ar X iv :2 00 4. 07 27 5v 2 [ m at h. L O ] 6 M ay 2 02 0 THE MODAL LOGICS OF KRIPKE-FEFERMAN TRUTH CARLO NICOLAI AND JOHANNES STERN Abstract. We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model M, or an axiomatization S thereof, we find a modal logic M such that a modal sentence φ is a theorem of M if and only if the sentence φ∗ obtained by translating the modal operator with the truth predicate is true in M or a theorem of S under all such translations. To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and establish the completeness of a family of non-congruent modal logics whose internal logic is subclassical with respect to this semantics. 1. Introduction In the aftermath of Gödel's incompleteness theorems, researchers became interested in general properties of the formalized provability predicate. Bernays distilled three such properties known as Hilbert-Bernays derivability conditions but it was Löb [Loe55] who, elaborating on Bernays work, proposed the three derivability conditions that are commonly thought to aptly characterize the properties of a "natural" provability predicate of recursively enumerable systems extending a sufficiently strong arithmetical theory.1 The striking feature of Löb's derivability conditions is that they are essentially principles of propositional modal logic, that is, if the provability predicate is replaced by a modal operator the three derivability conditions can be viewed as axioms of systems of modal logic. It is only natural then to ask which modal system is the exact modal logic of the provability predicate of recursively enumerable systems extending I∆0 + Exp – a question that was answered by Solovay's [Sol76] seminal arithmetical completeness result, which showed that logic to be the system GL. Interestingly, in his paper Solovay observes that by considering alternative modal principles and systems we can study the modal logic of other sentential predicates of interest. It is in this spirit that we investigate the modal logic of the truth predicate of fixed-point models of truth in the sense of [Kri75] and their axiomatizations by [Fef91, Fef08],2 that is, we propose Solovay-style completeness results for a family of truth theories. Let Σ be such a truth theory. Then we determine the modal logic S such that for all φ ∈ L✷ S ⊢ φ ⇔ for all realizations ∗ (Σ ⊢ I∗(φ)) Here ∗ a function that maps the propositional variable of the modal language to sentences of the language of the truth theory, while I is a translation that commutes 2000 Mathematics Subject Classification. Primary 03Bxx; Secondary 03B45; 03B50. Johannes Stern's research was funded by the ERC Starting Grant TRUST 803684. 1I∆0 + Exp can be considered a safe lower bound. Here exp stands for the sentence asserting the totality of the exponential function x 7→ 2x. See [HP98] for details. 2For a general overview of axiomatizations of fixed-point semantics, see [Hal14, §15-17]. 1 MODAL LOGICS OF KF 2 with the logical connectives but where the modal operator ✷ is translated by the truth predicate – we will call I the truth-interpretation: I∗(✷φ) = TpI∗(φ)q. Some initial research in this direction has already been undertaken by Czarnecki and Zdanowski [CZ19] and [Sta15] who study theories of truth inspired by revision theoretic approaches [GB93].3 Czarnecki and Zdanowski [CZ19] and [Sta15] prove, albeit in slightly different guise, the modal logic of nearly stable truth, that is, the modal logic of the axiomatic truth theory Friedman-Sheard [FS87, Hal94], to be the modal logic KDDc. As mentioned this paper is concerned with so-called Kripkean fixed-point theories of truth and their axiomatizations. The peculiar feature of these theories is that the truth predicate is in some respect nonclassical: depending on the particular version of the theory under consideration sentences may be neither true nor false, or both true and false. As a consequence, the so-called internal logic of the truth predicate, that is the logic holding within the scope of the truth predicate, will be nonclassical, which in turn forces the modal logic to be non-normal and, indeed, non-congruent.4 This has the further consequence that we cannot appeal to standard modal semantics for investigating the modal systems under consideration but have to introduce a novel version of possible world semantics, in which we allow for both classical and nonclassical worlds. Before we outline the structure of the paper and give some more background to our project it may be useful to stress why the project is an interesting one and, more to the point, why the question we ask is a non-trivial one. For it may seem that in the presence of suitable axiomatic theories of truth determining the corresponding modal logic is a simple matter: we obtain the modal logic by replacing the truth predicate by the modal operator and the individual variable ranging over sentences by propositional variables of the modal operator language in the axioms of the theory. Yet, this picture is too simplistic for two reasons: First, the truth theory will be formulated over a theory of syntax, usually some arithmetical theory such as I∆0 + Exp or extensions thereof, which will equip the truth theory with further expressive strength. For instance, in the truth theory we can prove the diagonal lemma, which is usually not possible within a modal logic. But by using the diagonal lemma we may prove further modal principles, which we cannot prove in the modal logic extracted in the simplistic way from our theory. Indeed, the formalized Löb's theorem provides a neat example to this effect: if we have the diagonal lemma, it can be proved on the basis of Löb's three derivability conditions but we cannot prove the modal Löb principle in the modal logic K4. Now, independently of whether a similar phenomenon arises when studying theories of truth, the observation highlights that determining the exact modal logic seems to be a worthwhile project. Second, in truth theories one usually stipulates the T-sentences for atomic sentences of the language without the truth predicate. But if we consider the parallel principle (TB) ✷p ↔ p in the context of a modal operator logic we run into trouble: modal logics are closed under the rule of uniform substitution, which means that (TB) would hold for all sentences φ of the language with the truth predicate. But then, as a consequence of the Liar paradox, no interesting classical truth theory can have a modal logic that 3For related work see also [Ste15] who connects various diagonal modal logics [Smo85] to truth theories via the truth interpretation. 4Non congruent modal logics are the non-classical modal logics in the sense of [Seg71] and [Che80]. MODAL LOGICS OF KF 3 assumes (TB).5 Nonetheless, already the T-sentences for atomic sentences in the language without the truth predicate – which are required in providing non-trivial axiomatizations of the truth predicate – have an impact on the modal logic of the theory and the exact impact cannot be immediately read off the axioms of the truth theory. In fact, we will show that such T-sentences force the modal logic of theories of Kripke-Feferman truth to comprise the axiom ✷φ ∧ ¬✷¬φ → φ,(faith✷) which does not appear in the usual list of axioms of these truth theories. Thus faith✷, for slightly different reasons, plays a similar role as Löb's principle in GL: it is easily derivable in our theories of truth, but it needs to be added to their modal logics as an additional axiom. In sum, we believe that determining the exact modal logic of a theory of truth is a worthwhile enterprise. 1.1. Plan of the paper. In §2 we introduce sub-classical modal logics and their semantics, which extend the main (fully structural) non classical logics considered in the literature on theories of truth. These include the four-valued logic of first-degree entailment and its three-valued paraconsistent and paracomplete extensions. As we will show later in the paper, the sub-classical modal logic are the modal logics of the internal theory or logic of the systems of Kripke-Feferman truth. §3 presents the classical modal logics that will be shown to be exactly the modal logics of KripkeFeferman theories. Since these logics are classical but have a sub-classical internal logics, providing a semantics for them is a nontrivial task. To this end, we introduce special frames in which classical worlds 'see' a unique nonclassical world, and prove the completeness of our modal logics with respect to this semantics. In §4, we move on to introduce Kripkean fixed point semantics and the systems of Kripke-Feferman truth. §5 contains the main results of the paper. We first establish the modal logic of the basic system KF and its extensions with completeness of consistency axioms for the truth predicate. Then we consider – by providing a different realization – a stronger form of truth theoretic completeness that holds for KF + CN, KF + CM, WKFC, DT, and all of their consistent extensions. We conclude the paper by pointing to some further research. 