UC Berkeley Working Papers Title One Modal Logic to Rule Them All? (Extended Technical Report) Permalink https://escholarship.org/uc/item/07v9360j Authors Holliday, Wesley Halcrow Litak, Tadeusz Publication Date 2018-03-26 License CC BY-NC-ND 4.0 eScholarship.org Powered by the California Digital Library University of California One Modal Logic to Rule Them All? (Extended Technical Report) Version of June 25, 2018. Abridged article forthcoming in Advances in Modal Logic, Vol. 12. Wesley H. Holliday University of California, Berkeley Tadeusz Litak Friedrich-Alexander-Universität Erlangen-Nürnberg Abstract In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same role as the compound modality ∀pA. Unlike the propositional quantifier by itself, the global quantificational modality can be straightforwardly interpreted in any Boolean Algebra Expansion (BAE). We present a logic GQM for this language and prove that it is complete with respect to the intended algebraic semantics. This logic enables a conceptual shift, as what have traditionally been called different “modal logics” now become [∀p]-universal theories over the base logic GQM: instead of defining a new logic with an axiom schema such as ✷ϕ → ✷✷ϕ, one reasons in GQM about what follows from the globally quantified formula [∀p](✷p → ✷✷p). Keywords: global quantificational modalities, propositional quantifiers, Boolean algebra expansions, Boolean algebras with operators, modal consequence. 1 Introduction In this paper, we investigate the effect of extending modal syntax with the global quantificational modality [∀p]. This proposal arose from our efforts to tackle two foundational problems in modal logic, which under careful inspection turn out to be related to each other. 1.1 The proliferation of modal “logics” According to a standard view, there is a striking contrast between first-order logic and modal logic: the former leads to a single system of First-Order Logic, 2 One Modal Logic to Rule Them All? while the latter leads to a vast landscape of different logics. In some contexts this has prompted the “suggestion. . . that the great proliferation of modal logics is an epidemy from which modal logic ought to be cured” [8, p. 25]. 1 One could object that first-order logic is not so monolithic, given the choice between classical, intuitionistic, superintuitionistic, or substructural bases. But even in the classical context, there are objections to the claimed contrast, now coming from the modal side. As van Benthem [4] writes: [T]hese systems are not “different modal logics”, but different special theories of particular kinds of accessibility relation. We do not speak of “different first-order logics” when we vary the underlying model class. There is no good reason for that here, either. (p. 93) Yet in modal logic there remains a distinction between theories and logics: the set of formulas satisfied at all points (or a point) in a model counts as a theory, but not a logic, while the set of formulas validated at all points (or a point) of a frame counts as a logic. Here we will not address the question in the philosophy of logic about what should count as a “logic”. Instead, we answer the question: is there a mathematically appealing way in which what are traditionally called “modal logics” are special theories relative to one logical system? 1.2 The riddle of propositional quantification The other problem we are concerned with is that of conservatively handling propositional quantifiers. Historically, propositional quantifiers were considered in modal logic from the very beginning. Most of the literature quotes references such as Kripke [22], Bull [7], Fine [13], and Kaplan [20]. In fact, however, propositional quantifiers were already present not only in Ruth Barcan Marcus’s post-war papers [3], but even in a chapter about the “existence postulate” by C. I. Lewis in his famous 1932 monograph with Langford [23, § VI.6]. Lewis’s postulate is classically equivalent to ∃p(✸p ∧ ✸¬p), and he insisted that “it is only through such principles that the outlines of a logical system can be positively delineated” [23, p. 181]. The problem, however, is that the addition of propositional quantifiers is not necessarily conservative for a given logic. It appears most natural, for example, to interpret them using infinite meets and joins in (algebras dual to) a suitable semantics (see § 3). Unfortunately, there are logics that are not even weakly complete with respect to lattice-complete algebras [24,26,25,28,33]. Moreover, even for standard logics such algebras might well validate undesirable quantified principles, as shown by “Kaplan’s paradox” [21] for possible world semantics (see § 4). Logics with propositional quantifiers also tend to display very bad computational behavior over the dual algebras of Kripke frames: even for logics as strong as S4.2, propositional quantification over Kripke frames produces a system as complex as full second-order logic [13,19]. 1 The authors of [8] attribute this suggestion to others rather than endorsing it themselves. Holliday, Litak 3 1.3 Our proposal In this paper, we intend to solve both problems by showing, on the one hand, how “different modal logics” can indeed be seen as different theories over a single base logic, and on the other hand, how each and every modal logic can be conservatively extended with a form of propositional quantification. This is made possible by extending the language with the global quantificational modality [∀p], which combines the propositional quantifier ∀p with the global modality A. Semi-formally, one can introduce it as [∀p]ϕ := ∀pAϕ. One can then think of the global modality as definable by taking a fresh variable in [∀p] and introduce further global quantificational modalities (GQMs) as follows: h∃piϕ := ¬[∀p]¬ϕ = ∃pEϕ [∃p]ϕ := h∃piAϕ = ∃pAϕ h∀piϕ := ¬[∃p]¬ϕ = ∀pEϕ. This language of global quantificational modalities, formally defined in § 2, will be our object of study. In § 3, we show that in a global sense, this language is as expressive as the standard language of second-order propositional modal logic over the lattice-complete algebras used to interpret that language; and in § 4, we show that the flexibility to interpret our language in incomplete algebras provides a response to “Kaplan’s paradox” for possible world semantics. In § 5, we introduce our logic GQM, and in § 6 we show that GQM solves the twin problems of proliferation (§ 1.1) and nonconservativity (§ 1.2). Toward proving the completeness of GQM with respect to its intended semantics, we establish prenex normal form results in § 7 and mutual translations with the first-order theory of “discriminator BAEs” in § 8. The storyline culminates with the completeness theorem at the end of § 8. We conclude in § 9. A number of proofs involving syntactic derivations are deferred to two appendices. An open-source git repository with formalizations of our proofs in Coq by Michael Sammler is available at https://gitlab.cs.fau.de/lo22tobe/GQM-Coq. 2 Language and semantics In § 1.3, we introduced the idea of [∀p] in terms of the propositional quantifier and the global modality. Officially, we take [∀p] as primitive. Fix a countably infinite set Prop of propositional variables and define: LGQM ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷ϕ | [∀p]ϕ, where p ∈ Prop. We treat ∨, →, ↔, and ✸ as abbreviations as usual and define: 2 • Aϕ := [∀r]ϕ for an r ∈ Prop not free (in the usual sense) in ϕ; Eϕ := ¬A¬ϕ; • h∃piϕ := ¬[∀p]¬ϕ, [∃p]ϕ := h∃piAϕ, and h∀piϕ := [∀p]Eϕ. • ⊥ := (p ∧ ¬p) and ⊤ := ¬⊥ for some p ∈ Prop; 2 Note that since ⊥ can be defined as [∀p]p, another elegant choice would be to have → as the only Boolean primitive. 