Lewis meets {B}rouwer: {C}onstructive strict implication

Lewis meets Brouwer: constructive strict implication Tadeusz Litaka , Albert Visserb arXiv:1708.02143v3 [cs.LO] 29 Oct 2017 a Informatik 8, FAU Erlangen-Nürnberg, Martensstraße 3, 91058 Erlangen [email protected] b Philosophy, Faculty of Humanities, Utrecht University, Janskerkhof 13, 3512BL Utrecht [email protected] Abstract C. I. Lewis invented modern modal logic as a theory of “strict implication” J. Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than ✷ and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the “strength” axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction has been discovered by the functional programming community in their study of “arrows” as contrasted with “idioms”. Our particular focus is on arithmetical interpretations of intuitionistic J in terms of preservativity in extensions of HA, i.e., Heyting’s Arithmetic. Contents 1 Introduction 2 2 The rise and fall of the house of Lewis 2.1 “The error of philosophers” . . . . . . . . . . . . . . . . . . . . . 2.2 Could Brouwerian inspiration help Lewis’ systems? . . . . . . . . 4 4 8 3 Strict implication in intuitionistic Kripke semantics 9 4 Axiomatizations 12 4.1 A fistful of logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 An armful of derivations . . . . . . . . . . . . . . . . . . . . . . . 16 5 Arithmetical interpretations: provability and preservativity « 5.1 Schematic logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Propositional logics of a theory . . . . . . . . . . . . . . . . . . . 5.3 Provability Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Preservativity Logic . . . . . . . . . . . . . . . . . . . . . . . . . 20 20 21 21 23 A draft of our contribution to the collection “L.E.J. Brouwer, 50 years later” October 31, 2017 6 Kripke completeness and correspondence ? 28 6.1 Notions of completeness . . . . . . . . . . . . . . . . . . . . . . . 29 6.2 Completeness and correspondence results . . . . . . . . . . . . . 31 6.3 Non-derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7 Strength: arrows, monads, idioms and 7.1 Notions of computation and arrows . . 7.2 Modalities for guarded (co)recursion . 7.3 Intuitionistic epistemic logic . . . . . . | 36 guards . . . . . . . . . . . . . . . 36 . . . . . . . . . . . . . . . 39 . . . . . . . . . . . . . . . 40 » 41 8 Applications of preservativity 8.1 NNIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8.2 On the falsity of Tertium non Datur . . . . . . . . . . . . . . . . 42 9 Conclusions 45 A A recap of realizability – 55 – 56 B Π01 -conservativity B.1 The classical case . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 B.2 The constructive case . . . . . . . . . . . . . . . . . . . . . . . . 57 C Interpretability – C.1 Basics . . . . . . . . . . . . . . . . C.2 Interpretability Logic introduced . C.3 Classical Interpretability Logic . . C.4 Constructive Interpretability Logic = D The problem of the Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 58 59 59 60 65 1. Introduction More is possible in the constructive realm than is dreamt of in classical philosophy. For example, we have nilpotent infinitesimals ([MR13]) and the categoricity of weak first-order theories of arithmetic ([McC88], [McC91], this paper Appendix C.4.2). We zoom in on one such possibility: the original modal connective of “strict implication” J proposed by C. I. Lewis [Lew18; LL32], and hence called here the Lewis arrow, does not reduce to the unary box ✷ over constructive logic. This simple insight opens the doors for a plethora of new intuitionistic modal logics that cannot be understood solely in terms of the box. To the best of our knowledge, this observation was originally made in the area of preservativity logic [Vis85; Vis94; Iem03; IDZ05] and metatheory of arithmetic provides perhaps the most interesting applications of intuitionistic J. However, one can claim that a similar discovery has been independently made in the study of functional programming in computer science (cf. § 7.1). 