2. Sub-classical modal logics for truth Our basic language is a standard modal operator language L✷, which is built over a countable set Prop of propositional variables p0, p1, p2, . . .: φ ::= pj | ⊥ | ⊤ | ¬φ |✷φ |φ ∧ φ |φ ∨ φ with pj ∈ Prop for j ∈ ω. The language L0 is the language obtained by removing ✷ from L✷. L։✷ is obtained by adding to L✷ a binary connective ։. In the context of our nonclassical modal logics, ✸ can be defined as usual as ¬✷¬, but this will not be the case in the classical modal logics investigated in the following sections. We will work with several sub-classical logics that support naïve truth and that amount to the internal logics of the classical systems of truth we are ultimately interested in. We formulate our systems in a sequent calculus, where Γ,∆,Θ,Λ . . . are finite sets of formulas of L0. We denote with Prop(φ) the set of propositional letters of φ (resp. Prop(Γ) for the propositional formulas in the set of formulas Γ). For X a set of sentences of L✷, we let ✷X := {✷φ | φ ∈ X}. 5It is precisely in this sense that our modal logics diverge from Feferman's [Fef84] 'type-free modal theory': while Feferman also picks up on the modal character of the various truth principles and formulates them using a modal operator, he does this over a specific theory and adopts TB, that is, Feferman is to be taken literally when he calls his system a 'modal theory' rather than a 'modal logic'. His 'modal theory' is not closed under uniform substitution and should therefore not be called a logic. MODAL LOGICS OF KF 4 We work with a sequent calculus formulations for first-degree entailment (FDE), symmetric Kleene logic (KS3), Strong Kleene (K3), the logic of paradox (LP), Weak Kleene (B3), and Feferman-Aczel logic (F3), which is a slight modification of B3 with a primitive conditional [AB75, Kle52, Cos74, CC13, Acz80, Fef08]. Details of the systems are given in Appendix A. We now introduce some modal extensions of our systems.6 Definition 1. For S ∈ {K3,B3, LP,KS3,FDE,F3}, the systems S✷ in the language L✷ (L ։ ✷ ) are defined by adding to S the rules: φ,Γ ⇒ ∆ (✷l) ✷φ,Γ ⇒ ∆ Γ ⇒ ∆, φ (✷r) Γ ⇒ ∆,✷φ ¬φ,Γ ⇒ ∆ (¬✷l) ¬✷φ,Γ ⇒ ∆ Γ ⇒ ∆,¬φ (¬✷r) Γ ⇒ ∆,¬✷φ The adequacy of the logics S✷ with respect to the possible worlds semantics introduced below follows from a more general result concerning a subclassical modal logic, which can be thought of as the modal analogue of the classical propositional logic K. Definition 2. For S ∈ {K3,B3, LP,KS3,FDE,F3}, the systems S in L✷ (L ։ ✷ ) are obtained by replacing (ref) with the initial sequent Γ, φ ⇒ φ,∆ for all φ ∈ L✷, and by adding to S the rules: Γ,¬φ ⇒ ¬∆ (l) ✷Γ,¬✷φ ⇒ ¬✷∆ Γ ⇒ φ,¬∆ (r) ✷Γ ⇒ ✷φ,¬✷∆ We apply the following notational conventions for T one of our modal systems: Derivations in T are at most binary branching finite trees labelled with sequents. Leaves are axioms – the relevant instance of reflexivity, (⊥), (⊤), and the remaining nodes are obtained by applications of the rules of inference of T . For T one of the logics above, T ⊢ Γ ⇒ ∆ stands for the existence of a derivation whose root is labelled by Γ ⇒ ∆. We can extend the above notion of derivability to arbitrary sets of formulas: for Γ,∆ arbitrary sets of formulas, we write T ⊢ Γ ⇒ ∆ iff there are finite Γ0 ⊆ Γ and ∆0 ⊆ ∆ such that T ⊢ Γ0 ⇒ ∆0. The length of a derivation can be defined as the number of nodes in the maximal branch of the derivation tree minus one. We write T ⊢n Γ ⇒ ∆ if the length of the derivation of Γ ⇒ ∆ in T is ≤ n. By our definition of sequent, contraction is trivially admissible in our logics. In addition, by straightforward induction on the length of the derivations in the appropriate systems, we have: Lemma 3 (Reflexivity, Weakening). (i) For S ∈ {K3, LP,KS3,FDE,B3} as in the previous definition, (a) For all φ ∈ L✷, S✷ ⊢ Γ, φ ⇒ φ,∆ (b) If S✷ ⊢n Γ ⇒ ∆, then S✷ ⊢n Γ,Γ0 ⇒ ∆,∆0 for Γ0,∆0 finite. (c) if S ⊢ n Γ ⇒ ∆, then S ⊢ n Γ,Γ0 ⇒ ∆,∆0 for Γ0,∆0 finite. (ii) For φ ∈ L։ ✷ , and Γ,∆ ⊂ L։ ✷ : 6Modal logics based on K3 are studied in [JT96]. Their framework forms the basis of the logical systems of nonclassical modal logic considered in this paper. An FDE-based version of K is extensively studied in [OW10] and labelled BK. Also, consistent and complete extensions of BK have been investigated by [OS20] and [OS16]. MODAL LOGICS OF KF 5 (a) F3✷ ⊢ Γ, φ ⇒ φ,∆ (b) If F3✷ ⊢n Γ ⇒ ∆, then F3✷ ⊢n Γ,Γ0 ⇒ ∆,∆0. (c) if F3 ⊢ n Γ ⇒ ∆, then F3 ⊢ n Γ,Γ0 ⇒ ∆,∆0. 2.1. Semantics. Next we introduce a possible worlds semantics for the propositional systems just defined. The main difference with standard possible models lies in the use of sub-classical valuation functions, which then give rise to nonclassical semantic clauses of the connectives and consequence relations [JT96, Pri08]. Definition 4. (i) A frame is a pair (Z,R) where Z is a nonempty set and R is a binary relation on Z. (ii) A four-valued valuation for L✷ is a function V : Prop × Z → {0, 1, n, b}, where {0, 1, n, b} is the set of truth values: • a consistent valuation is a function V : Prop × Z → {0, 1, n}; • a complete valuation is a function V : Prop × Z → {0, 1, b}; • a symmetric valuation assigns, at every z ∈ Z, values in exactly one of {1, 0, n} or {1, 0, b}. (iii) A model M is a triple (Z,R, V ), with (Z,R) a frame and V a valuation. A model so-defined is based on (Z,R). • A model M = (Z,R, V ) is consistent if V is consistent. • A model M = (Z,R, V ) is complete if V is complete. Let  be the ordering of the truth values {0, n, b, 1} displayed in the lattice 0 bn 1 Moreover, let 2 be the ordering n 2 0 2 1. Definition 5 (Truth). (i) Given a model M = (Z,R, V ), and s ∈ {fde, k3, lp, ks3}, an s-interpretation extends the valuation V by assigning to each sentence of L✷ a truth value: |p|M,zs = Vz(p) |¬φ| M,z s =      0 if |φ|M,zs = 1 1 if |φ|M,zs = 0 |φ|M,zs otherwise. |φ ∧ ψ|M,zs = inf{|φ| M,z s , |ψ| M,z s } |φ ∨ ψ| M,z s = sup{|φ| M,z s , |ψ| M,z s } |✷φ|M,zs = inf{|φ| M,z0 s | Rzz0} If V is four-valued, then s = fde; if V is consistent, then s = k3; if V is complete, then s = lp; finally, if V is symmetric, then s = ks3. (ii) Given a consistent M = (Z,R, V ), a B3-interpretation is given by: |p|M,zb3 = Vz(p) |¬φ| M,z b3 =      0 if |φ|M,zb3 = 1 1 if |φ|M,zb3 = 0 |φ|M,zb3 otherwise. |φ ∧ ψ|M,zb3 = min2(|φ| M,z b3 , |ψ| M,z b3 ) |φ ∨ ψ| M,z b3 = |¬(¬φ ∧ ¬ψ)| M,z b3 |✷φ|M,zb3 = inf2{|φ| M,z0 b3 | Rzz0} Notice the use of the ordering 2 in this clause. MODAL LOGICS OF KF 6 (iii) Again given a consistent M = (Z,R, V ), an F3-interpretation extends a B3-interpretation with the clause: |φ։ ψ|M,zf3 =      1, if |φ|M,zf3 = 0 or min2(|φ| M,z f3 , |ψ| M,z f3 ) = 1 0, if |φ|M,zf3 = 1 and |ψ| M,z f3 = 0 n, otherwise Notation. Given M = (Z,R, V ) and s ∈ {fde, lp, k3, ks3, b3, f3}, we write M, z s φ whenever |φ|M,zs ∈ {1, b}. In what follows, we will focus on the so-called local logical consequence relation in our semantics [BdRV01]. Definition 6 (Consequence). Let F = (Z,R) be an arbitrary frame, and F a class of models based on F . For Γ,∆ sets of sentences of L✷ and s ∈ {fde,lp,k3,ks3, b3, f3}, we have Γ Fs ∆ iff, for all M ∈ F and z ∈ Z: if ∀γ ∈ Γ M, z s γ for all γ ∈ Γ, then M, z s δ for some δ ∈ ∆. Remark 7. The notion of consequence for FDE and KS3 can be formulated with an extra clause for the anti-preservation of falsity. This is not the case for the stronger logics. The logics S are adequate with respect to the Kripke semantics introduced above. To prove this we generalize to different evaluation schemata the main strategy applied by [JT96] to modal logics extending K3. In particular, the notion of maximally consistent set is replaced with the one of saturated set. A saturated set is, roughly, a non-trivial set of formulas closed under the particular logic whose completeness is at stake.7 The detailed proof of the next claim is provided in Appendix B. Proposition 8 (Adequacy). Let F = (Z,R) be an arbitrary frame. Then for any Γ ⊆ L✷ and φ ∈ L✷: Γ Fs ∆ iff S ⊢ Γ ⇒ ∆ holds when F is the class of four-valued models based on F and S is FDE F is the class of consistent models based on F and S is K3, B3, or F3. F is the class of complete models based on F and S is LP F is the class of symmetric models based on F and S is KS3 Notice that, by