4 One Modal Logic to Rule Them All? • for each GQM h[ Qpi] ∈ {[∀p], [∃p], h∃pi, h∀pi}, its dual h[ Qpi] is defined in the obvious way, i.e., [∀p] is dual to h∃pi, and [∃p] is dual to h∀pi; • for ∗ ∈ {∧, ∨}, let G∗ be A if ∗ = ∧ and E otherwise, and let us use plain G to stand for A or E (uniformly in a formula) in results that hold for both; • for any formulas ϕ, ψ and propositional variable p, ϕpψ is the result of substituting ψ for all free occurrences of p in ϕ. Let L✷ (L✷A ) be the set of GQM formulas in which no global quantificational modalities (no global quantificational modalities other than A and E) appear. Definition 2.1 A formula ϕ is global iff each occurrence of ✷ and each propositional variable in ϕ is in the scope of a GQM (even if not bound by a GQM). E.g., [∀q]✷p is global; ✷[∀p]p is not. Remark 2.2 The use of a single unary ✷ is for simplicity only. What follows could instead be developed in a polymodal language with polyadic modalities. We now introduce the intended algebraic semantics for LGQM . Definition 2.3 A Boolean algebra expansion (BAE) is a tuple A = hA, ¬, ∧, ⊥, ⊤, ✷i where hA, ¬, ∧, ⊥, ⊤i is a Boolean algebra and ✷ : A → A. Definition 2.4 (i) A C-BAE (resp. A-BAE) is a BAE whose Boolean reduct is complete (resp. atomic). (ii) A BAO is a BAE with a normal ✷, i.e., ✷ distributes over all finite meets. (iii) A V-BAO is a BAO in which ✷ distributes over all existing meets. We may concatenate ‘C’, ‘A’, and ‘V’ to indicate multiple properties; e.g., a CA-BAE is a BAE whose Boolean reduct is both complete and atomic. This is a convention used in our earlier papers [26,25,28,16,15]. Definition 2.5 A valuation on a BAE A is a function θ : Prop → A that extends to a function θ̃ : LGQM → A as follows: θ̃(p) := θ(p) θ̃(¬ϕ) := ¬θ̃(ϕ) θ̃(ϕ ∧ ψ) := θ̃(ϕ) ∧ θ̃(ψ) θ̃(✷ϕ) := ✷θ̃(ϕ) ( ⊤ if γ̃(ϕ) = ⊤ for all valuations γ ∼p θ θ̃([∀p]ϕ) := ⊥ otherwise where γ ∼p θ iff γ and θ disagree at most at p. A formula ϕ is valid in A iff for every valuation θ on A, θ̃(ϕ) = ⊤. Let GQM ϕ iff ϕ is valid in all BAEs, in which case ϕ is simply valid. Lemma 2.6 For any valuation θ on a BAE A: Holliday, Litak θ̃(Aϕ) = ( ⊤ ⊥ if θ̃(ϕ) = ⊤ otherwise θ̃(h∃piϕ) = ( ⊤ ⊥ if ∃γ ∼p θ.γ̃(ϕ) 6= ⊥ otherwise θ̃(h∀piϕ) = ( ⊤ ⊥ if ∀γ ∼p θ.γ̃(ϕ) 6= ⊥ . otherwise 5 θ̃(Eϕ) = ( ⊤ ⊥ if θ̃(ϕ) 6= ⊥ otherwise θ̃([∃p]ϕ) = ( ⊤ ⊥ if ∃γ ∼p θ.γ̃(ϕ) = ⊤ otherwise Several definitions of semantic consequence are available, but as our default we pick the algebraic analogue of global model consequence [5, § 1.5]. Definition 2.7 Given Γ ∪ {ϕ} ⊆ LGQM , let Γ AGQM ϕ iff for any BAE A and θ : Prop → A, if θ̃(γ) = ⊤ for each γ ∈ Γ, then θ̃(ϕ) = ⊤. One of our main goals is to find a proof system complete with respect to AGQM . For this relation we have the following semantic deduction theorem. Lemma 2.8 For any formulas ϕ1 , . . . , ϕn , ψ ∈ LGQM and global formulas α1 , . . . , αn , β ∈ LGQM : (i) {ϕ1 , . . . , ϕn } AGQM ψ iff GQM A(ϕ1 ∧ · · · ∧ ϕn ) → Aψ. (ii) {α1 , . . . , αn } AGQM β iff GQM (α1 ∧ · · · ∧ αn ) → β. Proof. Part (i) is immediate from Definition 2.7 and Lemma 2.6. For part (ii), it is easy see that if ϕ is global, then θ̃(ϕ) = θ̃(Aϕ). ✷ We also distinguish two senses in which formulas may be equivalent. Definition 2.9 For any ϕ, ψ ∈ LGQM and class K of BAEs: (i) ϕ and ψ are equivalent over K iff for every A ∈ K and valuation θ on A, θ̃(ϕ) = θ̃(ψ) (or equivalently, ϕ ↔ ψ is valid in A); (ii) ϕ and ψ are globally equivalent over K iff for every A ∈ K and valuation θ on A, θ̃(ϕ) = ⊤ iff θ̃(ψ) = ⊤ (or equivalently, Aϕ ↔ Aψ is valid in A). (iii) ϕ and ψ are equivalent (resp. globally equivalent) iff they are equivalent (resp. globally equivalent) over the class of all BAEs. Remark 2.10 Since LGQM can be interpreted in arbitrary BAEs, it can be interpreted in any frames that give rise to BAEs, e.g.: Kripke frames (corresponding to CAV-BAOs); relational possibility frames [15] (corresponding to CV-BAOs); neighborhood frames (corresponding to CA-BAEs); neighborhood possibility frames [15] (corresponding to C-BAEs); discrete general frames [10] (corresponding to AV-BAOs); discrete general neighborhood frames (corresponding to A-BAEs); general neighborhood frames (corresponding to BAEs). Remark 2.11 The predicate lifting approach in coalgebraic logic can be seen as a way to reduce any set-based coalgebra to a neigborhood frame (see, e.g., [29, Rem. 3.10] and references therein). Thus, our solution to the problems discussed in § 1 should be of interest for coalgebraic logic. 6 One Modal Logic to Rule Them All? 3 Reduction of SOPML to GQM The standard language LSOPML of second-order propositional modal logic replaces [∀p] by the propositional quantifier ∀p. At first one might expect that the implicit global modality in [∀p] reduces the expressivity of LGQM relative to the language LSOPMLA of second-order propositional modal logic plus the global modality. In fact, we will show that every SOPMLA formula is globally equivalent to a GQM formula over standard semantics. For convenience, in this section we regard GQM formulas as SOPMLA formulas with [∀p] as ∀pA. First, let us recall the algebraic semantics for LSOPMLA that interprets ∀p using the meets in a C-BAE, as in, e.g., [16]. Definition 3.1 We extend a valuation θ on a C-BAE A to a valuation θ̃ : LSOPMLA → A using the clauses for ¬, ∧, and ✷ from Definition 2.5 plus: ( ^ ⊤ if θ̃(ϕ) = ⊤ θ̃(∀pϕ) = {γ̃(ϕ) | γ ∼p θ} θ̃(Aϕ) = ⊥ otherwise. Dually, ∃pϕ is interpreted using the join. The definitions of local and global equivalence from Definition 2.9 transfer in the obvious way to LSOPMLA . We will reduce LSOPMLA to LGQM over C-BAEs using a prenex form result. In [9] it was shown that over CAV-BAOs, every LSOPML formula is equivalent to a prenex one, i.e., a formula of the form Q1 p1 . . . Qn pn ϕ where Qi ∈ {∀, ∃} and ϕ is quantifier-free. In fact, the following more general result holds. Proposition 3.2 (i) Over CV-BAOs, every SOPML formula is equivalent to a prenex SOPML formula. (ii) Over C-BAEs, every SOPMLA formula is equivalent to a prenex SOPMLA formula. Proof. The proof of part (i) is the same as in [9, Prop. 3] except that we give a different argument for pulling the quantifier out of ✸∀pϕ, which does not assume A. For consistency with [9], we work with ✸, though everything dualizes easily. For any normal ✷, we have the following equivalence: ✸ψ ⇔ ∃q(✸q ∧ ✷(q → ψ)) for a variable q that does not occur in ψ. Thus, we have the following equivalences where q 6= p and q does not occur in ϕ: ✸∀pϕ ⇔ ∃q(✸q ∧ ✷(q → ∀pϕ)) ⇔ ∃q(✸q ∧ ✷∀p(q → ϕ)) ⇔ ∃q(✸q ∧ ∀p✷(q → ϕ)) ⇔ ∃q∀p(✸q ∧ ✷(q → ϕ)) setting ψ := ∀pϕ because q 6= p by V for ✷ because q 6= p. For part (ii), we use the fact that A distributes over arbitrary meets, so the reasoning for part (i) shows that E∀pϕ is equivalent to ∃q∀p(Eq ∧ A(q → ϕ)), which implies (⋆): A∃pψ is equivalent to ∀q∃p(Eq → E(q ∧ ψ)). Holliday, Litak 7 Now for any ✷ in a C-BAE, we have the following equivalence: ✸ψ ⇔ ∃q(✸q ∧ A(q ↔ ψ)) for a variable q that does not occur in ψ. Thus, we have the following equivalences where q 6= p and q does not occur in ϕ: ✸∀pϕ ⇔ ∃q(✸q ∧ A(q ↔ ∀pϕ)) setting ψ := ∀pϕ ⇔ ∃q(✸q ∧ A(∀p(q → ϕ) ∧ ∃p(ϕ → q))) because q 6= p ⇔ ∃q(✸q ∧ A(∀p(q → ϕ) ∧ ∃r(ϕpr → q))) for a fresh r ⇔ ∃q(✸q ∧ A∀p∃r((q → ϕ) ∧ (ϕpr → q))) because r 6= p | {z } α ⇔ ∃q(✸q ∧ ∀pA∃rα) by V for A ⇔ ∃q(✸q ∧ ∀p∀q ′ ∃r(Eq ′ → E(q ′ ∧ α))) by (⋆) where q ′ is fresh ⇔ ∃q∀p∀q ′ ∃r(✸q ∧ (Eq ′ → E(q ′ ∧ α))) because q 6= p, q 6= q ′ , q 6= r. The rest of the proof is as in [9]. ✷ To prove Proposition 3.4 below, it is convenient to have another equivalence for A∃pψ (other than (⋆) in the preceding proof) in part (ii) of the following. Lemma 3.3 The following are valid in all C-BAEs: (i) A∀pψ ↔ ∀pAψ; (ii) A∃pψ ↔ ∀qA(Eq → ∃rA(Er ∧ (r → q) ∧ ∃pA(r → ψ))) where q and r do not occur in ψ. Proof. Part (i) again follows from the distribution of A over arbitrary meets. Part (ii) follows W from the Boolean algebraic fact that for any C-BA A and Y ⊆ A, we have Y = ⊤ (take Y = {γ̃(ψ) | γ ∼p θ}) iff for all x ∈ A (take x as the semantic value of q), if x 6= 0, then there exists a y ∈ Y such that x ∧ y 6= 0, which is equivalent to there being a z ∈ A (take z to be the semantic value of r) such that z 6= 0, z ≤ x, and z is under some element of Y . ✷ Proposition 3.4 If α is a prenex SOPMLA formula, then Aα is equivalent over C-BAEs to a GQM formula. Proof. We continue to regard GQM formulas as SOPMLA formulas as in § 3. The proof is by induction on the number of quantifiers in the prenex formula α := Q1 p1 . . . Qn pn χ. Let α′ := Q2 p2 . . . Qn pn χ. Case 1: Q1 = ∀. Then by Lemma 3.3.(i), Aα is equivalent to ∀p1 Aα′ , which is equivalent to ∀p1 AAα′ . Since α′ has fewer quantifiers than α, by the inductive hypothesis Aα′ is equivalent to a GQM formula β. Hence ∀p1 AAα′ is equivalent to the GQM formula ∀p1 Aβ. Case 2: Q1 = ∃. Then by Lemma 3.3.(ii), Aα is equivalent to (1) ∀qA(Eq → ∃rA(Er ∧ (r → q) ∧ ∃p1 A(r → α′ ))) and hence to (2) ∀qA(Eq → ∃rA(Er ∧ (r → q) ∧ ∃p1 AA(r → α′ ))). 8 One Modal Logic to Rule Them All? Since r is not among p2 , . . . , pn , (2) is equivalent to (3) ∀qA(Eq → ∃rA(Er ∧ (r → q) ∧ ∃p1 AAQ2 p2 . . . Qn pn (r → χ))). Since Q2 p2 . . . Qn pn (r → χ) has fewer quantifiers than α, by the inductive hypothesis AQ2 p2 . . . Qn pn (r → χ) is equivalent to a GQM formula γ. Hence (3) is equivalent to the GQM formula (4) ∀qA(Eq → ∃rA(Er ∧ (r → q) ∧ ∃p1 Aγ)). ✷ We can now prove our main claim about the reduction of SOPML to GQM. Theorem 3.5 Every SOPMLA formula is globally equivalent over C-BAEs to a GQM formula. Proof. By Proposition 3.2.