2 We begin in § 2 by recalling Lewis’ invention of strict implication, mostly remembered by historians; these days, modal logic is almost by default taken to be the theory of boxes and diamonds. After sketching how J fell into disuse and neglect, we speculate whether removing the law of excluded middle could have saved Lewis’ vision of modal logic. This is also a good opportunity to highlight some unexpected analogies between the fates of Brouwer’s and Lewis’ projects. In § 3, we clarify how the intuitionistic distinction between φ J ψ and ✷(φ → ψ) is reflected in Kripke semantics. This may well prove the most natural way of introducing this connective for many readers. In § 4, we present the minimal deduction system1 iA and numerous additional principles used in the remainder in the paper. In § 4.2, we clarify connections between them, i.e., the inclusion relation between corresponding logics. With the syntactic apparatus ready, we turn in § 5 to a major motivation for the study of J: logics of Σ01 -preservativity of arithmetical theories as contrasted with more standard logics of provability. In order to provide an umbrella notion for the study of arithmetical interpretations of modal connectives, we begin this section by setting up a general framework for schematic logics, which may prove of interest in its own right. In § 6, we are finally tying together the semantic setup of § 3 and the syntactic infrastructure of § 4 by providing a discussion of completeness and correspondence results. Some of them are well-known, others are new. Having a complete semantics for the logics under consideration allows us in § 6.3 to complement earlier syntactic derivations (given in § 4.2) with examples of non-derivations. In § 7, we are presenting other applications of strong arrows and strong boxes. In fact, what we call here “strong arrows” turns out to correspond directly to “arrows” in functional programming. We are also briefly discussing connections with logics of guarded (co)recursion and intuitionistic logics of knowledge. But while intuitionistic J can be (re)discovered in areas ranging from computer science to philosophy, in our view arithmetical interpretations are most developed and interesting. Thus, in § 8 we return to the theme of § 5 presenting some applications of the logic of preservativity. In § 8.1 we discuss the application of preservativity to the study of the provability logic of Heyting Arithmetic HA. In § 8.2, we show that preservativity allows a more satisfactory expression of the failure of Tertium non Datur. The paper has several appendices that offer some supporting material. Appendix A collects basic facts about realizability needed in other sections. In Appendices B and C, we provide some basic insights in Π01 -conservativity logics and interpretability logics. These insights strengthen our understanding of preservativity logic both by extending this understanding and by offering a contrast to this understanding. Finally, Appendix D discusses the collapse of J in Lewis’ first monograph, i.e., A Survey of Symbolic Logic [Lew18] from the perspective of our deductive systems. 1 It was baptised “iP” by Iemhoff and coauthors [Iem01b; Iem03; IDZ05], but this acronym ties J too tightly to preservativity. 3 Of course, we are of the opinion that the reader should carefully study everything we put in the paper. However, we realize that this expectation is not realistic. For this reason, we present several roadmaps through the paper.2 The basic option is to read §§ 2–4 to get the basics of motivational background, the Kripke semantics and an impression of possible reasoning systems. ? The reader who wants more solid treatment of Kripke semantics can extend the basic option with § 6. | The computer science package consists of the basic option and § 7. = The reader who wants to go somewhat more deeply into the history of the subject can extend the basic option with Appendix D. « The reader who wants to understand the basics of arithmetical interpretations can extend the basic option with § 5. » An extended package for arithmetical interpretations combines « with § 8. – The full arithmetical package extends » with Appendices A, B and C. 2. The rise and fall of the house of Lewis 2.1. “The error of philosophers” We are reflecting on L.E.J. Brouwer’s heritage half a century after his passing. Given his negative views on the rôle of logic and formalisms in mathematics, it seems somewhat paradoxical that these days the name of intuitionism survives mostly in the context of intuitionistic logic.3 One is reminded in this context of what Nietzsche called the error of philosophers: The philosopher believes that the value of his philosophy lies in the whole, in the structure. Posterity finds it in the stone with which he built and with which, from that time forth, men will build oftener and better—in other words, in the fact that the structure may be destroyed and yet have value as material.4 2 Note also that reading the electronic version may sometimes prove easier due to omnipresent hyperlinks: apart from all the usually clickable entities (citations or numbers of (sub)sections, footnotes and table- or theorem-like environments . . . ), even most names of logical systems can be clicked upon to retrieve their definition in Tables 4.1 and 4.2. When reading a hardcopy, we advise keeping these Tables handy, perhaps jointly with Figure 6.2. 