changing the parameter S in the claims above, we are simultaneously changing both the logic on the right-hand side of the equivalence, and the evaluation scheme on the left hand side. The logics S are, in a sense, the equivalent of the modal logic K in the nonclassical settings. Turning to the modal logics S✷ we observe that these logics are precisely the logics of so-called idiosyncratic frames (Figure 2.1). Definition 9. Let F = (Z,R) be a frame. F is called idiosyncratic iff (∀z0, z1 ∈ Z)(Rz0z1 ↔ z0 = z1). The strategy employed in obtaining the adequacy of the logics S – involving of course the appropriate re-definition of saturation relative to S✷ – yields: 7A set Γ of sentences is trivial in T iff T ⊢ Γ ⇒ ∅. MODAL LOGICS OF KF 7 z zzz z . . .. . . Figure 1. An idiosyncratic frame. Proposition 10 (Adequacy). Let F = (Z,R) be an idiosyncratic frame. Then for any Γ ⊆ L✷ and φ ∈ L✷: Γ Fs ∆ iff S✷ ⊢ Γ ⇒ ∆ holds when F is the class of four-valued models based on F and S is FDE F is the class of consistent models based on F and S is K3, B3, or F3 F is the class of complete models based on F and S is LP F is the class of symmetric models based on F and S is KS3 3. Modalized truth principles We now move on to introducing classical, non-congruent modal logics, which we will show to be the modal logics of the truth theories to be considered. These modal logics have the particular feature that their inner logic, that is, the logic governing the scope of the modal operator will be one of the logics S✷ discussed in the previous section. After presenting these modal logics we introduce a novel version of possible world semantics in which we have both classical and non-classical worlds. We show that our modal logics are the logics of the classical worlds of socalled mixed idiosyncratic frames. 3.1. Axioms and rules. We start with the most basic system, which modalizes the clauses for the positive inductive definition underlying the Kripke-Feferman approach to truth.8 Since it will not cause any trouble in what follows, we will list the principles our classical modal logics as axioms, even though strictly speaking our logics are formulated in classical sequent calculus suitably extending the subclassical systems of §2. Definition 11 (Modal logic BM). The modal logic BM extends classical propositional logic with:9 (⊤) ✷⊤ (⊥) ¬✷⊥ (¬) ✷φ ↔ ✷¬¬φ (∧1) ✷(φ ∧ ψ) ↔ ✷φ ∧ ✷ψ (∧2) ✷¬(φ ∧ ψ) ↔ ✷¬φ ∨ ✷¬ψ (∨1) ✷(φ ∨ ψ) ↔ ✷φ ∨ ✷ψ (∨2) ✷¬(φ ∨ ψ) ↔ ✷¬φ ∧ ✷¬ψ (✷1) ✷φ ↔ ✷✷φ 8As discussed in the introduction, [Fef84, §12] also considers modalizations of similar principles, but the presence of axioms of the form ✷P ↔ P for atomic P of the language with ✷ starkly contrasts with our approach, in which we want to preserve uniform substitution at the expense of such principles for atomic formulas. 9We omit the sequent arrow in the formulation of the principles of M. MODAL LOGICS OF KF 8 (✷2) ✷¬φ ↔ ✷¬✷φ ✷φ ∧ ¬✷¬φ → φ(faith✷) The modal logic BMis just like BM but does not assume (faith✷). As stressed in the Introduction, the axiom (faith✷) is, as we will show, the distinctive axiom of classical Kripke-style theories of truth such as the Kripke-Feferman theories. In a nutshell it asserts that if a sentence φ is non-classically true but not false, i.e., it is not both true and false, then φ is also classically the case. Preempting our modal semantics, it asserts that if φ has a classical truth value at a non-classical world then it will have the same truth value at the classical worlds that see it, i.e., it expresses that the valuation is faithful with respect to classical truth values. The logic BM can then be extended with principles corresponding to the consistency, completeness, and symmetry of ✷. Definition 12 (The logics M-, M, Mn, Mb). The modal logic Mn extends BM with (D) ¬✷¬φ ∨ ¬✷φ. The modal logic Mb extends BM with (Dc) ✷¬φ ∨ ✷φ. The modal logic M (M-) extend BM (BM-) with (DDc) (¬✷¬φ ∨ ¬✷φ) ∨ (✷¬ψ ∨ ✷ψ). Lemma 13. Let M, Mn, Mb be defined as in Definition 12. Then (i) M ⊢ (✷φ → φ) ∨ (ψ → ✷ψ) (ii) Mn ⊢ ✷φ → φ (iii) Mb ⊢ φ → ✷φ. Remark 14. It's easy to see that over BM−, φ → ✷φ entails Dc and faith✷, and ✷φ → φ entails D and faith✷. Proposition 24 and Corollary 29 below will entail that, in stark contrast with what happens in the truth-theoretic side, the converse implications do not hold. Next we turn to logic whose modalities are governed by the b3and f3-evaluation schemata. For notational convenience, we let ∇ φ := (✷φ ∨ ✷¬φ) ∇φ := ¬ ∇ φ ∇ (φ, ψ) := ( ∇ φ ∧ ∇ ψ) Definition 15 (Mw and Mf). (i) Mw extends classical propositional logic with (⊤), (⊥), (¬), (∧1), (∨2), (✷1), (✷2), (faith✷), (D), and ✷(φ ∨ ψ) ↔ ∇ (φ, ψ) ∧ (✷φ ∨ ✷ψ)(∨3) ✷¬(φ ∧ ψ) ↔ ∇ (φ, ψ) ∧ (✷¬φ ∨ ✷¬ψ)(∧3) We call Mw− the system Mw without (faith✷). (ii) Mf is formulated in L։ ✷ and extends Mw with: ✷(φ։ ψ) ↔ (✷¬φ ∨ (✷φ ∧ ✷ψ))(։1) ✷(¬(φ։ ψ)) ↔ (✷φ ∧ ✷¬ψ)(։2) We call Mf− the system Mf without (faith✷). MODAL LOGICS OF KF 9 Remark 16. The status of ։ in Mf and variants thereof is peculiar. Internally it characterises the non-material conditional of F3, whereas externally it collapses into →. It can be easily verified in fact that (φ ։ ψ) ↔ (φ → ψ) is derivable in Mf-like theories, whereas from the semantics provided in the next section one can easily see that ✷(φ։ ψ) ↔ ✷(φ → ψ) is not. The next Lemma will play a central role in what follows. It states that the derivability of sequents in the nonclassical logics of truth S✷ introduced in §2 entails the derivability of specific conditionals in the classical systems that we are currently investigating. It is in this sense that the logics S✷ are the internal logic of the modal logics introduced in this section. Lemma 17 (Connecting Lemma). For (S, T ) one of the pairs (K3,Mn), (LP,Mb), (KS3,M) (FDE,BM), (B3,Mw), (F3,Mf): if S✷ ⊢ Γ ⇒ ∆, then T ⊢ ∧ ✷Γ → ∨ ✷∆, where ✷X := {✷φ | φ ∈ X}. Proof. The proof is by induction on the length of the proof in the relevant logics. Crucially, the proof for the pairs (B3✷,M w) and (F3,Mf) rests on the following property: for all φ ∈ L✷(L։✷ ), and S ∈ {M w,Mf}, (1) S ⊢ ∇φ iff there is a p ∈ Prop(φ) such that S ⊢ ∇p. qed. It should be noticed that, since Lemma 17 does not employ faith✷, it can be generalized to the theories without such assumption. In the following section, it will be useful to refer directly to the inner, nonclassical logic of our classical modal logics of truth. The next definition makes this idea precise. Definition 18 (Inner Logic). Given Lemma 17, we set I(S) :=                   FDE✷, if S = BM − KS3✷, if S = M − K3✷, if S = BM − + D LP✷, if S = BM − + Dc B3✷, if S = M w− F3✷, if S = M f− and call I(S) the inner logic S. 3.2. Semantics. In this section we introduce the anticipated novel semantics for the logics described in the previous section. This amounts to considering frames endowed with classical and nonclassical worlds. In particular we are interested in what we call mixed, idiosyncratic frames (cf. Figure 2), that is, frames in which a classical world sees exactly one idiosyncratic non-classical world (in the sense of Definition 9).10 Definition 19 (mixed idiosyncratic frame). Let W,Z be disjoint nonempty sets and R ⊆ W ∪ Z × Z. A mixed idiosyncratic frame satisfies: ∀w ∈ W ∃!v ∈ Z(wRv)(functionality) ∀u, v ∈ Z(uRv ↔ u = v)(idiosyncracy) We call a mixed idiosyncratic frame single-rooted if W is a singleton. 