(ii), ϕ is globally equivalent over C-BAEs to a prenex SOPMLA formula ψ. Then since ψ is globally equivalent to Aψ, it follows by Proposition 3.4 that ψ is globally equivalent to a GQM formula. ✷ Theorem 3.6 The set of GQM formulas valid over any class of C-BAEs containing the class of CAV-BAOs validating S4.2 is not recursively enumerable. Proof. [Sketch] Fine [13, Prop. 6] (cf. [19]) showed that the set of SOPML sentences valid in CAV-BAOs validating S4.2 is not recursively enumerable. The property of a BAE being an AV-BAO is expressible in LGQM ; we leave this to the reader as an exercise (cf. § 8, [17, § 9]). Let χAV be the corresponding sentence. The validity of an SOPML sentence ϕ over S4.2 CAV-BAOs is equivalent to the validity of the GQM sentence (χAV ∧ [∀]S4.2) → ϕ∗ over C-BAEs, where ϕ∗ is obtained from Aϕ by Theorem 3.5 and [∀]S4.2 is the GQM statement of the S4.2 axioms. Thus, the existence of a semi-decision procedure contradicting the statement would yield a semi-decision procedure contradicting [13,19]. ✷ Another route would be to use results of Thomason [32]. Both [32] and [19] deal with a stronger property: reducibility of full second-order consequence. We postpone the details to a sequel paper. 4 Interlude: “Kaplan’s paradox” In a festschrift for Ruth Barcan Marcus, Kaplan [21] posed a problem for possible world semantics involving propositional quantification. In brief, Kaplan claimed that the following should be consistent for a non-monotonic ✷: • κ := ∀pE∀q(✷q ↔ A(p ↔ q)). For example, if ✷ means “it is entertained at time t that. . . ”, then κ says that for all propositions p, it could have been that p was the unique proposition entertained at time t. Kaplan argued that logic should not rule this out. Yet he noted that κ is unsatisfiable in possible world semantics with ∀ quantifying over the powerset. For the truth of κ would yield an injection from the powerset of the set of worlds to the set of worlds. In fact, as Yifeng Ding (p.c.) observed, it is unsatisfiable in any C-BAE. The truth of κ would yield (a) an injection from the algebra to an antichain of elements. But since the algebra is complete, Holliday, Litak 9 (b) every subset of the antichain has a join, and all such joins are distinct. Together (a) and (b) contradict Cantor’s theorem. We will show there is a GQM formula ϕ, regarded as an SOPMLA formula as in § 3, such that (i) in any logic that derives some plausible equivalences, ϕ is provably equivalent to (the A-necessitation of) Kaplan’s formula, so intuitively the truth of ϕ implies the truth of Kaplan’s formula, and (ii) ϕ can be made true in an incomplete BAE. First, in any modal logic with propositional quantifiers in which the equivalences in the proof of Proposition 3.2.(ii) are provable, (the A-necessitation of) Kaplan’s formula is provably equivalent to • A∀p∃r(Er ∧ ∀qA(r → (✷q ↔ A(p ↔ q)))). 3 Then using Barcan’s equivalence A∀pψ ↔ ∀pAψ and S5 reasoning with E and A, the preceding formula is provably equivalent to • ∀pA∃rA(Er ∧ ∀qA(r → (✷q ↔ A(p ↔ q)))), which becomes the GQM formula • [∀p][∃r](Er ∧ [∀q](r → (✷q ↔ A(p ↔ q)))). Now this formula can be made true in a BAE. Pick any infinite set X, and let A be the Boolean algebra of its finite and cofinite subsets. Clearly, not only is X identifiable with the set At(A) of atoms of A, but also there is an injective (and hence bijective) map ✷ : A → At(A). It is easy to see that the formula above evaluates to ⊤ in A with this interpretation of ✷. Thus, according to the logic GQM, the GQM translation of Kaplan’s formula is consistent. 4 5 Axiomatization Let us now turn to formulating a complete proof system for LGQM . Definition 5.1 The logic GQM is the smallest set of formulas containing the following axioms and closed under the following rules. propositional axioms • all classical propositional tautologies. axioms for [∀p] • distribution: [∀p](ϕ → ψ) → ([∀p]ϕ → [∀p]ψ); • instantiation: [∀p]ϕ → ϕpψ where ψ is substitutable for p in ϕ; 5 • global instantiation: [∀p]ϕ → [∀r]ϕpψ where ψ is substitutable for p in ϕ and r is not free in ϕpψ ; 3 Of course the quantifier ∀q can be pulled to the front for a prenex form, but here we opt for better human readability. 4 After submitting the AiML version of this paper, we learned from John Hawthorne of the paper [2] in which the finite-cofinite algebra is also used in response to Kaplan’s paradox. 5 The definition of ψ being substitutable for p in ϕ is the obvious analogue of the definition of a term t being substitutable for a variable x in a first-order formula [12, p. 113]: no propositional variable in ψ should be captured by a quantifier in ϕ upon substituting ψ for p. 10 One Modal Logic to Rule Them All? • quantificational 5 axiom: ¬[∀p]ϕ → [∀r]¬[∀p]ϕ where r is not free in [∀p]ϕ. axioms linking [∀p] and ✷ • ✷-congruence: [∀p](ϕ ↔ ψ) → (✷ϕ ↔ ✷ψ). rules • modus ponens: if ⊢GQM ϕ and ⊢GQM ϕ → ψ, then ⊢GQM ψ; • [∀p]-necessitation: if ⊢GQM ϕ, then ⊢GQM [∀p]ϕ; • universal generalization: if ⊢GQM α → [∀p]ϕ and q is not free in α, then ⊢GQM α → [∀q][∀p]ϕ. Here ‘ ⊢GQM ϕ ’ means ϕ ∈ GQM. We write ‘ ⊢ ϕ ’ when no confusion will arise. Let us now record some useful theorems and metatheorems. Lemma 5.2 If q is substitutable for p in ϕ, and q is not free in ϕ, then ⊢ [∀p]ϕ ↔ [∀q]ϕpq . Proof. Proved in Appendix A.1 as Lemma A.3. ✷ Lemma 5.3 If ⊢ ϕ → ψ, then ⊢ h[ Qpi]ϕ → h[ Qpi]ψ. Proof. Proved in Appendix A.1 as Lemma A.5. ✷ Lemma 5.4 (i) ⊢ A(ϕ → ψ) → (Aϕ → Aψ); (vii) ⊢ EAϕ ↔ Aϕ; (ii) ⊢ G∗ (ϕ ∗ ψ) ↔ (G∗ ϕ ∗ G∗ ψ); (viii) ⊢ GGϕ ↔ Gϕ; (iii) if ⊢ ϕ → ψ, then ⊢ Gϕ → Gψ; (ix) ⊢ h[ Qpi]Aψ ↔ [Qp]ψ; (iv) ⊢ Aϕ → ϕ; (x) ⊢ h[ Qpi]Eψ ↔ hQpiψ; (v) ⊢ ϕ → Eϕ; (xi) ⊢ h[ Qpi]ψ ↔ A[hQpi]ψ; (vi) ⊢ Eϕ ↔ AEϕ; (xii) ⊢ h[ Qpi]ψ ↔ E[hQpi]ψ. Proof. Proved in Appendix A.2 as Lemma A.7. ✷ Let us now introduce a relation of syntactic consequence. In the following definition, Γ may be regarded as a set of globally true premises. Definition 5.5 Given Γ ∪ {ϕ} ⊆ LGQM , let Γ ⊢AGQM ϕ iff ϕ belongs to the smallest set Λ of GQM formulas that includes Γ ∪ GQM and is closed under modus ponens and A-necessitation: if ψ ∈ Λ, then Aψ ∈ Λ. Now we obtain a syntactic deduction theorem parallel to Lemma 2.8. Lemma 5.6 For any formulas ϕ1 , . . . , ϕn , ψ ∈ LGQM and global formulas α1 , . . . , αn ∈ LGQM : (i) {ϕ1 , . . . , ϕn } ⊢AGQM ψ iff ⊢GQM A(ϕ1 ∧ · · · ∧ ϕn ) → Aψ. (ii) {α1 , . . . , αn } ⊢AGQM ψ iff ⊢GQM (α1 ∧ · · · ∧ αn ) → ψ. Proof. Part (i) uses Lemma 5.4. Part (ii) then uses Lemma 5.4.(xi). By design, ⊢AGQM is sound with respect to AGQM . ✷ Holliday, Litak 11 Lemma 5.7 For Γ ∪ {ϕ} ⊆ LGQM , Γ ⊢AGQM ϕ implies Γ AGQM ϕ. Proof. Straightforward induction. ✷ It follows from Lemma 5.7 and the example in § 4 (or Theorem 3.6) that GQM is incomplete with respect to validity over the class of C-BAEs. However, we will see in § 8 that GQM is complete with respect to validity over all BAEs. 6 Conservativity and modal logics as GQM theories Before proving completeness, we show that GQM solves our two problems from § 1: the proliferation problem and the nonconservativity problem. A congruential modal logic is a set L ⊆ L✷ containing all propositional tautologies and closed under uniform substitution, modus ponens, and the rule that if ϕ ↔ ψ ∈ L, then ✷ϕ ↔ ✷ψ ∈ L. Let GQM-L be the smallest set of formulas that includes GQM ∪ L and is closed under all three rules of GQM. Proposition 6.1 (Conservativity) For any ϕ ∈ L✷ , ϕ ∈ GQM-L iff ϕ ∈ L. Proof. The Lindenbaum-Tarski algebra for L is a BAE in which every ϕ ∈ GQM-L is valid and in which any L✷ formula not in L can be refuted. ✷ A set Σ ⊆ L✷ axiomatizes a congruential modal logic L iff L is the smallest congruential modal logic such that Σ ⊆ L. Theorem 6.2 If Σ axiomatizes L, then we have the following equivalence: ϕ ∈ L iff there are ψ1 , . . . , ψn ∈ Σ such that ⊢GQM [∀p](ψ1 ∧ · · · ∧ ψn ) → ϕ, where p is the tuple of variables occurring in ψ1 , . . . , ψn . Proof. From right to left, we have: ⊢GQM [∀p](ψ1 ∧ · · · ∧ ψn ) → ϕ ⇒ ϕ ∈ GQM-L by [∀]-necessitation to ψ1 ∧ · · · ∧ ψn ∈ L and modus ponens ⇒ ϕ ∈ L by Proposition 6.1. From left to right, suppose ϕ ∈ L. Hence there is a sequence hϕ1 , . . . , ϕk i of L✷ formulas with ϕk = ϕ such that each ϕi is either a tautology, an axiom from Σ, or is obtained from previous formulas by either uniform substitution, modus ponens, or the congruence rule. It is easy to show that there is such a sequence in which uniform substitution is applied only to propositional tautologies or axioms from Σ, so let us assume that hϕ1 , . . . , ϕk i has this property. Further suppose that the tautologies and axioms from Σ occur at the beginning of the sequence, so the sequence is of the form π = hα1 , . . . , αm , ϕm+1 , . . . , ϕk i where each αi is a tautology or an axiom from Σ. We prove by induction that for m ≤ ℓ ≤ k, there is a sequence of GQM formulas containing each of [∀p]α1 , . . . , [∀p]αm , Aϕm+1 , . . . , Aϕℓ such that each formula of the sequence is one of the [∀p]αi , or is a theorem of GQM, or follows from previous formulas by modus ponens. It is easy to see that the existence of such a sequence for ϕk implies that ⊢GQM ([∀p]ψ1 ∧ · · · ∧ [∀p]ψk ) → Aϕk , which in turn implies ⊢GQM [∀p](ψ1 ∧ · · · ∧ ψn ) → ϕk by Lemma 5.4.