3 A related and better-known paradox is that Brouwer’s own name survives in mainstream mathematics mostly in connection with his work on topology, which is confirmed by several contributions in this collection. This despite the fact that he rejected these results on philosophical grounds and was actively involved in topological research only for the period necessary to secure academic recognition and international status. Moreover, it seems a myth that the non-constructive character of his most famous topological publications turned Brouwer into an intuitionist. There is ample evidence that while the exact form of his intuitionism evolved somewhat, his philosophical beliefs predate these results. Cf. van Stigt [van90b] for a detailed discussion of all these points. 4 Human, All-Too-Human, Part II, translated by Paul V. Cohn. 4 We feel thus excused to focus on propositional logics based on the intuitionistic propositional calculus (IPC). More specifically, our interest lies in an intuitionistic take on a formal language developed by an author nearly perfectly contemporary with Brouwer: Clarence Irving Lewis5 , the father of modern modal logic. And this time, the reason for this does not come from the well-known Gödel(-McKinsey-Tarski) translation of IPC into the system Lewis denoted as S4, which is discussed elsewhere in this collection. One can also see a certain irony in the fate of Lewis’ systems. They were explicitly designed to give an account of “strict implication” J. The unary ✷ can be introduced using ✷φ ↔ (⊤ J φ). (1) In fact, Lewis designed J and ✷ as mutually definable,6 setting φ J ψ := ✷(φ → ψ) (2) and over subsequent decades, modal logic in a narrow sense turned into the theory of unary ✷ and/or ✸. In a broader sense, pretty much any intensional operator extending the usual supply of connectives can be called a modality. Modalities came to represent not only necessity, but also arithmetical provability, knowledge, belief, obligation, and various forms of guarded quantification: validity after all possible program executions, in all accessible states, in all future time instants or at every point in an open neighbourhood (the list, of course, is far from being exhaustive). Just like in the case of intuitionistic logic, a wide range of semantics for modalities have been investigated, the most prominent being the Kripke semantics (relational structures), but also topologies, coalgebras, monoidal endofunctors on categories or more recent “possibility semantics”. Thus, Lewis’ dissatisfaction with material or extensional implication and disjunction, expressed first in a short 1912 article [Lew12], has ultimately led to the spectacular success story of modal logic, much like Brouwer’s7 dissatisfaction with non-constructive usage of implication and disjunction has ultimately led to the spectacular success story of intuitionistic logic. And yet, while Lewis did 5 He was born two years later than Brouwer and died two years earlier. be precise, in his books Lewis did not use ✷ as a primitive. His exact formulation of φ J ψ was ¬✸(φ ∧ ¬ψ). However, in the classical setting, this one is obviously equivalent to the one given by (2), and the reliance of Lewis’ formulation on involutive negation would be a major problem over IPC. See Appendix D for a more detailed examination of the rôle of involutive/classical negation in Lewis’ original system. 7 Speaking of Brouwer, note again the parallelism of dates: 1912, the year when Lewis fired his first shots for intensional connectives by publishing Implication and the Algebra of Logic [Lew12], is also the year when Brouwer obtained his position at the University of Amsterdam, was elected to the Royal Netherlands Academy of Arts and Sciences, delivered his famous inaugural address Intuitionism and Formalism and became liberated to pursue his own program. We refrain here from investigating further analogies, such as the fact that Lewis wrote his 1910 PhD on The Place of Intuition in Knowledge (cf. Murphey [Mur05, Ch. 1] for an extended discussion), that he had a solid background in idealism and Kant and that he remained under strong influence of these philosophical positions throughout his career. 