10Notice that, in Figure 2, the subframe ({z}, {〈z, z〉}) is not a mixed idiosyncratic frame in its own right. MODAL LOGICS OF KF 10 w z z wiw0 wj z . . . . . . . . . Figure 2. Example of a mixed idiosyncratic frame Mixed, idiosyncratic frames give rise to suitable models, once they are coupled with suitable valuations: in this context, a valuation takes a classical or a nonclassical world and a propositional atom and returns values in a set X with {0, 1} ⊆ X ⊆ {0, 1, b, n}. Definition 20 (mixed valuations). For (W,Z,R) a mixed, idiosyncratic frame, a valuation takes a member of W ∪ Z and a p ∈ Prop and returns a value in X with {0, 1} ⊆ X ⊆ {0, 1, b, n} and X = {0, 1} for w ∈ W . In particular we call a valuation: • four-valued, if (∀w ∈ W )(∀p ∈ Prop)(Vw(p) ∈ {0, 1}) and (∀z ∈ Z)(∀p ∈ Prop)(Vz(p) ∈ {0, 1, b, n}. • consistent, if (∀w ∈ W )(∀p ∈ Prop)(Vw(p) ∈ {0, 1}) and (∀z ∈ Z)(∀p ∈ Prop)( Vz(p) ∈ {1, 0, n}); • complete, if (∀w ∈ W )(∀p ∈ Prop)(Vw(p) ∈ {0, 1}) and (∀z ∈ Z)(∀p ∈ Prop)( Vz(p) ∈ {1, 0, b}); • symmetric, if (∀w ∈ W )(∀p ∈ Prop)(Vw(p) ∈ {0, 1}) and for all z ∈ Z, either (∀p ∈ Prop)( Vz ∈ {1, 0, b}) or (∀p ∈ Prop)( Vz(p) ∈ {1, 0, n}) but not both. • faithful, if for all w ∈ W , z ∈ Z, and p ∈ Prop, Rwz entails that Vw(p) = Vz(p) if Vz(p) ∈ {0, 1}. Definition 21 (mixed idiosyncratic model). With (W,Z,R) a mixed, idiosyncratic frame and V a mixed valuation, the tuple M := (W,Z,R, V ) is mixed idiosyncratic model. M is consistent (complete, symmetric, faithful), if V is consistent (complete, symmetric, faithful). A model is single-rooted if it is based on a single-rooted frame. The definition of truth in a mixed, idiosyncratic model combines the clauses of classical and nonclassical satisfaction. Definition 22 (truth in mixed idiosyncratic models). Let (W,Z,R) be a mixed idiosyncratic frame and M := (W,Z,R, V ) a mixed idiosyncratic model and s ∈ {fde, k3, b3, f3, lp, ks3}. In defining truth in a model we distinguish between truth in a model at a classical world and at a nonclassical world. (i) Let z ∈ Z. Then |φ|M,zs is defined as in Definition 5. As in Definition 5 we write M, z s φ iff |φ|M,zs ∈ {1, b} MODAL LOGICS OF KF 11 (ii) Let w ∈ W . Then |φ|M,ws is also defined using the clauses of Definition 5 with the exception of φ is ✷ψ for which we have: |φ|M,ws = { 1, if M, z s ψ, for all z with Rwz; 0, otherwise. Again we write M, w s φ iff |φ|M,ws ∈ {1, b}. Notice that in fact a formula φ will not receive a nonclassical truth value at a classical world. Given the nature of mixed models, the definition of consequence now splits into two notions of truth preservation, one at classical worlds, and one at nonclassical worlds. The latter is simply a reformulation in the present context of the notion of consequence from Definition 6. Definition 23 (consequence). With S a class of L✷-models based on a mixed, idiosyncratic frame (W,Z,R), s ∈ {fde, k3, b3, lp, f3, ks3}, Γ ⊆ SentL✷ , and φ ∈ SentL✷ , we let (i) Γ Ss φ :⇔ for all M ∈ S and w ∈ W , if M, w s γ for all γ ∈ Γ, then M, w s φ. (ii) Γ Ss ∆ :⇔ for all M ∈ S and z ∈ Z, if M, z s γ for all γ ∈ Γ, then M, z s δ for all δ ∈ ∆. In the rest of the section we will prove the adequacy of our classical modal logics with respect to the semantics just introduced. As we have already seen with the notion of consequence, the adequacy theorem we are about to prove splits in two: one clause concerns the adequacy of the internal logics with respect to the nonclassical semantics of the previous section; the other clause concerns directly our classical modal logics. However, the proof of the former claim follows directly from Proposition 10. Therefore we will be mainly occupied with establishing the latter clause. The following adequacy result will not only tell us that the non-congruent modal logics we introduced are complete with respect to the classical worlds at mixed idiosyncratic frames but also that the logic that governs the transformations inside the scope of the modal operator in these logics is precisely their inner logic in the sense of Definition 18. Proposition 24 (adequacy). Let F = (W,Z,R) be a mixed, idiosyncratic frame. Then for any Γ,∆ ⊆ L✷ and φ ∈ L✷, the claims (i) Γ Fs φ iff Γ ⊢S φ (ii) Γ Fs ∆ iff I(S) ⊢ Γ ⇒ ∆ hold when F is the class of mixed, idiosyncratic models and (s, S) are (fde,BM−); F is the class of mixed, idiosyncratic, symmetric models and (s, S) are (ks3,M−); F is the class of mixed, idiosyncratic, consistent models and (s, S) are either (k3,BM− + D), (b3,Mw−), or (f3,Mf−); F is the class of mixed, idiosyncratic, complete models and (s, S) is (lp, BM− +Dc) A fundamental ingredient of the proof of the adequacy theorem is the definition of canonical models for our logics. Such models will reflect the mixed nature of our frames: for each logic S, classical worlds will be maximally S-consistent sets of sentences, whereas nonclassical worlds will be I(S)-saturated sets. Definition 25 (canonical model). MODAL LOGICS OF KF 12 (i) The canonical model for S – S ∈ {BM−,M−,BM− + D,BM− + Dc,Mw−,Mf−} – is the structure MS := (WS , ZS, RS , V S), where WS is the set of maximally S-consistent sets of sentences ZS is the set of I(S)-saturated sets, where I(S) is the internal logic of S RS(x, y) :⇔ ( x ∈ WS ∧ {φ ∈ L✷ | ✷φ ∈ x} ⊆ y ) ∨ ( x ∈ ZS ∧ {φ | ✷φ ∈ x} = y = {¬φ | ¬ ⋄ φ ∈ x} ) V S := V I(S) (cf. definition 64) (ii) As before, (WS , ZS , RS) is the canonical frame for S. Lemma 26 (existence). With S ∈ {BM−,M−,BM− + D,BM− + Dc,Mw−,Mf−}, v ∈ WS ∪ ZS, and φ ∈ L✷, if ✷φ /∈ v, then there is a z ∈ ZS such that RSvz and φ /∈ z. Proof. The proof essentially employs the connecting Lemma 17. The case in which v ∈ ZS is immediate. If v ∈ WS , then ¬✷φ ∈ v. Next, we notice that for any Θ ⊆ {ψ ∈ L✷ | ✷ψ ∈ v}, (2) I(S) 0 Θ ⇒ φ. In fact, if Θ ⇒ φ were derivable, by Lemma 17 we would have ∧ ✷Θ → ✷φ, and therefore ✷φ ∈ v, contradicting our assumption. By following the blueprint of Lemmata 62 and 65, we construct an I(S)-saturated set z such that RSvz and φ /∈ z. qed. Lemma 27 (truth). With S ∈ {BM−,M−,BM− + D,BM− + Dc,Mw−,Mf−}, v ∈ WS ∪ ZS, and φ ∈ L✷, φ ∈ v if and only if MS , v s φ Proof. By induction on the positive complexity of φ. In the case in which φ is ¬✷ψ, and v ∈ WS , the right-to-left direction is immediate by the definition of truth and RS . The left-to-right direction follows from Lemma 26. qed. Proof of Proposition 24. The proof of (ii) proceeds essentially unchanged from the previous section, Proposition 8. For (i), the soundness direction is obtained by induction on the length of the proof in S. For the completeness direction, we first notice that, for any relevant S, the claims (i) any S-consistent set of sentences is satisfiable in F (ii) if Γ Fs φ then Γ ⊢S φ. are equivalent. Now let X ⊆ L✷ be S-consistent. It then suffices to find a model M in F and a w ∈ WS such that M, w φ for any φ ∈ X . We can then simply choose MS and any w ∈ W such that X ⊆ w. qed. Proposition 24 yields a completeness result for the classical, non-congruent modal logics that do not assume the faithfulness axiom (faith✷). The axiom states that if a formula φ of L✷ receives a classical truth value at a nonclassical world, then φ will have the same truth value at all classical worlds that see the nonclassical world. This informal claim is made rigorous in Lemma 28 below, which relative to mixed, idiosyncratic frames forces the valuation to be faithful in the sense of Definition 20. Lemma 28 thus allows to transform Proposition 24 into a completeness result for the modal logics that assume (faith✷). The respective modal logics will be complete with respect to the class of faithful models based on mixed, idiosyncratic frames. MODAL LOGICS OF KF 13 Lemma 28 (Faithful Models). Let F be a mixed, idiosyncratic frame, V a mixed valuation on F and M = (F, V ) the resulting model. Then ∀w ∈ W (M, w s ✷φ ∧ ¬✷¬φ → φ) iff V is a faithful valuation. Proof. We leave it to the reader to verify that (faith✷) is true at all classical worlds in faithful models based on mixed, idiosyncratic frames. For the converse direction we assume for reductio that ∀w ∈ W (M, w s ✷φ ∧ ¬✷¬φ → φ) on some nonfaithful model based on a mixed, idiosyncratic frame, that is, for some p ∈ Prop and w ∈ W and z ∈ Z with Rwz: Vz(p) ∈ {0, 1} but Vw(p) 6= Vz(p). There are two cases: • Vz(p) = 1 and Vw(p) = 0. Then M, w s ✷p ∧ ¬✷¬p and M, w 1s p, that is, M, w 1s ✷p ∧ ¬✷¬p → p. Contradiction. • Vz(p) = 0 and Vw(p) = 1. Then M, w s ✷¬p ∧ ¬✷¬¬p and M, w 1s ¬p, that is, M, w 1s ✷¬p ∧ ¬✷¬¬p → ¬p. Contradiction. qed. We can now state the adequacy of the faithful modal logics, which, as we will show in Section , will serve as the modal logics of the Kripke-Feferman truth theories. Corollary 29 (adequacy). Let F = (W,Z,R) be a mixed, idiosyncratic frame. Then for any Γ,∆ ⊆ L✷ and φ ∈ L✷, the claims (i) Γ Fs φ iff Γ ⊢S φ (ii) Γ Fs ∆ iff I(S) ⊢ Γ ⇒ ∆ holds when F is the class of mixed, idiosyncratic, faithful models and (s, S) are (fde, BM); F is the class of mixed, idiosyncratic, symmetric models and (s, S) are (ks3,M); F is the class of mixed, idiosyncratic, consistent models and (s, S) are either (k3,Mn), (b3,Mw), or (f3,Mf); F is the class of mixed, idiosyncratic, complete models and (s, S) is (lp,Mb) When proving the truth-theoretical completeness of these modal logics, it will be useful to restrict our attention to models based on unique root models. Corollary 30 (single-rooted frames). The adequacy results of Propositions 24 and 29 also hold for models based on single-rooted, mixed, idiosyncratic frames. Proof. The generated subframe of the canonical frame is a single-rooted mixed, idiosyncratic frame. qed. 4. Kripke-Feferman truth In this section we introduce the relevant truth-theoretic background. We start with the basics of fixed-point semantics, then we introduce the base theory for our axiomatic systems of truth, and finally we define the collections of axioms of these systems. 