(iv). 12 One Modal Logic to Rule Them All? For the base case, take the sequence [∀p]α1 , . . . , [∀p]αm . For the inductive hypothesis, assume hγ1 , . . . , γj i is an appropriate sequence for ℓ. For ℓ + 1, in π above, ϕℓ+1 is obtained either from some αi by uniform substitution or from previous formulas by modus ponens or the congruence rule. Case 1: ϕℓ+1 comes from αi by substitution. Observe that ⊢GQM [∀p]αi → Aϕℓ+1 by repeated application of instantiation and then global instantiation. Hence we can extend hγ1 , . . . , γj i to hγ1 , . . . , γj , [∀p]αi → Aϕℓ+1 , Aϕℓ+1 i, and Aϕℓ+1 follows from [∀p]αi and [∀p]αi → Aϕℓ+1 by modus ponens. Case 2: ϕℓ+1 comes from a previous χ in π by the congruence rule, so χ is a biconditional χ1 ↔ χ2 and ϕℓ+1 is ✷χ1 ↔ ✷χ2 . Case 2a: χ is among the αi . Observe that ⊢GQM [∀p](χ1 ↔ χ2 ) → A(✷χ1 ↔ ✷χ2 ) by instantiation, ✷-congruence, and Lemma 5.4.(iii) and 5.4.(ix). Hence we can extend hγ1 , . . . , γj i to hγ1 , . . . , γj , [∀p]αi → Aϕℓ+1 , Aϕℓ+1 i, and Aϕℓ+1 follows from [∀p]αi and [∀p]αi → Aϕℓ+1 by modus ponens. Case 2b: χ is among ϕm+1 , . . . , ϕn . Observe that ⊢GQM A(χ1 ↔ χ2 ) → A(✷χ1 ↔ ✷χ2 ) by ✷-congruence and Lemma 5.4.(iii) and 5.4.(viii). Hence we can extend hγ1 , . . . , γj i to hγ1 , . . . , γj , Aχ → Aϕℓ+1 , Aϕℓ+1 i, and Aϕℓ+1 follows from Aχ and Aχ → Aϕℓ+1 by modus ponens. Case 3: ϕℓ+1 comes from two previous formulas χ and χ′ in π by modus ponens. Then the reasoning is similar to the previous cases, using distribution.✷ We can easily rephrase Theorem 6.2 in the language of “theories.” Definition 6.3 A ⊢GQM -theory is a set of GQM formulas that includes GQM and is closed under modus ponens. Corollary 6.4 If Σ ⊆ L✷ axiomatizes a congruential modal logic L, then we have the following equivalence: ϕ ∈ L iff ϕ belongs to the smallest ⊢GQM -theory that includes [∀]Σ = {[∀p]ϕ | ϕ ∈ Σ and p are the variables in ϕ}. Given this reduction of modal logics to ⊢GQM -theories, we have the following. Corollary 6.5 GQM theoremhood is undecidable. Proof. In light of Theorem 6.2, a decision procedure for GQM would yield a decision procedure for every finitely axiomatizable modal logic. But there are undecidable logics with finite axiomatizations [11, § 16.4]. ✷ Theorem 3.6 showed that the set of GQM formulas valid over C-BAEs is not recursively enumerable. Our completeness result in Theorem 8.6 will yield that the situation is better over general algebraic semantics. 7 Prenex forms Our path to completeness begins with suitable normal and prenex forms. 7.1 Weak prenex forms Definition 7.1 (i) A nontrivial weak prenex (NWP) formula is a formula of the form h[ Qpi] ϕ, where h[ Qpi] is a nonempty sequence of GQMs and ϕ is a L✷A -formula. Holliday, Litak 13 (ii) A normal clause is a disjunction each disjunct of which is either (a) a literal, (b) of the form ✷ψ or ✸ψ with ψ quantifier free, or (c) a formula in NWP form. (iii) A conjunctive normal form weak prenex (CNFWP) formula is a conjunction of normal clauses. The following are key lemmas for the purposes of showing that formulas can be transformed into equivalent CNFWP formulas. Lemma 7.2 (i) ⊢ (G∗ α ∗ h[ Qpi]β) ↔ h[ Qpi](G∗ α ∗ β) where p is not free in α; (ii) ⊢ A(ϕ ∨ h[ Qpi]ψ) ↔ (Aϕ ∨ h[ Qpi]ψ); (iii) ⊢ ✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (([hQpi]ψ ∧ ✷α) ∨ (¬[hQpi]ψ ∧ ✷(α ∧ ϕ))). Proof. Parts (i), (ii), and (iii) are proved in Appendix A.3 as Propositions A.12, A.13, and A.14, respectively. ✷ Lemma 7.3 (i) If α1 , . . . , αm are each NWP formulas, then α1 ∗ · · · ∗ αm is provably equivalent to G∗ (α1 ∗ · · · ∗ αm ). (ii) If α1 , . . . , αn are each NWP formulas, then α1 ∗ · · · ∗ αn is provably equivalent to an NWP formula. (iii) If α is a normal clause, then Aα is provably equivalent to an NWP formula. (iv) If ϕ is a CNFWP formula, then Aϕ is provably equivalent to an NWP formula. Proof. (i) We have: (1) ⊢ (α1 ∗ · · · ∗ αn ) ↔ (G∗ α1 ∗ · · · ∗ G∗ αn ) αi is an NWP formula by Lemma 5.4.(xi)-(xii) since each (2) ⊢ (G∗ α1 ∗ · · · ∗ G∗ αn ) ↔ G∗ (α1 ∗ · · · ∗ αn ) (3) ⊢ (α1 ∗ · · · ∗ αn ) ↔ G∗ (α1 ∗ · · · ∗ αn ) by Lemma 5.4.(ii) from (1) and (2) by PL. (ii) By Lemma 5.2, we may assume without loss of generality that (a) no propositional variable occurs both free and bound in α1 , . . . , αn . The proof is by induction on the number of nonvacuous GQMs (i.e., GQMs binding variables, unlike A and E) occurring in α1 , . . . , αn . First, by Lemma 5.4.(xi)-(xii), we may replace each αi with an equivalent NWP formula αi′ containing no more GQMs and in which no vacuous GQM occurs before a nonvacuous GQM. Thus, if no αi′ begins with a nonvacuous GQM, then α1′ ∗ · · · ∗ αn′ is already an NWP formula, so we are done. Now suppose that some αi′ , say αn′ , is of the form ′ h[ Qpi]ϕ where h[ Qpi] is nonvacuous. Since α1′ , . . . , αn−1 are each NWP formulas, ′ ) is equivalent to G∗ α by part (i). Then we have: α := (α1′ ∗ · · · ∗ αn−1 (4) ⊢ α ↔ G∗ α (5) ⊢ (α1′ ∗ · · · ∗ αn′ ) ↔ (G∗ α ∗ h[ Qpi]ϕ) by (4) and PL 14 One Modal Logic to Rule Them All? (6) ⊢ (G∗ α ∗ h[ Qpi]ϕ) ↔ (G∗ α ∗ h[ Qpi]h[ Qri]ϕ) Lemma 5.4.(ix)-(x) where r is not free in ϕ or α, by (7) ⊢ (G∗ α ∗ h[ Qpi]h[ ∀ri]ϕ) ↔ h[ Qpi](G∗ α ∗ h[ Qri]ϕ) not free in α by (a) above (8) ⊢ h[ Qpi](G∗ α ∗ h[ ∀ri]ϕ) ↔ h[ Qpi](α ∗ h[ Qri]ϕ) by Lemma 7.2.(i), since p is by (4), PL, and Lemma 5.3 ′ (9) ⊢ (α1′ ∗ · · · ∗ αn′ ) ↔ h[ Qpi](α1′ ∗ · · · ∗ αn−1 ∗ h[ Qri]ϕ) by (5)–(8) by PL. ′ ′ Since α1 , . . . , αn−1 , h[ Qri]ϕ are each NWP formulas, and there is one fewer non′ vacuous GQM in α1′ , . . . , αn−1 , h[ Qri]ϕ than in α1′ , . . . , αn′ , the inductive hypoth′ ′ esis implies that α1 ∗ · · · ∗ αn−1 ∗ h[ Qri]ϕ is equivalent to an NWP formula. It ′ ∗ h[ Qri]ϕ) is equivalent to an follows by Lemma 5.3 that h[ Qpi](α1′ ∗ · · · ∗ αn−1 ′ ′ NWP formula, so by (9), α1 ∗ · · · ∗ αn is equivalent to an NWP formula, which means that α1 ∗ · · · ∗ αn is equivalent to an NWP formula. (iii) The proof is by induction on the number of disjuncts in a normal clause. Suppose α is β1 ∨ · · · ∨ βm . If no βk is an NWP formula, then Aα is already an NWP formula. Suppose βm := h[ Qpi]γ is an NWP formula, and let β := β1 ∨· · ·∨βm−1 . Then β is a normal clause, so by the inductive hypothesis, Aβ is equivalent to an NWP formula δ. Now we have: (10) ⊢ α ↔ (β ∨ h[ Qpi]γ) by our assumption of what α is (11) ⊢ Aα ↔ A(β ∨ h[ Qpi]γ) from (10) by Lemma 5.4.(iii) (12) ⊢ A(β ∨ h[ Qpi]γ) ↔ (Aβ ∨ h[ Qpi]γ) (13) ⊢ Aβ ↔ δ by Lemma 7.2.(ii) by the inductive hypothesis (14) ⊢ (Aβ ∨ h[ Qpi]γ) ↔ (δ ∨ h[ Qpi]γ) (15) ⊢ Aα ↔ (δ ∨ h[ Qpi]γ) from (12) and (13) by PL from (11), (12), and (14) by PL. Since δ is an NWP formula, δ ∨[hQpi]γ is a disjunction of NWP formulas. Thus, by part (ii), δ ∨ h[ Qpi]γ is equivalent to an NWP formula, and hence by (15), Aα is equivalent to an NWP formula. (iv) Suppose ϕ is a CNFWP formula α1 ∧ · · · ∧ αn . Hence Aϕ is equivalent to Aα1 ∧· · ·∧Aαn by Lemma 5.4.(ii). Since each αi is a normal clause, part (iii) implies that each Aαi is equivalent to an NWP formula χi . Hence Aα1 ∧· · ·∧Aαn is equivalent to χ1 ∧· · ·∧χn , which by part (ii) is equivalent to an NWP formula. Thus, ϕ is equivalent to an NWP formula. ✷ Our main normal form result is the following. Theorem 7.4 For every ϕ ∈ LGQM : (i) ϕ is provably equivalent to a CNFWP formula; (ii) Aϕ is provably equivalent to an NWP formula. Proof. We prove part (i) by induction on ϕ. The base case for propositional variables is immediate. Suppose ϕ is ¬ψ. By the inductive hypothesis, ψ is equivalent to a CNFWP formula. One then uses de Morgan and distributive laws to show that ¬ψ is also equivalent to a CNFWP formula. Suppose ϕ is ψ1 ∧ψ2 . By the inductive hypothesis, ψ1 and ψ2 are both equivalent to CNFWP Holliday, Litak 15 formulas ψ1′ and ψ2′ . Then ψ1 ∧ψ2 is equivalent to the CNFWP formula ψ1′ ∧ψ2′ . Suppose ϕ is [∀p]ψ. By the inductive hypothesis, ψ is equivalent to a CNFWP formula χ, which implies that Aψ is equivalent to Aχ by Lemma 5.3. Hence [∀p]ψ, which is equivalent to [∀p]Aψ by Lemma 5.4.(ix), is equivalent to [∀p]Aχ by Lemma 5.3. By Lemma 7.3.