6 To 5 not write much on formal logic after Symbolic Logic 8 published in 1932 [LL32], his occasional remarks do not suggest he would approve of the scattering of his Strict Implication systems into a bewildering galaxy of unimodal calculi. Indeed, he was not only opposed to the very name modal logic, but believed that his formalisms is the exact opposite of real “modal” logic, which in his view was . . . the extensional system of Principia Mathematica: There is a logic restricted to indicatives; the truth-value logic most impressively developed in Principia Mathematica. But those who adhere to it usually have thought of it—so far as they understood what they were doing—as being the universal logic of propositions which is independent of mode. And when that universal logic was first formulated in exact terms, they failed to recognize it as the only logic which is independent of the mode in which propositions are entertained and dubbed it “modal logic”. (Cf. [Mur05, p. 203]) His own belief was that the relation of strict implication expresses precisely that relation which holds when valid deduction is possible [emphasis ours]. It fails to hold when valid deduction is not possible. In that sense, the system of Strict Implication may be said to provide that canon and critique of deductive inference which is the desideratum of logical investigation [LL32, p. 247] and that Strict Implication explains the paradoxes incident to truth-implication. [LL32, p. 247] While the failure of Lewis’ systems to conquer this intended territory had to do with philosophical prejudices of the following decades, they were also simply less suited for these purposes than Lewis thought. The original system of A Survey of Symbolic Logic in 1918 [Lew18]—stemming back to a 1914 paper [Lew14]— was plagued by a number of issues, the most famous one pointed out by Post: the combination of an axiom equivalent to (in an updated notation) (✷φ J ✷ψ) J (¬ψ J ¬φ) with other axioms and classical negation laws trivialized the modality and collapsed strict implication to material implication [Lew20]. We provide an extended analysis of Lewis’ SSL problem in Appendix D; we believe it is an interesting application of the intuitionistic theory of J discussed in this paper.9 In 8 Symbolic Logic was a collaboration between C. I. Lewis and C. H. Langford. The authors, however, made it clear in the preface who wrote and is “ultimately responsible” for which chapter, a practice rather uncommon today. All the passages quoted in this paper come from chapters written by Lewis. As Murray G. Murphey says in his monograph on C. I. Lewis: “Symbolic Logic was less a cooperative venue than a coauthored book . . . To what extent each advised the other on their separate chapters is left unclear, but probably there was not much of an attempt to harmonize . . . Langford’s theory of propositions, for example, in Chapter IX is clearly not Lewis’s theory.” [Mur05, p.183]. 9 Cf. also the discussion by Murphey [Mur05, pp. 101–102] or Parry [Par70]. 6 Symbolic Logic [LL32]—more precisely, in its famous Appendix II—Lewis was more cautious, creating several “lines of retreat” (as Parry [Par70] described it) in the form of S3, S2 and S1. At least on the technical front, this time things went better. Immediate polemics focused on possibility of definability of intensional connectives in extensional systems, but none of the authors involved proposed anything resembling what we much later came to know as the Standard Translation of modal logic into predicate logic.10 There were, however, subtler problems, pointed out in in the post-war period by Ruth Barcan Marcus:11 it is plausible to maintain that if strict implication is intended to systematize the familiar concept of deducibility or entailment, then some form of the deduction theorem should hold for it. [Bar53] She showed [Bar46; Bar53] that S1 to S3 fail this criterion, for several conceivable formulations of the Deduction Theorem. And those which behave somewhat better in this respect, i.e., from S4 upwards are too strong to capture a general notion of strict implication which Lewis would approve of. In fact, S4 and S5, which we came to count among