4.1. Peano arithmetic. We start with an arithmetical language LN that includes the standard signature {0, S,+,×}, and extend it with a unary truth predicate T. We assume a canonical, monotone Gödel numbering for LT. For e an LTexpression, we write #e for its Gödel code and peq for the LN-term representing MODAL LOGICS OF KF 14 peq. LT features finitely many function symbols for suitable primitive recursive functions for syntactic operations on (codes of) expressions, such as:11 Operation Function symbol #e1,#e2 7→ #(e1 = e2) eq #t 7→ the value of the closed term #t val #e 7→ #¬e ng #e1,#e2 7→ #(e1 ∧ e2) and #e1,#e2 7→ #(e1 ։ e2) fc #v,#φ 7→ #(∀vφ) all #t,#φ(v) 7→ #φ(t/v) sub n 7→ #n num We assume in particular that the evaluation function is defined for the finitely many primitive recursive functions other than itself – in particular, its defining equations are part of our base theory. In this way it remains primitive recursive. Definition 31 (Peano Arithmetic). Peano arithmetic is the first-order system in LN whose axioms are: - ∀x(0 6= S(x)) - ∀x∀y(S(x) = S(y) → x = y) the recursive equations for +,× and the finitely many additional primitive recursive function symbols; the axiom schema of induction: (IND(LN)) φ(0) ∧ ∀x(φ(x) → φ(Sx)) → ∀xφ(x) for all formulas φ(v) of LN. Definition 32. The system PAT in LN ∪ {T} extends the basic axioms of PA with all instances of induction in LT. 4.2. Fixed Point Semantics. Kripke-Feferman truth can be seen as axiomatizing a collection of inductive constructions of the sets of S-true sentences of LT, where S is one of the subclassical logics considered above [Mar84, Kri75, Vis89, Fef08]. Let M  PA and LMT be LT expanded with constants a, b, c, . . . for all elements of its domain M .12 Let True0 the PA-definable set of true LN-equations, and False0 the PA-definable set of false LN-equations. We define operators on sets S ⊆ M satisfying (M, S)  SentLT(all(v, a)) ∧ CtermN(b) ∧ CtermN(c) ∧ val(b) = val(c)(reg) → ( Tsub(a, v, b) ↔ Tsub(a, v, c) ) Sets satisfying reg are called regular [Can89]: they simply state that the truth predicate is sensitive to substitutions of identicals. Definition 33 (Kripke jumps). Given some M  PA, we are interested in two main operators on regular S ⊆ M . 11In addition, we assume a function symbols for the proper subtraction function to avoid certain unintended propoerties of the Weak Kleene schema defined below [Spe17, CD91]. 12The language expansion is not needed in the case of the standard model N, which contains names for all natural numbers. MODAL LOGICS OF KF 15 (i) The Strong-Kleene jump Φ: P(M) → P(M) is such that a ∈ Φ(S) if and only if M  SentLT(a), and ( M  True0(a), or M  a = ng(b) ∧ False0(b), ; or (M, S)  Cterm(b) ∧ a = sub(pTvq, pvq, b) ∧ Tval(b), or (M, S)  Cterm(b) ∧ a = sub(p¬Tvq, pvq, b) ∧ Tng(val(b)), or (M, S)  a = ng(ng(b)) ∧ T(b), or (M, S)  a = and(b, c) ∧ Tb ∧ Tc, or (M, S)  a = ng(and(b, c)) ∧ Tng(b) ∨ Tng(c), or (M, S)  a = all(u, b) ∧ ∀x(Cterm(x) → Tsub(b, u, x)), or (M, S)  a = ng(all(u, b)) ∧ ∃x(Cterm(x) ∧ Tsub(ng(b), u, x)) ) . (ii) Let D(x) :↔ (Tx ∨ Tng(x)). The Weak-Kleene jump Ψ: P(M) → P(M) replaces in the definition of Φ the clauses for the negated conjunction and quantifiers with: M SentLT(a), and (M, S)  a = ng(and(b, c)) ∧ (Tng(b) ∨ Tng(c)) ∧ D(a) ∧ D(b) or (M, S)  a = all(u, b) ∧ ∀xD(sub(b, u, x))∧ ∀x(Cterm(x) → Tsub(b, u, x)), or (M, S)  a = ng(all(u, b)) ∧ ∀xD(sub(b, u, x))∧ ∃x(Cterm(x) ∧ Tsub(ng(b), u, x)) ) . (iii) The Aczel-Feferman jump Ξ: P(M) → P(M) is then defined for formulas of L։T and follows the blueprint of the definition of Ψ modulo replacing SentLT with SentL։ T and the addition, to the second main conjunct, of the disjuncts: (M, S)  a = fc(b, c) ∧ Tng(b) ∨ (Tb ∧ Tc) (M, S)  a = ng(fc(b, c)) ∧ Tb ∧ Tng(c) The subsets of SentMLT := {a ∈ M | M  SentLT(a)} and Sent M L։ T := {a ∈ M | M  SentL։ T (a)} satisfying reg form a complete lattice and the operator Φ(*) is monotone, therefore, by the Tarski-Knaster theorem [Tar55]: Lemma 34. The operators Φ,Ψ,Ξ give rise to complete lattice of fixed points with minimal and maximal elements the sets obtained by iterating the operators on ∅ and on the set of sentences of the relevant language respectively. In what follows, when referring to a fixed point, we will always refer to a fixed point of Φ,Ψ,Ξ. Any fixed point X will have the property that: φ ∈ X iff Tpφq ∈ X for any sentence φ of LT, where the bi-conditional is necessarily metatheoretic. This property approximates the naïve truth schema and, since (Tpφq ↔ φ) ∈ X for φ ∈ LN, it improves on the standard Tarskian solutions [Tar56] in a substantial way. This partially explains why this semantic construction is the basis of several contemporary approaches to the Liar paradox. Different such approaches often diverge on which class of fixed points they accept. Definition 35. A fixed point X on M is called: MODAL LOGICS OF KF 16 • consistent, if there is no a ∈ SentMLT (Sent M L։ T ) such that a ∈ X and ngM(a) ∈ X; • complete, if for all a ∈ SentMLT (Sent M L։ T ), either a ∈ X or ngM(a) ∈ X. It can be easily verified that the least fixed points are consistent, and the greatest ones are complete. It follows from the definitions that there will also be fixed points X ∈ L that are neither consistent nor complete. By the diagonal Lemma, for any M  PA, fixed point X ⊆ M , and any i ∈ ω, we can find sentences τ(i, ~x) (truth-teller sentences) such that (3) M  Tpτ(i, x1, . . . , xn)q ↔ τ(i, x1, . . . , xn). In the following we use τi(~x) as short for τ(i, x1, . . . , xn). The analysis of paradoxicality in [Kri75] revealed that truth-teller sentences are free to assume different truth-values in different fixed-points of Φ,Ψ,Ξ.13 Lemma 36. Let M  PA. For i ∈ ω and any Y, Z ⊆ ω there is a fixed point X ⊆ SentMLT (Sent M L։ T ) such that #τi ∈ X iff i ∈ Y ;(i) #¬τi ∈ X iff i ∈ Z.(ii) If Y ∩ Z = ∅ [Y ∪ Z = ω] then X can be chosen to be consistent [complete]. 4.3. Axioms and Rules. In this paper Kripke-Feferman theories of truth are extensions of PA by a finite collection of axioms for the truth predicate, and, possibly, additional instances of induction in LT. The truth axioms are required to be sound with respect to the fixed-point semantics just introduced and to have the additional feature that, for M  PA and A a the conjunction of such axioms: X ⊆ M is a fixed point iff (M, X)  PA +A.14 We call KF the most basic system, whose truth predicate does not rule out interpretation in which the truth predicate is both partial and inconsistent. Definition 37 (KF). KF extends PAT with the axioms: ∀x, y ( Cterm(x) ∧ Cterm(y) → (Teq(x, y) ↔ val(x) = val(y)) ) (KF1) ∀x, y ( Cterm(x) ∧ Cterm(y) → (Tng(eq(x, y)) ↔ val(x) 6= val(y)) ) (KF2) ∀x(SentLT(x) → (Tng(ng(x)) ↔ Tx))(KF3) ∀x∀y(SentLT(and(x, y)) → (Tand(x, y) ↔ Tx ∧ Ty))(KF4) ∀x∀y(SentLT(and(x, y)) → (Tng(and(x, y)) ↔ Tng(x) ∨ Tng(y)))(KF5) ∀u∀x ( SentLT (all(u, x)) → (Tall(u, x) ↔ ∀yTsub(x, u, num(y))) ) (KF6) ∀u∀x ( SentLT (all(u, x)) → (Tng(all(u, x)) ↔ ∃yTsub(ng(x), u, num(y))) ) (KF7) 13The case of Weak Kleene is as usual a bit more complex: if one lacks the means for direct self-reference such as a primitive substitution function, one may not be free in assigning arbitrary truth values to truth-tellers. If for instance truth-tellers are obtained by means of existential quantification, the existence of a non determined instance would render the quantification non determined [CD91]. In our case we assume the means for direct self-reference and sidestep these subtle issues. 