(iv), Aχ is equivalent to an NWP formula, from which it follows by Lemma 5.3 that [∀p]Aχ is equivalent to an NWP formula. Such a formula is in CNFWP. Suppose ϕ is ✷ψ. By the inductive hypothesis, ψ is equivalent to a CNFWP formula α1 ∧ · · · ∧ αn . We will prove that for any CNFWP formula σ1 ∧ · · · ∧ σk , ✷(σ1 ∧ · · · ∧ σk ) is equivalent to a formula in CNFWP, by induction on the number of GQMs occurring in σ1 ∧ · · · ∧ σk . If no σi contains a disjunct in NWP, then no σi contains a GQM, which means ✷(σ1 ∧ · · · ∧ σk ) is already in CNFWP. So suppose that some σi , say σk , contains as a disjunct an NWP formula h[ Qpi]γ. Hence σk is equivalent to β ∨ h[ Qpi]γ for a normal clause β. Let σ := σ1 ∧ · · · ∧ σk−1 . Thus, ✷(σ1 ∧ · · · ∧ σk ) is equivalent to ✷(σ ∧ (β ∨ h[ Qpi]γ)), which by Lemma 7.2.(iii) is equivalent to (1) ([hQpi]γ ∧ ✷σ) ∨ (¬[hQpi]γ ∧ ✷(σ ∧ β)). Now σ and σ∧β are CNFWP formulas containing fewer GQMs than σ1 ∧· · ·∧σk . Hence by the inductive hypothesis, there are CNFWP formulas χ1 and χ2 such that ✷σ is equivalent to χ1 and ✷(σ ∧ β) is equivalent to χ2 . Thus, ✷(σ1 ∧ · · · ∧ σk ) is equivalent to (2) ([hQpi]γ ∧ χ1 ) ∨ (¬[hQpi]γ ∧ χ2 ). Since h[ Qpi]γ is an NWP formula and χ1 and χ2 are CNFWP formulas, (2) can be transformed into an equivalent CNFWP using distributive laws and the fact that ¬[hQpi]γ is equivalent to the NWP formula h[ Qpi]¬γ. Part (ii) follows from part (i) and Lemma 7.3.(iv). ✷ 7.2 Pure weak prenex forms The following special case of NWP form will be essential in relating LGQM to the first-order language in § 8. Definition 7.5 A formula is in pure weak prenex form (PWP) iff it is of the form h[ Qpi]Gϕ where h[ Qpi] is a sequence of [∀pi ] and h∃pi i GQMs only, G is either A or E, and ϕ is a L✷A -formula. Theorem 7.6 Every NWP formula is provably equivalent to a PWP formula. Proof. By induction on the length of the quantifier prefix. Assuming ϕ is a PWP formula, we must show that h[ Qpi]ϕ is equivalent to a PWP formula. If h[ Qpi] ∈ {[∀p], h∃pi}, there is nothing to do. Case 1: h[ Qpi] := h∀pi. By Lemma 5.4.(ix)-(x), where r is not free in ϕ, h∀piϕ is equivalent to [∀p]h∃riϕ, which is a PWP formula. Case 2: h[ Qpi] := [∃p]. By Lemma 5.4.(ix)-(x), where r is not free in ϕ, [∃p]ϕ is equivalent to h∃pi[∀r]ϕ, which is a PWP formula. ✷ 16 8 One Modal Logic to Rule Them All? Completeness via FO-theory of discriminator BAEs Using the prenex results of § 7, we will now prove the completeness of GQM via mutual translations with the first-order theory of discriminator BAEs. Definition 8.1 A Boolean algebra expansion with a discriminator (BAEA ) is a tuple A = hA, ¬, ∧, ⊥, ⊤, ✷, Ai where hA, ¬, ∧, ⊥, ⊤, ✷i is a BAE and A is the dual form of the unary discriminator term [18], i.e., an algebraic counterpart of the global modality: Aa = ⊤ if a = ⊤, and Aa = ⊥ otherwise. Let FOBAEA (resp. FOBAE ) be the set of first-order formulas in the BAEA (resp. BAE) signature (recycling Prop for our set of first-order variables). The class of all BAEA s is elementary, although it is not exactly a variety (an equationally definable class): rather, it is the class of all simple members of the corresponding variety [18, Thm. 3]. BAEs and BAEA s are in 1-1 correspondence: BAEA s have BAEs as reducts; every BAE A can be trivially extended to a BAEA AA ; and both operations are mutual inverses. In a similar way, we can assign to every formula of FOBAEA a formula equivalent to a PWP formula (where ∼ and & are the negation and conjunction connectives in the first-order language, whereas ¬ and ∧ in the first-order language are function symbols for the Boolean algebraic operations): (ϕ ≈ ψ)∗ := A(ϕ ↔ ψ) (α & β)∗ := ((α)∗ ∧ (β)∗ ) (∼α)∗ := ¬(α)∗ (∀pα)∗ := [∀p](α)∗ . Note that the terms in the FOBAEA formula become formulas of LGQM , with the Boolean function symbols becoming propositional connectives. In the reverse direction, define for each PWP formula: (Aϕ)∗ := ϕ ≈ ⊤ ([∀p]ϕ)∗ := ∀p(ϕ)∗ (Eϕ)∗ := ϕ 6≈ ⊥ (h∃piϕ)∗ := ∃p(ϕ)∗ . Any A or E GQMs inside ϕ become function symbols in the FOBAEA translation. Lemma 8.2 For any nontrivial 6 BAE A, θ : Prop → A, and α ∈ FOBAEA : A, θ  α iff θ̃((α)∗ ) = ⊤ and A, θ 2 α iff θ̃((α)∗ ) = ⊥. Proof. By induction on the complexity of α. The atomic case follows directly from properties of the connective ↔, Lemma 2.6, and the fact that in a nontrivial Boolean algebra, ⊤ and ⊥ are distinct. The Boolean cases follow from the first-order satisfaction definition, the inductive hypothesis, and the algebraic ✷ behavior of ⊤ and ⊥. The GQM case is by Definition 2.5. Corollary 8.3 For any ∆∪{α} ⊆ FOBAEA , ∆ FOBAEA α iff (∆)∗ AGQM (α)∗ . Proof. Immediate from Lemma 8.2 and the definitions of consequence. ✷ Theorem 8.4 (i) For any PWP formula ϕ ∈ LGQM , ϕ ⊢AGQM ((ϕ)∗ )∗ and ((ϕ)∗ )∗ ⊢AGQM ϕ. 6 By a nontrivial BAE, we mean a BAE in which ⊤ 6= ⊥. Holliday, Litak 17 (ii) For any ∆ ∪ {α} ⊆ FOBAEA , ∆ ⊢FOBAEA α implies (∆)∗ ⊢AGQM (α)∗ . (iii) For any ∆ ∪ {α} ⊆ FOBAEA , ∆ ⊢FOBAEA α iff (∆)∗ ⊢AGQM (α)∗ . Proof. For part (i), given h∃piψ := ¬[∀p]¬ψ, we have (for a fresh q): ([hQpi]Aϕ)∗ = Qp∀q(ϕ ≈ ⊤) (([hQpi]Aϕ)∗ )∗ = h[ Qpi]A(ϕ ↔ ⊤) ([Qp]Eϕ)∗ = Qp∃q(ϕ 6≈ ⊥) (([hQpi]Eϕ)∗ )∗ = h[ Qpi]E¬(ϕ ↔ ⊥). It is an easy exercise using Lemmas 5.3 and 5.4 to show that h[ Qpi]Aϕ is GQMequivalent to h[ Qpi]A(ϕ ↔ ⊤) and h[ Qpi]Eϕ to h[ Qpi]E¬(ϕ ↔ ⊥). For part (ii), see Appendix B. Part (iii) is obtained from (ii) by noting that the opposite direction follows from the soundness of GQM (Lemma 5.7), ✷ Corollary 8.3, and the completeness of FOBAEA . An astute reader will note here that even though (·)∗ and (·)∗ are mutual inverses up to equivalence, the matrix of ((α)∗ )∗ consists of a single equation or its negation, including for those α whose matrix is a nontrivial conjunction of disjunctions. This is in keeping with general discriminator theory [34]. Corollary 8.5 (i) For any ∆ ∪ {α} ⊆ FOBAEA , ∆ FOBAEA α iff (∆)∗ ⊢AGQM (α)∗ . (ii) For any set of PWP fomulas Γ ∪ {ϕ} ⊆ LGQM , Γ ⊢AGQM ϕ iff (Γ)∗ FOBAEA (ϕ)∗ . Proof. For part (i), we proceed as follows: ∆ FOBAEA α ⇔ ∆ ⊢FOBAEA α ⇔ (∆)∗ ⊢AGQM by completeness of FOBAEA (ϕ)∗ by Theorem 8.4.(iii). For part (ii), we have: (Γ)∗ FOBAEA (ϕ)∗ ⇔ ((Γ)∗ )∗ ⊢AGQM ((ϕ)∗ )∗ ⇔Γ ⊢AGQM ϕ by part (i) by Theorem 8.4.(i). ✷ Theorem 8.6 (Completeness) For any Γ ∪ {ϕ} ⊆ LGQM , Γ ⊢AGQM ϕ iff Γ AGQM ϕ. Proof. First, as far as the consequence relation ⊢AGQM is concerned, we can prefix all formulas in Γ∪{ϕ} by A (by Lemma 5.6) and then transform them into equivalent PWP formulas (Theorems 7.4.(ii) and 7.6). Corollary 8.5 established that Γ ⊢AGQM ϕ iff (Γ)∗ FOBAEA (ϕ)∗ . By Corollary 8.3, this is equivalent to ✷ ((Γ)∗ )∗ FOBAEA ((ϕ)∗ )∗ . The result then follows by Theorem 8.4.(i). Remark 8.7 The use of Theorem 8.4 in this section should be compared with [6, Thm. 3.7] and [27, Lem. 19]. Our success in establishing the equivalence between the (global) GQM-consequence relation and that of FOBAE means that we can internalize the metatheory of modal logics concisely and in a generic way 18 One Modal Logic to Rule Them All? using “bridge theorems” of abstract algebraic logic [6,1,14]. For lack of space, we are not pursuing this option further in this paper, but a good illustration of how GQM can be used in such a generalization can be found in our recent paper [17, § 9], which in fact led us to the invention of this formalism. Bases of admissible rules (see, e.g., [31]) seem to provide another promising candidate. Details and more examples will be provided in a sequel paper. 