normal systems (unlike S1–S3) and for which the advantage of switching to the unary setting is most obvious, for Lewis himself were foster children he was forced to adopt. As is well-known, it was Oskar Becker12 [Bec30] who proposed these axioms, even calling one of them the Brouwersche Axiom; let us not discuss the adequacy of this name here, but not only does it provide us with another excuse to mention Brouwer in this paper, it has also survived until today in names of systems like KB or KTB. Becker intended to cut the number of non-equivalent modalities in the calculus, a goal which seems rather orthogonal to Lewis’ plans: Those interested in the merely mathematical properties of such systems of symbolic logic tend to prefer more comprehensive and less strict systems such as S5 and material implication. The interests of logical study would probably be best served by an exactly opposite tendency. [LL32, p. 502] Kurt Gödel did review Becker’s work [Göd86, p. 216–217] and was familiar with William T. Parry’s early analysis of the notion of analytic implication based on J [Göd86, p. 266–267].13 This apparently led14 to his landmark 1933 10 Cf., e.g., the attempts of Bronstein&Tarter or Abraham addressed, respectively, by McKinsey and Fitch; see Murphey [Mur05, Ch. 6] for references. It is worth pointing out that Lewis himself [LL14] dealt with this question in a paper published only posthumously (with Langford as a “nominal” coauthor, see editor’s note [Mar14] for a contemporary perspective). 11 Her earliest papers [Bar46] are signed by her maiden surname, Ruth Barcan, which survives until today in the name of the Barcan formula. 12 Although many developments discussed in this subsection—in particular proposing and justifying S4 axioms with an explicit Brouwerian motivation—had their forerunner in a neglected 1928 paper by Ivan E. Orlov, cf. [Doš92; Baz03]. 13 As another small example how modal and intuitionistic inspirations tended to work handin-hand for Gödel: his proof that IPC is not characterized by any finite algebra [Göd86, p. 268–271] is presented as an answer to a question posed by Otto Hahn during a discussion following Parry’s presentation. 14 His short review of Becker points out that Becker’s attempts to relate modal logic to “the intuitionistic logic of Brouwer and Heyting” and claims that steps taken by Becker to “deal with this problem on a formal plane” are unlikely to succeed; Orlov (cf. Footnote 12) was more insightful, but it does not appear that Gödel was familiar with his paper. 7 paper [Göd86, p. 296–303] translating the nascent intuitionistic calculus into what turns out to be a notational variant of S4 formulated with unary box as a primitive. Thus, immediately after Symbolic Logic was published, Gödel pretty much doomed the fate of J and condemned non-normal systems to at most secondary status: his paper not only provided an independent motivation (in terms of “the intuitionistic logic of Brouwer and Heyting” . . . ) for the study of extensions of S4 rather than subsystems of S3, but also highlighted the elegance and conciseness of ✷-based axiomatizations for these logics. In short, it appears that regardless of the fact that historical circumstances did not favour Lewis, none of his systems was destined to success or genuinely free of design or conceptual issues. Nevertheless, the idea of providing an implication connective yielding tautologies only when the antecedent is genuinely relevant for the consequent proved prescient.15 In fact, one can easily argue that even the later enterprise of relevance logic would not satisfy Lewis’ expectations: he wanted to supplement material implication with a strict one, not replace it altogether. In this sense, still more recent resource-aware formalisms with computer-science motivation where both a substructural and an intuitionistic/classical implication are present (either as an abbreviation or directly in the signature) like linear logic [Gir87; Tro92; Abr93; Bie94] or the logic of bunched implications BI [OP99; Pym02; POY04] seem closer to Lewis’ original idea. 2.2. Could Brouwerian inspiration help Lewis’ systems? At the time of publication of Symbolic Logic, Lewis was both familiar with and open to non-boolean extensional connectives. The chapters he wrote for that monograph deal in detail with n-valued systems of Lukasiewicz.16 At the same time, he published a paper on Alternative systems of logic [Lew32]. In both these references, he