14We notice that this criterion for defining Kripke-Feferman systems of truth is more selective than the N-categoricity criterion from [FHKS15]. In fact, N-categoricity criterion would consider a schematic version of the theories considered below – i.e. where the compositional clauses are given in schematic and not universally quantified form –, or disquotational theories in the style of PUTB (see [Hal14]) in a negation-free language, as axiomatization of suitable Kripkean fixed points. Our truth-theoretic completeness results clearly extends to schematic, compositional versions of Kripke-Feferman systems, and it should easily extend to suitable disquotational systems. MODAL LOGICS OF KF 17 ∀x(Cterm(x) → (Tsub(pTvq, pvq, x) ↔ Tval(x)))(KF8) ∀x ( Cterm(x) →(KF9) (Tsub(p¬Tvq, pvq, x) ↔ (Tng(val(x)) ∨ ¬SentLT (val(x)))) ) Stronger systems are obtained by adding to KF some axioms forcing a consistent or a complete truth predicate: ∀x ( SentLT (x) → (Tng(x) → ¬Tx) ) (CN) ∀x ( SentLT (x) → (¬Tx → Tng(x)) ) (CM) The next system, called WKFC from 'Weak-Kleene Kripke Feferman with consistency', is based on a modfication of the truth clauses for connectives and quantifiers inspired to B3. We abbreviate D(x) :↔ Tx ∨ Tng(x) D(x, y) :↔ D(x) ∧ D(y) D1(x) :↔ ∀y(Tsub(x, num(y)) ∨ Tsub(ng(x), u, num(y))) Definition 38 (WKFC). The system WKFC extends PAT with KF1-4,KF6,KF8-9, CN, and ∀x∀y(SentLT(and(x, y)) → (Tng(and(x, y))) ↔ D(x, y) ∧ (Tng(x) ∨ Tng(y)))) (WKFC∧) ∀u∀x ( SentLT (all(u, x)) → (Tng(all(u, x)) ↔ D 1(x) ∧ ∃yTsub(ng(x), u, num(y))) ) (WKFC∀) The last collection of axioms results in the system DT introduced by Feferman in [Fef08] – and also Feferman's preferred system of truth – whose truth predicate is based in the logic F3.15 Definition 39 (DT). The system DT extends WKFC with ∀x∀y(SentL։ T (fc(x, y)) → (Tfc(x, y) ↔ (Tng(x) ∨ (Tx ∧ Ty))))(DT։) ∀x∀y(SentL։ T (fc(x, y)) → (Tng(fc(x, y)) ↔ (Tx ∧ Tng(y))))(DT¬։) Remark 40. Similarly to what is described by Remark 16, → and ։ are externally equivalent in DT and variants thereof, whereas by the properties of the fixed-point semantics introduced Tpφ → ψq ↔ Tpφ։ ψq is not provable in such theories. The next lemma collects some simple facts concerning the provability and unprovability of Liar sentences in the Kripke-Feferman systems just introduced. Lemma 41. Let l be a LN term such that PAT  l = p¬Tlq, and let λ :↔ ¬Tl. We have: (i) Σ 0 λ, Σ 0 ¬λ for Σ ∈ {KF}; (ii) Σ ⊢ λ ∧ ¬Tp¬λq ∧ ¬Tpλq, for Σ ∈ {KF + CN,WKFC,DT}; (iii) KF + CM ⊢ Tpλq ∧ ¬λ. 5. Truth-theoretic Completeness In this section we establish the fundamental link between the classical modal logics introduced in §3.1 and the Kripke-Feferman theories of truth in the form of Solovay-style completeness results. 15To be precise, we are not presenting here the original axiomatization by Feferman, but a variant of it considered in [Fuj10]. MODAL LOGICS OF KF 18 5.1. The modal logic of KF. We start by establishing the Solovay-completeness of the basic Kripke-Feferman system KF. Definition 42 (Truth-realization, Truth-interpretation). A truth-realization is a function ⋆ : Prop → SentLT . Each realization induces a truth-interpretation, i.e. a function I⋆ : SentL✷ → SentLT such that: I⋆(φ) =                  p⋆i , if φ := pi 0 = 0, if φ = ⊤ 0 = 1, if φ = ⊥ ¬I⋆(ψ), if φ = ¬ψ I⋆(ψ) ∧ I⋆(χ), if φ = ψ ∧ χ T (pI⋆(ψ)q), if φ = ✷ψ. The definition can easily be extended to the case of L։ ✷ and L։T by adding an extra clause for the truth interpretation: I⋆(ψ) ։ I⋆(χ), if φ = ψ ։ χ Since this will be clear from the context, we will use the term truth-interpretation for both translations. The following is the main result of the present subsection, and establishes that BM is the modal logic of the basic system KF. Theorem 43. For all φ ∈ SentL✷ , BM ⊢ φ if and only if for all realizations ⋆, KF ⊢ I⋆(φ). The proof of theorem 43 consists of two parts: the soundness and the completeness of BM. The soundness direction is established via a straightforward induction on the length of the proof in BM. Lemma 44 (Soundness of BM). For all φ ∈ SentL✷ , if BM ⊢ φ, then for all realizations ⋆, KF ⊢ I⋆(φ). The converse direction will be proven in its contrapositive form. Lemma 45 (Truth-Completeness). For all φ ∈ SentL✷ , if BM 6⊢ φ, then there exists a realization ⋆, KF 6⊢ I⋆(φ). Before proving Lemma 45, we briefly sketch the general proof strategy. We start the proof by assuming BM 6⊢ φ. By the modal completeness theorem and, in particular, Corollaries 29 and 30, we know that there is a mixed, faithful single rooted idiosyncratic FDE-model M such that at its classical root w, M, w fde ¬φ. We then choose a particular truth-realization, which allows us to "code up" or "mimic" the valuation of the FDE-model by choosing an appropriate KF-model. In the KF-model the truth interpretation of ¬φ will be true. Hence the truthinterpretation of φ under the chosen truth-realization, which we label the Witness Realization, is not a theorem of KF. The idea behind the Witness Realization is to interpret propositional variables as a conjunction of truth tellers. As we have seen in Lemma 36 for any collection of truth tellers we can find KF-models that declare truth tellers of the particular collection true (false). The following Lemma, which follows easily from Lemma 36, can be seen as a particular application of this fact. Lemma 46. Let Φ be as above. Then for any mixed, faithful, single-rooted idiosyncratic model M based on an evaluation scheme e ∈ {fde, ks3, k3, lp} and any N  PA we can find a fixed point S ⊂ N (N being the domain of N ) of Φ such that for all pj ∈ Prop with j ∈ ω: MODAL LOGICS OF KF 19 (i) #τ2j ∈ S iff M, w e pj or M, z e pj; (ii) #¬τ2j ∈ S iff M, z e ¬pj; (iii) #τ2j+1 ∈ S iff M, w e ¬pj and M, z e pj; (iv) #¬τ2j+1 ∈ S iff M, z e pj. Moreover, for k3 ( lp) we can find a consistent (complete) fixed point S; for ks3 depending on the model we can choose either a consistent or a complete fixed point S. Definition 47 (Witness Realization). Let • : Prop → SentLT be a truth realization such that for all j ∈ ω p•j = τ2j ∧ ¬τ2j+1. • is called the Witness Realization. Our next claim is key to main lemma of this section, and describes the behaviour of the witness realization at the nonclassical world. Lemma 48. M, S and e be as in Lemma 46 and • be the Witness Realization. Then for all φ ∈ L✷: (i) if M, z e φ, then I•(φ) ∈ S; (ii) if M, z 6 e φ, then I•(φ) /∈ S. Proof. Both cases are proved by an induction on the positive complexity of φ. We discuss the base cases, the remaining cases are easily obtained by induction hypothesis. We start with item (i). Suppose φ = pj for some j ∈ ω and M, z e φ. Then, by Lemma 46(i), #τ2j ∈ S and, by (iv), #¬τ2j+1 ∈ S. Since S is a fixed point of Φ, this implies that #(τ2j ∧ ¬τ2j+1) ∈ S, i.e., #p•j ∈ S. Similarly, if φ = ¬pj for some j ∈ ω and M, z e φ, then by Lemma 46(ii) #¬τ2j ∈ S. From this we may conclude that #(¬τ2j ∨ ¬¬τ2j+1) ∈ S, that is #¬(τ2j ∧ ¬τ2j+1) ∈ S. The latter is just #I•(¬pj) ∈ S. For item (ii) we assume φ = pj for some j ∈ ω and M, z 6 e φ. Then by Lemma 46 (iv) #¬τ2j+1 6∈ S which implies that #(τ2j ∧¬τ2j+1) 6∈ S, i.e., #I•(pj) 6∈ S. We now assume φ = ¬pj for some j ∈ ω and M, z 6 e φ. By Lemma 46 (ii) we infer #¬τ2j 6∈ S. Now, we distinguish between two cases: in the first case M, z 6 e pj and by (iii) #τ2j+1 6∈ S. Alternatively, M, z e pj but then, since we are working in a faithful model M, w e pj , i.e., M, w 6 e ¬pj and again by (iii) it follows that #τ2j+1 6∈ S. We can conclude that #¬(τ2j ∧ ¬τ2j+1) 6∈ S. But the latter is just #I•(¬pj) 6∈ S. qed. We can then establish the main lemma to the truth-completeness of KF. Lemma 49 (Main Lemma). Let M, S and e be as in Lemma 46 and • be the Witness Realization. Then, for all φ ∈ L✷, M, w e φ only if (N , S)  I•(φ). Proof. The proof is again by induction on the positive complexity of φ. We cover the base cases and the cases of the modal operator. The induction step for the remaining operators and quantifiers is immediate by the properties of KF-models. We assume φ = pj and M, w e φ. We know by Lemma 46 (i) that #τ2j ∈ S and by (iii) that #τ2j+1 6∈ S. By the properties of truth tellers this implies (N , S)  I•(pj). We now assume φ = ¬pj and M, w e φ. We distinguish between case (a) M, z 6 e pj and case (b) where M, z e pj . In case (a) we can infer Lemma 46 (i) that #τ2j 6∈ S, which suffices to show that (N , S)  I•(¬pj). In case (b) we infer by 46 (iii) that #τ2j+1 ∈ S, which again suffices to show that (N , S)  I•(¬pj). MODAL LOGICS OF KF 20 Let φ = ✷ψ and M, w e φ. We know that M, w e φ if and only if M, z e ψ. But from M, z e ψ we infer by Lemma 48 that #I•(ψ) ∈ S. Thus (N , S)  TpI•(ψ)q, which by Definition 42 is just (N , S)  I•(φ). Finally, let φ = ¬✷ψ and M, w e φ. We know that M, w e φ if and only if M, z 6 e ψ. But from M, z 6 e ψ we infer by Lemma 48 that #I•(ψ) 6∈ S. Thus (N , S)  ¬TpI•(ψ)q, which by Definition 42 is just (N , S)  I•(φ). qed. We can now prove the truth-completeness of BM. Proof of Lemma 45. Assume BM 6⊢ φ. Then by Corollaries 29 and 30 we know that there is a mixed, faithful single-rooted idiosyncratic model M such that at its root w, M, w fde ¬φ. By Lemma 46 we then choose an appropriate fixed-point model (N , S) of KF – e.g. a fixed-point model based on N – such that by the Main Lemma, i.e. Lemma 49, (N , S) |= I•(¬φ), where • is the Witness Realization. The latter implies KF 0 I•(φ) and hence that there is truth-realization ⋆ such that KF 0 I⋆(φ). qed. Proof of theorem 43. By Lemma 44 and Lemma 45. qed. Corollary 50. For all φ ∈ SentL✷ M ⊢ φ if and only if for all realizations ⋆ (KF + CM ∨ CN ⊢ I⋆(φ));(ks3) Mn ⊢ φ if and only if for all realizations ⋆ (KF + CN ⊢ I⋆(φ));(k3) Mb ⊢ φ if and only if for all realizations ⋆ (KF + CM ⊢ I⋆(φ)).