9 Conclusions We have seen that GQM provides the sought-after way of viewing “modal logics” as theories relative to one logical system, while also offering a generic and conservative way to enrich any modal logic with propositional quantifiers, including coalgebraic logics (see Remark 2.11). This study led us to new perspectives on the first-order correspondence language for BAEs (§ 8) on the one hand and SOPML on the other hand (§ 3). In the first case, the equivalence between FOBAEA -consequence and ⊢AGQM -consequence illustrates a curious use of techniques from abstract algebraic logic (cf. Remark 8.7) beyond their usual scope. In the second case, we were led to new prenex normal form results. We also believe that focusing on the syntax of GQM and its algebraic semantics can lead to a clarification of philosophical problems concerning propositional quantification, such as Kaplan’s paradox in § 4. Along the way, a number of issues have been postponed to a follow-up paper. In particular, we mentioned that over dual, set-based semantics GQMconsequence may be intractable, as it is over Kripke frames. On the other hand, given that modal logics can be identified with (fragments of) universal GQMtheories, the existence of a rich modal completeness apparatus indicates that for suitable fragments of GQM and formulas of specific syntactic shapes, developing GQM model theory is not hopeless and may yield additional insights in modal logic. This will be a subject of future investigation, as will the systematic internalization of “bridge theorems” mentioned in Remark 8.7. Another issue we have not touched on is that of Gentzen-style proof theory for (well-behaved fragments of) GQM. Since the difficulty of developing Gentzen systems for many modal logics was behind the idea “that the great proliferation of modal logics is an epidemy from which modal logic ought to be cured” [8, p. 25], it would be of interest to see if GQM could help here as well. A further intriguing possibility is that of weakening the classical base of GQM to an intuitionistic one. Just as modal logics are [∀]-universal theories in classical GQM, intermediate (modal) logics could be [∀]-universal theories in intuitionistic GQM. There is a connection here with the origins of modal logic: not only was C. I. Lewis a proponent of propositional quantification in modal logic, as well as perhaps the earliest opponent of modal proliferation, but also he seemed interested in the idea of strict implication on an intuitionistic base (see [30] for discussion). It is argued in [30] that moving Lewis’s strict implication to an intuitionistic base is indeed a conceptually fruitful step. The enrichment of that system with GQM may be the ultimate Lewisian logic. One need not stop at intuitionistic logic. With the power of the global Holliday, Litak 19 quantificational modality, there is the possibility that even vaster swaths of “logics” could become special theories over a generalized version of GQM. Alternatively, the classical base of GQM could be retained, while the connectives of different “logics” are treated as modal operators in BAEs, whose behavior is governed by [∀]-universal GQM formulas. Under one of these approaches, a version of GQM could bring us closer to the idea of “one logic to rule them all.” Acknowledgement We thank Johan van Benthem, Yifeng Ding, Peter Fritz, Lloyd Humberstone, Michael Sammler, and the referees for AiML for their helpful feedback. References [1] Andréka, H., I. Németi and I. Sain, Algebraic logic, in: D. M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, Kluwer, Dordrecht, 2001, 2nd edition pp. 133– 249. [2] Bacon, A., J. Hawthorne and G. Uzquiano, Higher-order free logic and the Prior-Kaplan paradox, Canadian Journal of Philosophy 46 (2016), pp. 493–541. [3] Barcan, R. C., The identity of individuals in a strict functional calculus of second order, The Journal of Symbolic Logic 12 (1947), pp. 12–15. [4] van Benthem, J., “Modal Logic for Open Minds,” CSLI Publications, Stanford, 2010. [5] Blackburn, P., M. de Rijke and Y. Venema, “Modal Logic,” Cambridge Tracts in Theoretical Computer Science 53, Cambridge University Press, Cambridge, 2001. [6] Blok, W. J. and B. Jónsson, Equivalence of consequence operations, Studia Logica 83 (2006), pp. 91–110. [7] Bull, R. A., On modal logic with propositional quantifiers, The Journal of Symbolic Logic 34 (1969), pp. 257–263. [8] Bull, R. A. and K. Segerberg, Basic modal logic, in: D. M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, Vol. 3, Kluwer Academic Publishers, Dordrecht, 2001, 2nd edition pp. 1–82. [9] ten Cate, B., Expressivity of second order propositional modal logic, Journal of Philosophical Logic 35 (2006), pp. 209–223. [10] ten Cate, B. and T. Litak, The importance of being discrete, Technical Report PP-200739, Institute for Logic, Language and Computation, University of Amsterdam (2007). [11] Chagrov, A. V. and M. Zakharyaschev, “Modal Logic,” Oxford Logic Guides, Clarendon Press, Oxford, 1997. [12] Enderton, H. B., “A Mathematical Introduction to Logic,” Harcourt Academic Press, 2001, 2nd edition. [13] Fine, K., Propositional quantifiers in modal logic, Theoria 36 (1970), pp. 336–346. [14] Font, J. M., R. Jansana and D. Pigozzi, A survey of abstract algebraic logic, Studia Logica 74 (2003), pp. 13–97. [15] Holliday, W. H., Possibility frames and forcing for modal logic (February 2018) (2018), UC Berkeley Working Paper in Logic and the Methodology of Science. URL https://escholarship.org/uc/item/0tm6b30q [16] Holliday, W. H., A note on algebraic semantics for S5 with propositional quantifiers, Notre Dame Journal of Formal Logic (Forthcoming). URL https://escholarship.org/uc/item/303338xr [17] Holliday, W. H. and T. Litak, Complete additivity and modal incompleteness, Review of Symbolic Logic (Forthcoming), URL http://www.escholarship.org/uc/item/8pp4d94t. [18] Jipsen, P., Discriminator varieties of Boolean algebras with residuated operators, in: Algebraic Methods in Logic and in Computer Science, Banach Center Publications 28, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1993 pp. 239–252. 20 One Modal Logic to Rule Them All? [19] Kaminski, M. and M. Tiomkin, The expressive power of second-order propositional modal logic, Notre Dame Journal of Formal Logic 37 (1996), pp. 35–43. [20] Kaplan, D., S5 with quantifiable propositional variables, The Journal of Symbolic Logic 35 (1970), p. 355. [21] Kaplan, D., A problem in possible world semantics, in: W. Sinnott-Armstrong, D. Raffman and N. Asher, editors, Modality, morality, and belief: essays in honor of Ruth Barcan Marcus, Cambridge University Press, Cambridge, 1995 pp. 41–52. [22] Kripke, S., A completeness theorem in modal logic, Journal of Symbolic Logic 24 (1959), pp. 1–14. [23] Lewis, C. and C. Langford, “Symbolic Logic,” The Century Company, New York, 1932. [24] Litak, T., Modal incompleteness revisited, Studia Logica 76 (2004), pp. 329–342. [25] Litak, T., “An Algebraic Approach to Incompleteness in Modal Logic,” Ph.D. thesis, Japan Advanced Institute of Science and Technology (2005). [26] Litak, T., On notions of completeness weaker than Kripke completeness, in: R. Schmidt, I. Pratt-Hartmann, M. Reynolds and H. Wansing, editors, Advances in Modal Logic, Vol. 5, College Publications, London, 2005 pp. 149–169. [27] Litak, T., Isomorphism via translation, in: G. Governatori, I. M. Hodkinson and Y. Venema, editors, Advances in Modal Logic, Vol. 6, College Publications, London, 2006 pp. 333–351. [28] Litak, T., Stability of the Blok theorem, Algebra Universalis 58 (2008), pp. 385–411. [29] Litak, T., D. Pattinson, K. Sano and L. Schröder, Model Theory and Proof Theory of Coalgebraic Predicate Logic, Logical Methods in Computer Science 14 (2018). URL https://lmcs.episciences.org/4390 [30] Litak, T. and A. Visser, Lewis meets Brouwer: constructive strict implication, Indagationes Mathematicae 29 (2018), pp. 36–90, a special issue on “L.E.J. Brouwer, fifty years later”. URL https://arxiv.org/abs/1708.02143. [31] Rybakov, V. V., “Admissibility of Logical Inference Rules,” Elsevier, Amsterdam, 1997. [32] Thomason, S. K., Reduction of second-order logic to modal logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21 (1975), pp. 107–114. [33] Vosmaer, J., A new proof of an old incompleteness theorem, Bulletin of the Section of Logic 39 (2010), pp. 199–204. [34] Werner, H., “Discriminator algebras,” Akademie Verlag, Berlin, 1978. A Syntactic Derivations A.1 Proofs of Lemmas 5.2 and 5.3 Lemma A.1 If ⊢ ϕ → ψ, then ⊢ [∀p]ϕ → [∀p]ψ. Proof. By [∀p]-necessitation, distribution, and modus ponens. ✷ Lemma A.2 ⊢ h[ Qpi]ϕ ↔ ¬[hQpi]¬ϕ. Proof. For h[ Qpi] = [∀p], by definition h∃pi := ¬[∀p]¬. For h[ Qpi] = [∃p], the definitions h∀piϕ := [∀p]¬A¬ϕ and [∃p]ϕ := ¬[∀p]¬Aϕ imply ⊢ h∀piϕ ↔ ¬[∃p]¬ϕ. The cases of h[ Qpi] = h∃pi and h[ Qpi] = h∀pi easily follow. ✷ We can now prove Lemma 5.2. Lemma A.3 If q is substitutable for p in ϕ, and q is not free in ϕ, then ⊢ [∀p]ϕ ↔ [∀q]ϕpq . Proof. First, observe that if q is substitutable for p in ϕ, and q is not free in ϕ, then p is substitutable for q in ϕpq , and p is not free in ϕpq . In addition, (ϕpq )qp = ϕ. Thus, to show that the biconditional in the lemma are provable it suffices to show that the left-to-right direction is provable for any q, p, and ϕ satisfying the assumptions. We have: Holliday, Litak (1) ⊢ [∀p]ϕ → Aϕpq by global instantiation since q is substitutable for p in ϕ (2) ⊢ [∀p]ϕ → [∀q]Aϕpq in [∀p]ϕ (3) ⊢ Aϕpq → ϕpq (4) ⊢ [∀q]Aϕpq (5) ⊢ [∀p]ϕ → 21 from (1) by universal generalization since q is not free by instantiation → [∀q]ϕpq [∀q]ϕpq from (3) by Lemma A.1 from (2) and (4) by PL ✷ Lemma A.4 (i) existential generalization: ⊢ [∀r]ϕpψ → [∃p]ϕ where ψ is substitutable for p in ϕ. (ii) existential elimination: if ⊢ (χ ∧ [∀r]ϕpq ) → α, where q is substitutable for p in ϕ, q is not free in χ, ϕ, or α, r 6= q, and