discusses possible definitions of “truth-implications” [LL32] or “implication-relations” [Lew32] one can entertain in finite, but not necessarily 15 The connection between modal logics and relevance logics has been always actively debated, see, e.g., Mares [Mar04, Ch. 6] for an extended presentation, including a reminder that Ackermann’s 1956 paper which “began the study of relevant entailment” took issue with some tautologies valid for Lewis’ J, in particular ex falso quodlibet. But in fact the relationship can be traced back at least to 1933, when Parry in his work on analytic implication based on J proposed what relevance logicians came to know as the variable sharing criterion: much later, Dunn [Dun72] noted that Parry’s system is contained in S4 and proposed a “demodalization” of Parry’s original system still preserving that criterion. As another connection with Gödel, let us note that his discussion [Göd86, p. 266–267] of the work of Parry suggested a completeness result that was only proved in 1986 by Fine [Fin86]. Moreover, one can push the clock back even beyond Parry and Gödel, to the paper of Orlov (cf. Footnote 12), which seems the first attempt to relate relevance, intuitionistic, and modal principles, including the first axiomatization of what came to be known as the implicative-negative fragment of the relevance logic R [Doš92]. Let us note here the view of van Atten [van07] that “logic as Brouwer sees it is a relevance logic”, rejecting in particular ex falso (absent also in earliest versions of formalizations of intuitionistic logic by Kolmogorov and Glivenko), which subverts the standard understanding of the BHK interpretation (cf § 7.1 below). 16 At the time, Lewis still attributed it to a collaboration between Lukasiewicz and Tarski. 8 binary matrices. The latter paper also contains a rare (perhaps the only one) reference to Brouwer in his writings: [T]he mathematical logician Brouwer has maintained that the law of the Excluded Middle is not a valid principle at all. The issues of so difficult a question could not be discussed here; but let us suggest a point of view at least something like his. . . . The law of the Excluded Middle is not writ in the heavens: it but reflects our rather stubborn adherence to the simplest of all possible modes of division, and our predominant interest in concrete objects as opposed to abstract concepts. The reasons for the choice of our logical categories are not themselves reasons of logic any more than the reasons for choosing Cartesian, as against polar or Gaussian coördinates, are themselves principles of mathematics, or the reason for the radix 10 is of the essence of number. [Lew32, p. 505] Of course, the question of Lewis’ own potential take on combining IPC and J remains speculative: it does not seem he was familiar with the work of Kolmogorov, Glivenko and Heyting, turning Brouwer’s philosophical insights into a propositional calculus. Nevertheless, let us note two points: • even the collapse of Lewis’ original system [Lew14; Lew18] was caused by classical laws combined with a misguided boolean inspiration, namely the insistence on involutivity of the strict negation (cf. Appendix D); • even when considering classical Kripke frames, the negation-free logic obtained by replacing → with J is a sublogic of the intuitionistic logic [Cor87; Doš93; CJ01; CJ05] (see also Question 4.3). Our paper, however, focuses on an even more fundamental advantage of studying the theory of J over IPC. Whatever is there to be said about the universal logic of propositions which is independent of mode and its extensional basis, defining J using (2) is premature in the constructive setting. Furthermore, instances of such a “constructive strict implication” can be seen in areas ranging from metatheory of intuitionistic arithmetic to functional programming, often satisfying very different laws to those strict implication was supposed to obey; indeed, sometimes rather meaningless classically. For example, Sa (φ → ψ) → (φ J ψ) holds in numerous logics justified from a computational/Curry-Howard (§ 7), arithmetical (§ 5.4.4) or even philosophical (§ 7.3) point of view.17 3. Strict implication in intuitionistic Kripke semantics It is time to begin a more systematic discussion, starting with the relational interpretation of J. In this paper, we are concerned with the following propo17 From a Lewisian point of view, would intuitionistic → be the “strict” implication and J be the “material” implication in such systems? 