(lp) Proof. The soundness of M (Mn, Mb) with respect to KF + CM ∨ CN (KF + CN, KF + CM) follows again by an induction on the length of a proof in M (Mn,Mb). For the converse direction, i.e. the truth-completeness, we adopt the strategy employed in proving Lemma 45: we assume that some formula φ is not provable in the modal logic at stake. We then apply the modal completeness theorem to find a suitable faithful, mixed single rooted idiosyncratic model that falsifies φ. Then using Lemma 46 we can find suitable fixed-point models of KF + CM ∨ CN (KF + CN, KF + CM) in which the truth-interpretation based on our Witness Realization of φ is false. qed. Before moving to the next section, in which we consider strengthenings of some of the claims just obtained, we notice that Lemma 48 provides also a direct proof of the truth-theoretic completeness of the modal logics S✷ with the respects to the corresponding – in the sense of the underlying non-classical logics – non-classical axiomatizations of Kripkean truth in the style of PKF from [HH06]. 5.2. The modal logics of WKFC, DT, and of Kripke's fixed points. In this section we determine the modal logics of the truth-theories based on Weak Kleene. In doing so, we will employ an alternative arguments to the one employed in the previous subsection, that will also deliver alternative truth-completeness proofs for KF+CN and KF+CM. However, such alternative strategies are not different proofs of the same results, but in fact yield much stronger claims, namely they determine the modal logic of all consistent extensions (not necessarily recursively enumerable) of the truth systems considered. For example, [Bur14] proposed an extension of KF in LT with a minimality schema – called KFμ –, that was intended to axiomatize Kripke's minimal fixed point model. Our result will show that Mn is the modal logic not only of KFμ, but also to stronger extensions of KF such as the set of sentences of LT satisfied in the model (N, IΦ), where IΦ is the minimal fixed point of Φ. Theorem 43 establishes that BM is the modal logic of KF. However, KF is not the only first-order theory whose modal logic is BM. MODAL LOGICS OF KF 21 Observation 1. There are 2א0 recursive, consistent, and mutually inconsistent extensions of KF in LT whose modal logic is BM. Proof. By Gödel's incompleteness theorem, one constructs copy of the full binary tree starting with KF: each node, say labelled with T , has two children labelled with T + γT and T + ¬γT , where γT is an arithmetical sentence undecidable in T . Each node will then be consistent (assuming KF is), and inconsistent with the other non-root nodes. The strategy leading to theorem 43 can then be employed to show that BM is the modal logic of the resulting theories. It is worth noting that for arithmetically unsound theories the required countermodel in the proof of theorem 43 can only be nonstandard. qed. At the same time, Corollary 50 shows the existence of consistent, recursively enumerable extensions of KF in LT whose modal logic is not BM. A natural question is then whether there are extensions of KF in LT whose modal logic is stable under further extensions. As anticipated, the answer turns out to be positive: to establish this, we present an alternative proof of Corollary 50. As a consequence, we will obtain a truth completeness proof for WKFC and DT which will also be stable under consistent extensions, including the LT-sentences true in fixed-point models. Lemma 51. (i) Let M = ({w}, {z}, R, V ) be a mixed, idiosyncratic, faithful, single-rooted, consistent model of L✷. Then: a. For all φ ∈ L✷, if M, w k3 φ, then there is some truth-realization ⋆ such that KF + CN ⊢ I⋆(φ). b. For all φ ∈ L✷, if M, w b3 φ, then there is some truth-realization ⋆ such that WKFC ⊢ I⋆(φ). c. For all φ ∈ L։ ✷ , if M, w f3 φ, then there is some truth-realization ⋆ such that DT ⊢ I⋆(φ). (ii) Let M = ({w}, {z}, R, V ) be a mixed, idiosyncratic, faithful, single-rooted, complete model of L✷. Then for all φ ∈ L✷, if Mb, w lp φ, then there is some truth-realization ⋆ such that KF + CM ⊢ I⋆(φ). Lemma 51 relies in turn on the following, crucial Lemmata, which are stronger versions of Lemma 48. Lemma 52. Let M = ({w}, {z}, R, V ) be as in Lemma 51(i). Then there is a realization ◦ such that: (i) for all φ ∈ L✷, a. if M, z k3 φ, then KF + CN ⊢ TpI◦(φ)q; b. if M, z 6 k3 φ, then KF + CN ⊢ ¬TpI◦(φ)q. (ii) for all φ ∈ L✷, a. if M, z b3 φ, then WKFC ⊢ TpI◦(φ)q; b. if M, z 6 b3 φ, then WKFC ⊢ ¬TpI◦(φ)q. (iii) for all for all φ ∈ L։ ✷ a. if M, z f3 φ, then DT ⊢ TpI◦(φ)q; b. if M, z 6 f3 φ, then DT ⊢ ¬TpI◦(φ)q. Proof. Let ◦ be the realization p◦ =         0 = 0, if Vz(p) = 1, λ, if Vw(p) = 1 and Vz(p) = n, ¬λ, if Vw(p) = 0 and Vz(p) = n, 0 = 1, otherwise. MODAL LOGICS OF KF 22 We verify in some detail (ii) and (iii), because (i) easily follows from the axioms of KF + CN and Lemma 41. Both cases are inductions on the positive complexity of φ. (ii) The noteworthy cases for a. are the cases in which φ is p or ¬p, in which one employs the properties of liar sentences in WKFC, and the one in which φ := ¬(ψ ∧ χ). Since M, z b3 φ, there are three cases: M, z b3 ¬ψ and M, z b3 ¬χ M, z b3 ψ and M, z b3 ¬χ M, z b3 ¬ψ and M, z b3 χ By induction hypothesis, in all cases on obtains in WKFC that D(pI◦(ψ)q, pI◦(χ)q) and TpI◦(¬ψ)q ∨ TpI◦(¬χ)q. Therefore, WKFC ⊢ TpI◦(¬(ψ ∧ χ))q. Symmetrically, b.'s atomic cases follow from the properties of the liar sentence in WKFC. For the crucial case of φ := ¬(ψ ∧ χ), the assumption yields two main cases: |φ|M,zb3 = n or |φ| M,z b3 = 0. The latter case is readily obtained: by induction hypothesis, WKFC proves TpI◦(ψ ∧ χ)q), and so ¬TpI◦(¬(ψ ∧ χ))q. For the first case, |ψ|M,zb3 = |¬ψ| M,z b3 = n or |χ| M,z b3 = |¬χ| M,z b3 = n. If |ψ|M,zb3 = n, the induction hypothesis entails that WKFC ⊢ ¬TpI ◦(ψ)q and WKFC ⊢ ¬TpI◦(¬ψ)q. Therefore, WKFC ⊢ ¬D(pI◦(ψ)q, pI◦(χ)q) and WKFC ⊢ ¬TpI◦(¬(ψ ∧ χ))q. (iii) All cases are analogous to the proof of (ii) except of course the cases in which φ = ψ ։ ψ or φ = ¬(ψ ։ χ), which we now consider. a: if |ψ ։ χ|M,zf3 = 1, then either |ψ| M,z f3 = 0 or |ψ| M,z f3 = |χ| M,z f3 = 1. In either case, by induction hypothesis and (DT ։), we have DT ⊢ TpI◦(ψ) ։ I◦(χ)q, and therefore DT ⊢ TpI◦(ψ ։ χ)q. If |ψ ։ χ|M,zf3 = 0, then |ψ| M,z f3 = 1 and |χ|M,zf3 = 0. The induction hypothesis yields that DT ⊢ TpI ◦(ψ)q ∧ Tp¬I◦(χ)q, so the claim follows by (DT¬։). b: if M, z 6 f3 ψ ։ χ, then M, z 6 f3 ¬ψ and M, z 6 f3 ψ ∧ χ. Therefore, either M, z 6 f3 ¬ψ and M, z 6 f3 ψ, or M, z 6 f3 ¬ψ and M, z 6 f3 χ. Thus, by induction hypothesis, DT ⊢ ¬Tp¬I◦(ψ)q ∧ ¬TpI◦(ψ)q or DT ⊢ ¬Tp¬I◦(ψ)q ∧ ¬TpI◦(χ)q By (DT։), we can in either case conclude that DT ⊢ ¬TpI◦(ψ) ։ I◦(χ)q, that is DT ⊢ ¬TpI◦(ψ ։ χ)q. Finally, if M, z 6 f3 ¬(ψ ։ χ), then either M, z 6 f3 ψ or M, z 6 f3 ¬χ. Therefore, by induction hypothesis, DT ⊢ ¬TpI◦(ψ)q or DT ⊢ ¬TpI◦(χ)q In either case, by (DT¬։) and the definition of I◦, one obtains that ¬TpI◦(ψ ։ χ)q is provable in DT. qed. Lemma 53. Let M = ({w}, {z}, R, V ) be as in Lemma 51(ii). Then there is a realization † such that: (i) if M, z lp φ, then KF + CM ⊢ TpI†(φ)q; (ii) if M, z 6 lp φ, then KF + CM ⊢ ¬TpI†(φ)q. Proof. Let † be p† =         0 = 1, if Vz(p) = 0, ¬λ, if Vw(p) = 1 and Vz(p) = b, λ, if Vw(p) = 0 and Vz(p) = b, 0 = 1, otherwise. The proof is again by induction on φ both in (i) and (ii). qed. MODAL LOGICS OF KF 23 We can now prove Lemma 51. Proof of Lemma 51. The proofs are again by induction on the positive build up of φ. In cases (i)a, (i)b and (i)c one employs ◦. We only provide some details for (i)c mainly because of the peculiar nature of the interaction between → and ։. The other claims are easier. If φ is p, then by assumption either p◦ = (0 = 0) or p◦ = λ, so DT ⊢ p◦ by the DT axioms and Lemma 41. If φ is ¬p, then p◦ = (0 = 1) or p◦ = λ, and in either case DT ⊢ ¬p◦. If φ is ✷ψ or ¬✷ψ, the claims follow from Lemma 52. If φ is ψ ։ χ, by Remark 40 it is sufficient to show that DT ⊢ I◦(pψq) → I◦(pχq). If M, w f3 ψ ։ χ, then either M, w f3 ¬ψ or M, w f3 ¬ψ∧χ. By induction hypothesis, in either case we obtain the desired claim. Similarly, if M, z f3 ¬(ψ ։ χ), then M, w f3 ψ and M, w f3 ¬χ, therefore by induction hypothesis DT ⊢ I ◦(ψ) ∧ I◦(¬χ). The claim is then obtained by Remark 40 and by definition of I◦. For (ii) one employs †. The rest is analogous. qed. We finally establish the main result of this section. Proposition 54. (i) Let S be a consistent, first-order extension of KF + CN in LT, then Mn ⊢ φ if and only if for all realizations ⋆, S ⊢ I⋆(φ). (ii) Let S be a consistent, first-order extension of WKFC in LT, then Mw ⊢ φ if and only if for all