r is not free in ϕ, then ⊢ (χ ∧ [∃p]ϕ) → α. Proof. By definition [∃p]ϕ is ¬[∀p]¬[∀s]ϕ for a variable s not free in ϕ. For part (i), suppose s′ is a propositional variable that does not occur free in ϕ or in ψ. Then since ψ is assumed to be substitutable for p in ϕ, it follows that ψ is substitutable for p in [∀s′ ]ϕ. Now we have: (1) ⊢ ¬[∀s]ϕ → ¬[∀s′ ]ϕ by Lemma A.3 (2) ⊢ [∀p]¬[∀s]ϕ → [∀p]¬[∀s′ ]ϕ ′ (3) ⊢ [∀p]¬[∀s ]ϕ → (¬[∀s p in [∀s′ ]ϕ ′ (6) ⊢ [∀r]ϕpψ by instantiation, since ψ is substitutable for from (3), since (¬[∀s′ ]ϕ)pψ = ¬[∀s′ ]ϕpψ . (4) ⊢ [∀p]¬[∀s′ ]ϕ → ¬[∀s′ ]ϕpψ (5) ⊢ [∀r]ϕpψ → [∀s′ ]ϕpψ from (1) by Lemma A.1 ]ϕ)pψ by global instantiation → ¬[∀p]¬[∀s]ϕ from (2)–(5) by PL. For part (ii), we have: (1) ⊢ (χ ∧ [∀r]ϕpq ) → α by assumption (2) ⊢ (χ ∧ ¬α) → ¬[∀r]ϕpq (3) ⊢ ¬[∀r]ϕpq → from (1) by PL [∀r]¬[∀r]ϕpq (4) ⊢ (χ ∧ ¬α) → [∀r]¬[∀r]ϕpq by quantificational 5 axiom from (2)–(3) by PL (5) ⊢ (χ ∧ ¬α) → [∀q][∀r]¬[∀r]ϕpq q is not free in χ or α (6) ⊢ [∀r]¬[∀r]ϕpq → ¬[∀r]ϕpq (7) ⊢ [∀q][∀r]¬[∀r]ϕpq (8) ⊢ (χ ∧ ¬α) → by instantiation → [∀q]¬[∀r]ϕpq [∀q]¬[∀r]ϕpq from (4) by universal generalization, since from (6) by Lemma A.1 from (5) and (7) by PL [∀q]¬[∀r]ϕpq (9) ⊢ → [∀p]¬[∀r]ϕ by Lemma A.3, because q is substitutable for p in ϕ, and q is not free in ϕ (10) ⊢ ¬[∀r]ϕ → ¬[∀s]ϕ by Lemma A.3, since neither r nor s is free in ϕ (11) ⊢ [∀p]¬[∀r]ϕ → [∀p]¬[∀s]ϕ from (10) by Lemma A.1 22 One Modal Logic to Rule Them All? (12) ⊢ (χ ∧ ¬α) → [∀p]¬[∀s]ϕ from (8), (9), and (11) by PL (13) ⊢ (χ ∧ ¬[∀p]¬[∀s]ϕ) → α from (12) by PL. ✷ We can now prove Lemma 5.3, split into two parts for clarity. Lemma A.5 (i) if ⊢ ϕ → ψ, then ⊢ [Qp]ϕ → [Qp]ψ; (ii) if ⊢ ϕ → ψ, then ⊢ hQpiϕ → hQpiψ. Proof. Part (i) with Q := ∀ is Lemma A.1. For part (i) with Q := ∃, we have: (1) ⊢ ϕ → ψ assumption (2) ⊢ [∀r]ϕ → [∀r]ψ for r not free in ϕ or ψ, from (1) by part (i) for Q := ∀ (3) ⊢ [∀r]ψ → [∃p]ψ by existential generalization (4) ⊢ [∀r]ϕ → [∃p]ψ from (2) and (3) by PL (5) ⊢ [∃p]ϕ → [∃p]ψ from (4) by existential elimination, since r is not free in ϕ and p is not free in [∃p]ψ. For (ii), if ⊢ ϕ → ψ, then ⊢ ¬ψ → ¬ϕ, so ⊢ [Qp]¬ψ → [Qp]¬ϕ by (i). Then ⊢ ¬[Qp]¬ϕ → ¬[Qp]¬ψ by PL and hence ⊢ hQpiϕ → hQpiψ by Lemma A.2. ✷ The next lemma will be useful in the appendices to follow. Lemma A.6 (i) ⊢ h[ ∀pi]ϕ → h[ ∃pi]ϕ; (ii) ⊢ h[ ∃ri]ϕ → h[ ∀ri]ϕ if r not free in ϕ; (iii) ⊢ [Qp]ϕ → hQpiϕ. Proof. For part (i), by existential generalization, we have ⊢ [∀p]ϕ → [∃p]ϕ. Then by contraposition with ϕ := ¬ψ, we obtain ⊢ ¬[∃p]¬ψ → ¬[∀p]¬ψ, which implies ⊢ h∀piψ → h∃piψ. For part (ii), from ⊢ [∀r]ϕ → [∀r]ϕ we obtain ⊢ [∃r]ϕ → [∀r]ϕ by existential elimination since r is not free in ϕ. Then by contraposition with ϕ := ¬ψ, we obtain ⊢ ¬[∀r]¬ψ → ¬[∃r]¬ψ, which implies ⊢ h∃riψ → h∀riψ. For part (iii) for Q := ∀, h∀piψ is [∀p]Eψ. By instantiation and contraposition, we have ⊢ ψ → Eψ, so ⊢ [∀p]ψ → [∀p]Eψ and hence ⊢ [∀p]ψ → h∀piψ. For Q := ∃, [∃p]ψ is [∀p]Eψ and h∃piψ is ¬[∀p]¬ψ. Since [∀p]Eψ → Eψ by instantiation and ⊢ [∀p]¬ψ → A¬ψ by global instantiation, we have ⊢ [∀p]Eψ → ¬[∀p]¬ψ by definition of E and PL. Hence ⊢ [∃p]ψ → h∃piψ.✷ A.2 Proof of Lemma 5.4 Lemma A.7 (i) ⊢ A(ϕ → ψ) → (Aϕ → Aψ); (v) ⊢ ϕ → Eϕ; (ii) ⊢ G∗ (ϕ ∗ ψ) ↔ (G∗ ϕ ∗ G∗ ψ); (vi) ⊢ Eϕ ↔ AEϕ; (iii) if ⊢ ϕ → ψ, then ⊢ Gϕ → Gψ; (vii) ⊢ EAϕ ↔ Aϕ; (iv) ⊢ Aϕ → ϕ; (viii) ⊢ GGϕ ↔ Gϕ; Holliday, Litak (ix) ⊢ h[ Qpi]Aψ ↔ [Qp]ψ; (xi) ⊢ h[ Qpi]ψ ↔ A[hQpi]ψ; (x) ⊢ h[ Qpi]Eψ ↔ hQpiψ; (xii) ⊢ h[ Qpi]ψ ↔ E[hQpi]ψ. 23 Proof. The first eight are straightforward: (i) is by distribution. (ii) for ∗ = ∧ follows from part (i) and [∀p]-necessitation using PL. (ii) for ∗ = ∨ is by part (ii) for ∗ = ∧, the definition of E, and PL. (iii) for A is by Lemma A.5.(i). (iii) for E is by part (iii) for A, the definition of E, and PL. (iv) is by instantiation. (v) is by part (iv), the definition of E, and PL. (vi) is by the quantificational 5 axiom and PL for the left-to-right direction and by part (iv) for the right-to-left direction. (vii) is by part (vi), the definition of E, and PL. (viii) follows from parts (iv) and (vii) by standard modal reasoning. For part (ix), first suppose h[ Qpi] = [Qp]. Then the left-to-right direction is by part (iv) and then Lemma A.5.(i). For the right-to-left direction with Q = ∀, we have ⊢ [∀p]ψ → Aψ by global instantiation and hence ⊢ [∀p]ψ → [∀p]Aψ by universal generalization since p is not free in the antecedent. The right-to-left direction with Q = ∃ follows from the definition of [∃p]ψ as ¬[∀p]¬Aψ, part (viii), and Lemma A.5.(i). Now suppose that h[ Qpi] = hQpi. If Q = ∀, then we must show that ⊢ h∀piAψ ↔ [∀p]ψ. The right-to-left direction follows from ⊢ [∀p]ψ → [∀p]Aψ, established above, and Lemma A.6.(iii). From left to right, by definition h∀piAψ is [∀p]EAψ, which by parts (vii) and (iv) and Lemma A.5.(i) entails [∀p]ψ. Finally, suppose that Q = ∃. Since h∃pi is by definition ¬[∀p]¬ and [∃p] is ¬[∀p]¬A, we have ⊢ h∃piAψ ↔ [∃p]ψ. For part (x), first suppose that h[ Qpi] = hQpi. Then the right-to-left direction is by part (v) and Lemma A.5.(ii). Since h∃pi is defined as ¬[∀p]¬ and E as ¬A¬, the left-to-right direction for Q = ∃ follows from part (ix) and PL. For the left-to-right direction with Q = ∀, by definition h∀piEψ is [∀p]EEψ, which by part (v) and Lemma A.5.(i) entails [∀p]Eψ, which by definition h∀piψ. Now suppose h[ Qpi] = [Qp]. If Q = ∀, then by the definition of h∀piψ we have ⊢ [∀p]Eψ ↔ h∀piψ. If Q = ∃, then we must show ⊢ [∃p]Eψ ↔ h∃piψ. By definition, [∃p]Eψ is h∃piAEψ, which by parts (vi) and Lemma A.5.(ii) is equivalent to h∃piEψ, which we showed above to be equivalent to h∃piψ. For part (xi), the right-to-left direction is by part (iv). For the left-to-right direction in the case of [∀p], we have: (1) ⊢ [∀p]ψ → [∀p]ψ propositional tautology (2) ⊢ [∀p]ψ → A[∀p]ψ from (1) by universal generalization, since the variable in A is assumed not to be free in [∀p]. In the case of [∃p], the quantificational 5 axiom gives ⊢ ¬[∀p]¬Aψ → A¬[∀p]¬Aψ, which by definition of [∃p] yields ⊢ [∃p]ψ → A[∃p]ψ. For the left-to-right direction of part (xi) in the case of hQpi, we have: (3) ⊢ [Qp]¬ψ → A[Qp]¬ψ from part (xi) for [Qp] as above 24 One Modal Logic to Rule Them All? (4) ⊢ E[Qp]¬ψ → EA[Qp]¬ψ (5) ⊢ E[Qp]¬ψ → [Qp]¬ψ (6) ⊢ hQpiψ → AhQpiψ from (3) by part (iii) of the lemma from (4) by parts (vii) and (iv) of the lemma from (5) by PL and the definition of hQpi. Part (xii) follows from part (xi) by definition of E and PL. ✷ A.3 Proof of Lemma 7.2 To prove Lemma 7.2.(i), we begin with the following cases. Lemma A.8 If p is not free in α, then: (i) ⊢ (Aα ∧ [∀p]β) ↔ [∀p](α ∧ β); (ii) ⊢ (Aα ∧ [∃p]β) ↔ [∃p](α ∧ β); (iii) ⊢ (Eα ∨ [∀p]β) ↔ [∀p](Eα ∨ β); (iv) ⊢ (Eα ∨ [∃p]β) ↔ [∃p](Eα ∨ β). Proof. For the left-to-right direction of part (i), we have: (1) ⊢ [∀p]β → Aβ by global instantiation (2) ⊢ (Aα ∧ [∀p]β) → (Aα ∧ Aβ) (3) ⊢ (Aα ∧ [∀p]β) → A(α ∧ β) from (1) by PL from (2) by Lemma A.7.(ii) and PL (4) ⊢ (Aα ∧ [∀p]β) → [∀p]A(α ∧ β) from (3) by universal generalization, since p is not free in Aα ∧ [∀p]β (5) ⊢ (Aα ∧ [∀p]β) → [∀p](α ∧ β) from (4) by Lemma A.7.(ix). For the right-to-left direction of part (i), we have: (6) ⊢ (α ∧ β) → α propositional tautology (7) ⊢ [∀p](α ∧ β) → [∀p]α (8) ⊢ [∀p]α → Aα by global instantiation (9) ⊢ [∀p](α ∧ β) → Aα (10) ⊢ (α ∧ β) → β from (6) by Lemma A.5.(i) from (7) and (8) by PL propositional tautology (11) ⊢ [∀p](α ∧ β) → [∀p]β from (10) by Lemma A.5.(i) (12) ⊢ [∀p](α ∧ β) → (Aα ∧ [∀p]β) from (9) and (11) by PL. For the left-to-right direction of part (ii), we have: (13) ⊢ (Aα ∧ Aβ) → A(α ∧ β) by Lemma A.7.(ii) (14) ⊢ A(α ∧ β) → [∃p](α ∧ β) by existential generalization (15) ⊢ (Aα ∧ Aβ) → [∃p](α ∧ β) from (13) and (14) by PL (16) ⊢ (Aα ∧ [∃p]β) → [∃p](α ∧ β) from (15) by existential elimination since p is not free in α or [∃p](α ∧ β). For the right-to-left direction of part (ii), we have: (17) ⊢ (α ∧ β) → α propositional tautology (18) ⊢ [∃p](α ∧ β) → [∃p]α from (17) by Lemma A.5.(i) Holliday, Litak 25 (19) ⊢ [∃p]α → Aα by Lemma A.6.(ii) since p is not free in α (20) ⊢ (α ∧ β) → β propositional tautology (21) ⊢ [∃p](α ∧ β) → [∃p]β from (20) by Lemma A.5.(i) (22) ⊢ [∃p](α ∧ β) → (Aα ∧ [∃p]β) from (18), (19), and (21) by PL. For the left-to-right direction of part (iii), we have: (23) ⊢ Eα → (Eα ∨ β) propositional tautology (24) ⊢ AEα → A(Eα ∨ β) (25) ⊢ Eα → A(Eα ∨ β) from (23) by Lemma A.5.(i) from (24) by Lemma A.7.(vi) and PL (26) ⊢ Eα → [∀p]A(Eα ∨ β) free in Eα (27) ⊢ Eα → [∀p](Eα ∨ β) (28) ⊢ β → (Eα ∨ β) from (25) by universal generalization, since p is not from (26) by Lemma A.7.(ix) and PL propositional tautology (29) ⊢ [∀p]β → [∀p](Eα ∨ β) from (28) by Lemma A.5.