the relation fde σ . As for the truth-theoretic initial sequents, we consider three examples.
The soundness of logical rules follows by definition of fde σ . The rule of Remark 4.8. It may be worth noticing that, given a model N , (E, A) of a PKF-theory, the set A is definable via E as in point (iii) of Notation 4.5 above, that is, as the set of sentences (in the sense of N ) whose negation is in E, along with elements not coding any sentence. This is so due to the PKF initial sequents TSt L T and T¬. This means that Eˆis the unique A such that N , Similar remarks apply below. 20 We now show, with lemmata 4.9-4.10, that the same result holds for the pair (KF int -PKF). From N , (E, Eˆ) fde σ ϕ(u) ⇒ ϕ(u + 1), by Lemma 4.6, we get for t arbitrary. Again by Lemma 4.6, this yields which together with N , (E, A) fde σ ϕ(0) implies the desired conclusion.
The argument of Lemmata 4.9 and 4.10 can be adjusted to the pair (KF D -PKF D ), as it will be shown next.
then, for all terms t, The verification is similar to that of Lemma 4.9. Using Lemma 4.6 and commutation of T with quantifiers and connectives (which we have since The desired conclusion follows by IND int D and Lemma 4.6. Proof. Once again, the verification is similar to that of Lemma 4.10. Suppose that the following jointly hold, for some t such that N |= Fml 1 L T (t): From (3) we obtain and from (4), which yields: The conclusion follows by IND D and Lemma 4.6.
From lemmata 4.4, 4.7, 4.9, and 4.10 we obtain the following We next show that the results are stable under addition of truth-theoretic principles. That is: Proof. We consider (KF cs , PKF cs ). 21 In order to show that models of KF cs expanded by the anti-extension Eˆare models of PKF cs , we need to show that the additional rule ¬L of PKF cs is fde-sound in these models.  we would get #ϕ ∈ E. Hence we obtain (*), and we conclude that some sentence in Δ is fde-satisfied in N , (E, Eˆ) . Conversely, in order to show that the truth-theoretic initial sequent Cons of KF cs is classically satisfied in (N , E), we first notice that Now assume (omitting context for readability) It follows that The arguments for the pairs (KF cp , PKF cp ) and (KF S , PKF S ) are analogous.
In order to complete the above picture, we are left with showing that KF-systems with full induction and PKF-systems with additional transfinite induction up to < ε 0 have the same models. In the next Lemma, it is shown that every model of KF is indeed a model of PKF + . However, the question whether every model of PKF + classically satisfies the induction schema IND will be left open for future research. Since (N , E) σ TI ≺ [L T , < ε 0 ], for γ < ε 0 we have (recall that c is (the numeral of) the code of γ in our notation system) Observe that the argument for Lemma 4.15 can be generalized so to establish a more robust characterization of the relationship between bounded induction in KF-systems and transfinite induction in PKF-systems. To see this, given a formula ϕ ∈ L T , let Prog ≺ (ϕ) := ∀x(∀y ≺ x ϕ(y) → ϕ(x)), (ϕ(x)). Now consider the theory obtained from KF int by replacing IND int with Σ ninduction for L T , and call it KF n . Since it is known that IΣ n formulated in L T proves TI ≺ [T, < ω n+1 ], 23 we immediately obtain a lower bound for the derivability of transfinite induction within KF n . Let now (N , E) σ KF n .
With essentially the same argument as above, one can show that N , (E, Eˆ) is a structure fde-satisfying TI ≺ [L T , < ω n+1 ] and TI ≺ [L T , < ω n+1 ](C). But then N , (E, Eˆ) is a model of a PKF-theory obtained by extending PKF with TI ≺ [L T , < ω n+1 ] and TI ≺ [L T , < ω n+1 ](C). 24 These observations give rise to the following By transparency of T in PKF • , this is equivalent to either #γ i / ∈ E for some 1 ≤ i ≤ n, or #δ j ∈ E for some 1 ≤ j ≤ m. (5) Now let (N , E) be an arbitrary model of KF • . By Theorem 5.1, E is the extension of a model of PKF • . Hence, by (5), By completeness, it follows that KF • T Γ ⇒ T Δ .
For the converse direction, let KF • -PKF • be one of the above pairs except KF ( ) -PKF + ( ) . Suppose that KF • T Γ ⇒ T Δ . This is the case precisely if, for an arbitrary model (N , E) of KF • , (N , E) σ T Γ ⇒ T Δ , which is in turn the case precisely if either #γ i / ∈ E for some 1 ≤ i ≤ n, or #δ j ∈ E for some 1 ≤ j ≤ m. (6) Let N , (E, A) be an arbitrary model of PKF • . By Theorem 5.1, (N , E) is a model of KF • , and hence, by (6), By completeness, we obtain PKF • T Γ ⇒ T Δ , which yields the desired conclusion by transparency of T.
Acknowledgements. I would like to thank Martin Fischer, Carlo Nicolai, and Johannes Stern for very helpful discussions on the topic of this article, and two anonymous referee for their valuable comments. This work has been supported partly by the AHRC South, West and Wales Doctoral Training Partnership (SWW DTP), Grant No. AH/L503939/1-DTP1 and partly by the DAAD-Deutscher Akademischer Austauschdienst-German Academic Exchange Service, Grant No. 57552337.
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