realizations ⋆, S ⊢ I⋆(φ). (iii) Let S be a consistent, first-order extension of DT in LT, then Mf ⊢ φ if and only if for all realizations ⋆, S ⊢ I⋆(φ). (iv) Let S be a consistent, first-order extension of KF + CM in LT, then Mb ⊢ φ if and only if for all realizations ⋆, S ⊢ I⋆(φ). Proof. We consider the case for KF + CN, the other are analogous. Corollary 50 already gives us the soundness direction. For the completeness direction, if Mn 0 φ, then by Corollaries 29 and 30 there is a single-rooted mixed, idiosyncratic, consistent faithful model M such that M, w k3 ¬φ for w ∈ W . By Lemma 51 (i)a, KF + CN ⊢ I◦(¬φ). Therefore, there is a realization ◦ such that S 0 I◦(φ) for all S ⊇ KF + CN. qed. 6. Conclusion We have determined the (propositional) modal logic of Feferman's axiomatizations of Kripke's theory of truth. In the case of systems whose truth-predicates behave according to a paracomplete or paraconsistent three-valued logics, such modal logics amount to the modal logics of all of their consistent extensions, including the sentences satisfied by consistent and complete fixed point models. In the present paper we did not consider paraconsistent (threeor four-valued) theories of truth based on Weak Kleene or Feferman-Aczel logic. We expect that our results extend to such cases with little modification, but this would need to be verified in detail. 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Appendix A The basic nonclassical system we are interested in is the four-valued sub-classical logic known as first-degree entailment [AB75]. Definition 55 (FDE). (ref) Γ, φ ⇒ φ,∆ for φ a literal Γ ⇒ ∆, φ φ,Γ ⇒ ∆ (cut) Γ ⇒ ∆ (⊥) Γ,⊥ ⇒ ∆ (⊤) Γ ⇒ ⊤,∆ Γ, φ ⇒ ∆ (dn-l) Γ,¬¬φ ⇒ ∆ Γ ⇒ φ,∆ (dn-r) Γ ⇒ ¬¬φ,∆ Γ,¬φ ⇒ ∆ Γ ⇒ ¬ψ,∆ (¬∧l) Γ,¬(φ ∧ ψ) ⇒ ∆ Γ ⇒ ¬φi,∆ (¬∧r) i = 0, 1 Γ ⇒ ∆,¬(φ0 ∧ φ1) Γ, φi ⇒ ∆ (∧l) Γ, φ0 ∧ φ1 ⇒ ∆ Γ ⇒ φ,∆ Γ ⇒ ψ,∆ (∧r) Γ ⇒ ∆, φ ∧ ψ Γ,¬φi ⇒ ∆ (¬∨l) Γ,¬(φ0 ∨ φ1) ⇒ ∆ Γ ⇒ ¬φ,∆ Γ ⇒ ¬ψ,∆ (¬∨r) Γ ⇒ ¬(φ ∨ ψ),∆ MODAL LOGICS OF KF 26 Γ, φ ⇒ ∆ Γ, ψ ⇒ ∆ (∨l) Γ, φ ∨ ψ ⇒ ∆ Γ ⇒ φi,∆ (∨r) Γ ⇒ φ0 ∨ φ1,∆ FDE can be considered to be the basis of well-known three-valued paracomplete or paraconsistent systems [Kle52, Cos74]. Definition 56. (i) KS3 is obtained by adding to FDE the sequent: (sym) Γ, φ,¬φ ⇒ ψ,¬ψ,∆ (ii) K3 is obtained by adding to FDE the rule: Γ ⇒ ∆, φ (¬l) ¬φ,Γ ⇒ ∆ (iii) LP is obtained by adding to FDE the rule: φ,Γ ⇒ ∆ (¬r) Γ ⇒ ∆,¬φ The next logic we consider Weak Kleene logic. Our axiomatization is a variant of the one that can be found in [CC13]. Definition 57 (Weak Kleene, B3). (ref) Γ, p ⇒ p,∆ Γ ⇒ ∆, φ φ,Γ ⇒ ∆ (cut) Γ ⇒ ∆ (⊥) Γ,⊥ ⇒ ∆ (⊤) Γ ⇒ ⊤,∆ Γ ⇒ ∆, φ (¬l) ¬φ,Γ ⇒ ∆ Γ, φ ⇒ ∆ (¬r) Γ ⇒ ∆,¬φ with Prop(φ) ⊆ Prop(Γ) Γ, φi ⇒ ∆ (∧l) Γ, φ0 ∧ φ1 ⇒ ∆ Γ ⇒ φ,∆ Γ ⇒ ψ,∆ (∧r) Γ ⇒ ∆, φ ∧ ψ Γ, φ ⇒ ∆ Γ, ψ ⇒ ∆ (∨l) Γ, φ ∨ ψ ⇒ ∆ Γ ⇒ φi,∆ (∨r) Γ ⇒ φ0 ∨ φ1,∆ with Prop(φ0, φ1) ⊆ Prop(Γ) Remark 58. The restriction on the rules (∧r) and (¬r) can be seen as 'forcing' a determinate truth value on the principal formulas. The last logic we consider is the extension of Weak Kleene considered (semantically) by Peter Aczel for his Frege Structures [Acz80], and Feferman in [Fef08]. Definition 59 (Feferman Logic, F3). The language L։0 of F3, besides the connectives of B3, features a special conditional ։. The rules of F3 are the the rules of B3 plus: Γ ⇒ ∆, φ ψ,Γ ⇒ ∆ (։l) Γ, φ։ ψ ⇒ ∆ Γ ⇒ ¬φ,∆ (։r1) Γ ⇒ φ։ ψ,∆ Γ, φ ⇒ ψ,∆ (։r2) Γ ⇒ φ։ ψ,∆ with Prop(φ, ψ) ⊆ Prop(Γ) MODAL LOGICS OF KF 27 Remark 60. (։r2) is derivable in B3 for the material conditional defined by ¬ and ∨. Appendix B: Completeness of the Modal Nonclassical Systems As usual, the soundness direction is straightforward. The main idea of the completeness proof is to modify the standard Henkin strategy by replacing the notion of maximally consistent set with the one of saturated set. Definition 61 (S-Saturated set [JT96]). For S as above, a set Γ of L✷-sentences is S-saturated iff for all finite ∆ ⊆ SentL✷ : if S ⊢ Γ ⇒ ∆, then Γ ∩ ∆ 6= ∅. Lemma 62. If S 0 Γ ⇒ ∆, then there is a S-saturated Γ ∗ ⊇ Γ such that Γ∗ ∩ ∆ = ∅. Proof sketch. Starting with an enumeration of L✷-sentences in which every sentence occurs infinitely many times, one defines: Γ0 =: Γ Γn+1 =:      Γn ∪ {φn}, if S ⊢ Γn, φn ⇒ Θ entails Θ ∩ (SentL✷ \ ∆) 6= ∅ for all finite Θ ⊆ SentL✷ , Γn otherwise; Γ∗ := ⋃ n∈ω Γn Now Γ∗ ⊆ SentL✷ \ ∆. Therefore, Γ ∗ ∩ ∆ = ∅. It remains to be shown that Γ∗ is S-saturated. If S ⊢ Γ ∗ ⇒ Θ for some Θ then, since deductions are finite, there is an n and a finite Θ0 ⊆ Θ such that S ⊢ Γn ⇒ Θ0. By induction on the size of the finite set Θ0 ∩ (SentL✷ \ ∆) – [JT96, Lemma 4.3] –, one obtains that Γ∗ ∩ Θ 6= ∅. qed. Canonical models are then constructed from saturated sets is the usual way. However, in contrast to the classical case it no longer suffices to define the accessibility relation z0Rz1 simply by requiring z1 to be a superset of {φ | ✷φ ∈ z0}. Rather we also need to stipulate that z1 ⊆ {φ | ✸φ ∈ z0}. 16 In the classical setting these two conditions are equivalent since our worlds are assumed to be maximally consistent. Definition 63 (Canonical frame). For S as above, the canonical frame for S (ZS , RS) is specified by: ZS := {z | z is S-saturated} RSz0z1 :⇔ {φ | ✷φ ∈ z0} ⊆ z1 ⊆ {φ | ✸φ ∈ z0} The canonical model is obtained from the canonical frame by extending it with a suitable evaluation. The details of the evaluations vary depending on the kind of saturated set we are considering. We let, for S ∈ {FDE,K3, LP,B3,F3,KS3}: V Sz (p) =         1 if p ∈ z and ¬p /∈ z 0 if ¬p ∈ z and p /∈ z b if p ∈ z and ¬p ∈ z n otherwise 16Recall that in our nonclassical context ✸ := ¬✷¬, whereas this will not be true in the context of our classical modal logic. MODAL LOGICS OF KF 28 Definition 64 (Canonical model). For S ∈ {K3,B3,F3, LP,FDE,KS3}, the canonical model MS for S is the triple (Z S , RS , V S). Lemma 65 (Existence). Let z0 and z1 be S-saturated. Then the following implications hold: (i) if {φ | ✷φ ∈ z0} ⊆ z1, then there is an S-saturated z ⊆ z1 such that RSz0z; (ii) if z1 ⊆ {φ | ✸φ ∈ z0}, then there is an S-saturated z ⊇ z1 such that RSz0z. Proof. We start with (i). Since obviously SentL✷ \ (SentL✷ \ (z1 ∩ {φ | ✸φ ∈ z0})) = z1 ∩ {φ | ✸φ ∈ z0})) one starts by noticing that S 0 {φ | ✷φ ∈ z0} ⇒ SentL✷ \ (z1 ∩ {φ | ✸φ ∈ z0}). This is because, if S ⊢ {φ | ✷φ ∈ z0} ⇒ SentL✷ \ (z1 ∩ {φ | ✸φ ∈ z0}), then S ⊢ {φ | ✷φ ∈ z0} ⇒ Θ, for some finite Θ ⊆ SentL✷ \ (z1 ∩ {φ | ✸φ ∈ z0}). Since {φ | ✷φ ∈ z0} ⊆ z1 and z1 is S-saturated, z1 ∩ Θ 6= ∅. So we can divide up Θ in such a way that S ⊢ {φ | ✷φ ∈ z0} ⇒ ∨ (Θ \ z1),Θ ∩ z1. By the S rules, S ⊢ z0 ⇒ ✷ ∨ (Θ \ z1),✸(Θ ∩ z1). Since z0 is S-saturated, either ✷ ∨ (Θ \ z1) ∈ z0, or ✸(Θ ∩ z1) ∩ z0 6= ∅. If the former, then ∨ (Θ \ z1) ∈ z1, which is impossible. If the latter, then z1 ∩ {φ | ✸φ ∈ z0} ∩ Θ 6= ∅, which is also impossible. Therefore, by Lemma 62, we can construct an S-saturated z such that {φ | ✷φ ∈ z0} ⊆ z ⊆ z1 ∩ {φ | ✸φ ∈ z0}, which yields of course the claim. The proof of (ii) is similar to the previous case. Since z0 is S-saturated, S 0 {φ | ✷φ ∈ z0}, z1 ⇒ SentL✷ \ {φ | ✸φ ∈ z0}. Again by Lemma 62 there is a z such that {φ | ✷φ ∈ z0} ∪ z1 ⊆ z ⊆ {φ | ✸φ ∈ z0} as desired. qed. Lemma 66 (Truth Lemma). Let z ∈ ZS for S ∈ {K3,B3, LP,FDE,KS3}. Then for all φ ∈ L✷: MS , z s φ if and only if φ ∈ z. Proof Sketch. By induction on the positive complexity of φ. There are two nontrivial cases. The first is when φ is of the form ✷ψ. The right-to-left direction is obtained by induction hypothesis. In the left-to-right direction, starting with ✷ψ /∈ z, one finds a S-saturated set z1 ⊇ {φ | ✷φ ∈ z}. By the first part of Lemma 65, there is an S-saturated z0 ⊆ z1 such that R Szz0. Since φ /∈ z0, by induction hypothesis one obtains that MS , z0 1s φ, as required. MODAL LOGICS OF KF 29 The second non-trivial case, when φ is ¬✷ψ, is symmetric to the previous one and employs the second part of Lemma 65. For the right-to-left direction, suppose ¬✷ψ ∈ v. One then notices that (4) S 0 {φ | ✷φ ∈ v},¬ψ ⇒ ∅ Therefore, by Lemma 62, we can find a saturated z0 ⊇ {φ | ✷φ ∈ v},¬ψ. By Lemma 65, there is also a z1 ⊇ z0 with R Szz1. So, ¬ψ ∈ z1. The claim then follows by induction hypothesis. qed. We can finally prove the adequacy of our systems. Proof of Prop. 8. The soundness direction is obtained by a straightforward induction on the length of the proof in S. For the completeness direction, one assumes that S 0 Γ ⇒ ∆ and finds, by Lemma 62 an S-saturated z ⊇ Γ such that ∆ ∩ z = ∅. By the truth Lemma, MS , z s γ for all γ ∈ Γ and M S , z 1s δ for any δ ∈ ∆, as required. qed. King's College London and University of Bristol E-mail address: carlo.nicolai@kcl.ac.uk; johannes.stern@bristol.ac.uk