(i) (30) ⊢ (Eα ∨ [∀p]β) → [∀p](Eα ∨ β) from (27) and (29) by PL. For the right-to-left direction of part (iii), we have: (31) ⊢ [∀p](A¬α → β) → ([∀p]A¬α → [∀p]β) by distribution (32) ⊢ A¬α → [∀p]A¬α by universal generalization, since p is not free in A¬α (33) ⊢ [∀p](A¬α → β) → (A¬α → [∀p]β) (34) ⊢ [∀p](Eα ∨ β) → (Eα ∨ [∀p]β) from (31) and (32) by PL from (33) by definition of E and PL. For the left-to-right direction of part (iv), we have: (35) ⊢ Eα → AEα from Lemma A.7.(vi) by definition of E and PL (36) ⊢ AEα → [∃p]Eα (37) ⊢ Eα → [∃p]Eα by existential generalization from (35) and (36) by PL (38) ⊢ Eα → (Eα ∨ β) propositional tautology from (38) by Lemma A.5.(i) (39) ⊢ [∃p]Eα → [∃p](Eα ∨ β) (40) ⊢ Eα → [∃p](Eα ∨ β) (41) ⊢ β → (Eα ∨ β) from (37) and (39) by PL propositional tautology (42) ⊢ [∃p]β → [∃p](Eα ∨ β) from (41) by Lemma A.5.(i) (43) ⊢ (Eα ∨ [∃p]β) → [∃p](Eα ∨ β) from (40) and (42) by PL. For the right-to-left direction of part (iv), we have: (44) ⊢ A(¬Eα → β) → (A¬Eα → Aβ) (45) ⊢ A(Eα ∨ β) → (¬A¬Eα ∨ Aβ) by Lemma A.7.(i) from (44) by PL (46) ⊢ A(Eα ∨ β) → (EEα ∨ Aβ) from (45) by definition of E and PL (47) ⊢ A(Eα ∨ β) → (Eα ∨ Aβ) from (46) by Lemma A.7.(viii) and PL (48) ⊢ Aβ → [∃p]β by existential generalization 26 One Modal Logic to Rule Them All? (49) ⊢ A(Eα ∨ β) → (Eα ∨ [∃p]β) from (47) and (48) by PL (50) ⊢ [∃p](Eα ∨ β) → (Eα ∨ [∃p]β) is not free in (Eα ∨ [∃p]β). from (49) by existential elimination since p ✷ Lemma A.8 can be succinctly summarized as follows. Lemma A.9 If p is not free in α, then: (i) ⊢ (Aα ∧ [Qp]β) ↔ [Qp](α ∧ β); (ii) ⊢ (Eα ∨ [Qp]β) ↔ [Qp](Eα ∨ β). Lemma A.10 If p is not free in α, then: (i) ⊢ (Eα ∨ hQpiβ) ↔ hQpi(α ∨ β); (ii) ⊢ (Aα ∧ hQpiβ) ↔ hQpi(Aα ∧ β). Proof. From Lemma A.9 by PL and Lemma A.2. ✷ Proposition A.11 If p is not free in α, then: (i) ⊢ (Aα ∧ h[ Qpi]β) ↔ h[ Qpi](Aα ∧ β); (ii) ⊢ (Eα ∨ h[ Qpi]β) ↔ h[ Qpi](Eα ∨ β). Proof. Part (i) is given by Lemma A.9.(i), using the fact from Lemma A.7.(viii) that ⊢ Aα ↔ AAα, and Lemma A.10.(ii). Part (ii) is given by Lemmas A.9.(ii) and A.10.(i), using the fact from Lemma A.7.(viii) that ⊢ Eα ↔ EEα.✷ Proposition A.11 can be succinctly summarized by the following statement, which is Lemma 7.2.(i). Proposition A.12 ⊢ (G∗ α ∗ h[ Qpi]β) ↔ h[ Qpi](G∗ α ∗ β) where p is not free in α. Next we prove Lemma 7.2.(ii). Proposition A.13 ⊢ A(ϕ ∨ h[ Qpi]ψ) ↔ (Aϕ ∨ h[ Qpi]ψ). Proof. For the left-to-right direction, we have: (1) ⊢ E[hQpi]ψ ∨ ¬E[hQpi]ψ (2) ⊢ E[hQpi]ψ → h[ Qpi]ψ propositional tautology by Lemma A.7.(xii) (3) ⊢ A(¬[hQpi]ψ → ϕ) → (A¬[hQpi]ψ → Aϕ) (4) ⊢ (ϕ ∨ h[ Qpi]ψ) → (¬[hQpi]ψ → ϕ) by Lemma A.7.(i) propositional tautology (5) ⊢ A(ϕ ∨ h[ Qpi]ψ) → A(¬[hQpi]ψ → ϕ) from (4) by Lemma A.7.(iii) (6) ⊢ (A(ϕ ∨ h[ Qpi]ψ) ∧ ¬E[hQpi]ψ) → Aϕ from (5) and (3) by PL (7) ⊢ A(ϕ ∨ h[ Qpi]ψ) → (Aϕ ∨ h[ Qpi]ψ) from (1), (2), and (6) by PL. For the right-to-left direction, we have: (8) ⊢ ϕ → (ϕ ∨ h[ Qpi]ψ) propositional tautology from (8) by Lemma A.7.(iii) (9) ⊢ Aϕ → A(ϕ ∨ h[ Qpi]ψ) (10) ⊢ h[ Qpi]ψ → A[hQpi]ψ by Lemma A.7.(xi) (11) ⊢ h[ Qpi]ψ → (ϕ ∨ h[ Qpi]ψ) propositional tautology Holliday, Litak (12) ⊢ A[hQpi]ψ → A(ϕ ∨ h[ Qpi]ψ) 27 from (11) by Lemma A.7.(iii) (13) ⊢ (Aϕ ∨ h[ Qpi]ψ) → A(ϕ ∨ h[ Qpi]ψ) from (9), (10), and (12) by PL. ✷ Finally, we prove Lemma 7.2.(iii). Proposition A.14 ⊢ ✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (([hQpi]ψ ∧ ✷α) ∨ (¬[hQpi]ψ ∧ ✷(α ∧ ϕ))). Proof. First, we have: (1) ⊢ h[ Qpi]ψ → A[hQpi]ψ by Lemma A.7.(xi) (2) ⊢ h[ Qpi]ψ → ((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ α) propositional tautology (3) ⊢ A[hQpi]ψ → A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ α) (4) ⊢ h[ Qpi]ψ → A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ α) from (2) by Lemma A.7.(iii) from (1) and (3) by PL (5) ⊢ A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ α) → (✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ ✷α) congruence (6) ⊢ ¬[hQpi]ψ → A¬[hQpi]ψ by ✷- by Lemma A.7.(xii), definition of E, and PL (7) ⊢ ¬[hQpi]ψ → ((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (α ∧ ϕ)) propositional tautology (8) ⊢ A¬[hQpi]ψ → A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (α ∧ ϕ)) A.7.(iii) (9) ⊢ ¬[hQpi]ψ → A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (α ∧ ϕ)) from (7) by Lemma from (6) and (8) by PL (10) ⊢ A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (α ∧ ϕ)) → (✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ ✷(α ∧ ϕ)) by ✷-congruence. Now for the left-to-right direction of the formula to be proved, we have: (11) ⊢ h[ Qpi]ψ ∨ ¬[hQpi]ψ propositional tautology (12) ⊢ h[ Qpi]ψ → ([hQpi]ψ ∧ A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ α)) from (4) by PL (13) ⊢ ¬[hQpi]ψ → (¬[hQpi]ψ ∧ A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (α ∧ ϕ))) from (9) by PL (14) ⊢ ([hQpi]ψ ∧ A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ α)) ∨ (¬[hQpi]ψ ∧ A((α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ (α ∧ ϕ))) from (11), (12), and (13) by PL (15) ⊢ ([hQpi]ψ ∧ (✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ ✷α)) ∨ (¬[hQpi]ψ ∧ (✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) ↔ ✷(α ∧ ϕ))) from (14), (5), and (10) by PL (16) ⊢ ✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) → (([hQpi]ψ ∧ ✷α) ∨ (¬[hQpi]ψ ∧ ✷(α ∧ ϕ))) by PL. from (15) For the right-to-left direction, we have: (17) ⊢ ([hQpi]ψ ∧ ✷α) → ✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) from (4) and (5) by PL (18) ⊢ (¬[hQpi]ψ ∧ ✷(α ∧ ϕ)) → ✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) from (9) and (10) by PL (19) ⊢ (([hQpi]ψ ∧ ✷α) ∨ (¬[hQpi]ψ ∧ ✷(α ∧ ϕ))) → ✷(α ∧ (ϕ ∨ h[ Qpi]ψ)) and (18) by PL. from (17) ✷ 28 One Modal Logic to Rule Them All? Proof of Theorem 8.4.(ii) B In order to prove Theorem 8.4.(ii), we first recall the needed first-order apparatus, for which we follow Enderton [12]. A generalization of a first-order formula ϕ is any formula of the form ∀p1 . . . ∀pn ϕ for n ≥ 0. Enderton takes as axioms all generalizations of the following: • all substitution instances of propositional tautologies; • ∀pϕ → ϕpt where the term t is substitutable for p in ϕ; • ∀p(ϕ → ψ) → (∀pϕ → ∀pψ); • ϕ → ∀pϕ where p does not occur free in ϕ; • p ≈ p, and p ≈ q → (ϕ → ϕ′ ) where ϕ is atomic and ϕ′ is obtained from ϕ by replacing p in zero or more places by q. In addition, we add all generalizations of the following axioms for the elementary theory of nontrivial discriminator BAEs: • first-order axioms of Boolean algebras; • ∀p((p ≈ ⊤ & Ap ≈ ⊤) OR (p 6≈ ⊤ & Ap ≈ ⊥)) and ⊤ 6≈ ⊥. Let Γ ⊢FOBAEA ϕ iff ϕ belongs to the smallest set of FOBAEA formulas that includes all the axioms above, is closed under modus ponens, and includes Γ. Lemma B.1 For any ϕ ∈ FOBAEA , term t and variable p: (i) if p is not free in ϕ, then p is not free in (ϕ)∗ ; (ii) if t is substitutable for p in ϕ, then t is substitutable for p in (ϕ)∗ ; (iii) (ϕpt )∗ = ((ϕ)∗ )pt . Lemma B.2 For every ϕ ∈ FOBAEA , ⊢GQM (ϕ)∗ ↔ A(ϕ)∗ . Proof. A straightforward induction using Lemma 5.4. ✷ We are now ready to prove Theorem 8.4.(ii): for any ∆ ∪ {α} ⊆ FOBAEA , ∆ ⊢FOBAEA α implies (∆)∗ ⊢AGQM (α)∗ . Proof. The proof is by induction on the length of ⊢FOBAEA proofs. We first check that the translation of each axiom is a theorem of GQM. Since GQM has the [∀]-necessitation rule that if ⊢GQM ϕ, then ⊢GQM [∀p]ϕ, it suffices to check that each of the ungeneralized axioms translates to a theorem of GQM: • The translation of any propositional tautology is clearly also a propositional tautology. • By Lemma B.1.(iii), (∀pϕ → ϕpt )∗ = [∀p](ϕ)∗ → ((ϕ)∗ )pt , which by Lemma B.1.(ii) is an instance of instantiation. • (∀p(ϕ → ψ) → (∀pϕ → ∀pψ))∗ = [∀p]((ϕ)∗ → (ψ)∗ ) → ([∀p](ϕ)∗ → [∀p](ψ)∗ ), which is an instance of distribution. • (ϕ → ∀pϕ)∗ = (ϕ)∗ → [∀p](ϕ)∗ , and we have ⊢ (ϕ)∗ → A(ϕ)∗ by Lemma B.2 and hence ⊢ (ϕ)∗ → [∀p](ϕ)∗ by Lemma 5.2 since p is not free in (ϕ)∗ (by Lemma B.1). Holliday, Litak 29 • (p ≈ p)∗ := A(p ↔ p), which is obtained from the tautology p ↔ p by [∀]-necessitation. • (p ≈ q → (ϕ → ϕ′ ))∗ = A(p ↔ q) → ((ϕ)∗ → ((ϕ)∗ )′ ) where ((ϕ)∗ )′ is obtained from (ϕ)∗ by replacing the appropriate occurrences of p by q. Proving that ⊢GQM A(p ↔ q) → ((ϕ)∗ → ((ϕ)∗ )′ ) is routine. • The translation of any axiom of Boolean algebra is clearly derivable in GQM using PL and [∀]-necessitation. • (∀p((p ≈ ⊤ & Ap ≈ ⊤) OR (p 6≈ ⊤ & Ap ≈ ⊥)))∗ is
[∀p] (A(p ↔ ⊤) ∧ A(Ap ↔ ⊤)) ∨ (¬A(p ↔ ⊤) ∧ A(Ap ↔ ⊥)) , which is straightforward to derive using Lemma 5.4 and [∀]-necessitation. • (⊤ = 6 ⊥)∗ = ¬A(⊤ ↔ ⊥), which is derivable by instantiation and PL. Finally, any application of modus ponens for ⊢FOBAEA can be matched— using the inductive hypothesis—by an application of modus ponens for ⊢AGQM . ✷