9 sitional languages: LJ (with Lewis’ arrow), L✷ (the unimodal one, identified with a fragment of LJ ) and L (the propositional language of IPC): LJ L✷ L φ ::= ⊥ | ⊤ | p | (φ ∧ φ) | (φ ∨ φ) | (φ → φ) | (φ J φ), φ ::= ⊥ | ⊤ | p | (φ ∧ φ) | (φ ∨ φ) | (φ → φ) | (✷φ), φ ::= ⊥ | ⊤ | p | (φ ∧ φ) | (φ ∨ φ) | (φ → φ). As usual, ¬φ abbreviates φ → ⊥. For the sake of clarity, the binding priorities are as follows: unary connectives ¬ and ✷ bind strongest, next comes J, then ∧ and ∨, and finally →. Regarding associativity, it is used tacitly for ∧ and ∨, just like commutativity. Regarding → and J, they are commonly assumed to associate to the right, but we will be careful not to overuse this convention, as it can be confusing. We begin with recalling the basic setup of intuitionistic Kripke frames for L✷ .18 They come equipped with two accessibility relations. One of them, which we will denote by , is a partial ordering19 interpreting intuitionistic implication: k φ → ψ if, for all ℓ  k, if ℓ φ, then ℓ ψ. (3) This forces the denotation of → to be -persistent or, as some authors say, “monotone” or “upward-closed”. It is enough to impose (3) and require -persistence of atoms to ensure persistence for all L-formulas. The other accessibility relation ❁ is the modal one. There are two choices one can make to ensure -persistence for ✷: One is to modify the satisfaction clauses. This might be a reasonable thing to do, for one might wish to use the partial order to give a more intuitionistic reading of the modalities. The other remedy is to impose conditions on models that ensure that the monotonicity lemma does hold. [Sim94, §3.3] In fact, in a unimodal language the difference between these two strategies is not essential; it becomes more consequential when a single accessibility relation is used to interpret, for example, both ✷ and ✸ (see [Sim94, §3.3] for a discussion and more references). Still, most references choose the latter one, i.e., keeping the same reading of ✷ as in the classical case and imposing conditions on the interaction of  and ❁ to ensure persistence. Boz̆ić and Dos̆en [BD84] have established that in the presence of unary ✷ with semantics defined by 18 As far as L ✷ is concerned, our discussion largely follows Litak [Lit14]. The reader is referred there for more details and references. 19 In fact, it is essential only that the relation is a preorder (i.e., a reflexive and transitive relation), but such a generalization brings no tangible benefits from the point of view of expressivity, definability and completeness of propositional logics. 10 @ℓ k /o /o / mO ℓO ? /o / ℓ′ /m ? ? @ℓ t4 4t k k ✷-p J-p 4t 4t 4t 4 > n >~  m ~> > O /ℓ k brilliancy m mix Figure 3.1: Minimal conditions one can impose on ✷-frames and J-frames. See Figure 6.2 for a visual representation of other conditions corresponding to additional axioms. k ✷φ if, for all ℓ ❂ k, ℓ ψ persistence is equivalent to the condition ✷-p if k  ℓ ❁ m, then, for some ℓ′ , we have k ❁ ℓ′  m (i.e.,  · ❁ ⊆ ❁ · , where “ · ” denotes relational composition). However, most references require tighter interaction. On certain occasions, like in Goldblatt [Gol81], one sees a strengthening to J-p if k  ℓ ❁ m, then k ❁ m (i.e.,  · ❁ ⊆ ❁). But the most common one (see, e.g., [Sot84; WZ97; WZ98]) is the still stronger mix if k  ℓ ❁ m  n, then k ❁ n (i.e.,  · ❁ ·  ⊆ ❁). This condition naturally obtains in a canonical model construction à la Stone and Jónsson-Tarski for prime filters of (reducts of) Heyting algebras with normal ✷ [BD84; Sot84; Köh81; BJ13]. Moreover, mix is “mostly harmless” for ✷: it can be obtained from ✷-p by adding the requirement that for any ℓ, the set of its ❁-successors is -upward closed, that is, brilliancy if k ❁ ℓ  m, then k ❁ m (i.e., ❁ ·  ⊆ ❁). The name, to the best of our knowledge, has been proposed by Iemhoff [Iem01b; Iem01a; Iem03; IDZ05], another one being strongly condensed [BD84]. As noted in standard references [BD84; Gol81], not only brilliancy cannot be defined using ✷, but any model satisfying ✷-p can be made brilliant without changing the satisfaction relation for ✷-formulas in a straightforward way: by replacing ❁ by its composition with . Consider now the Lewisian strict implication φ J ψ. Here is the natural satisfaction clause in this semantics, directly transferring the classical one: k φ J ψ if, for all ℓ ❂ k, if ℓ φ, then ℓ ψ. (4) The first consequence of such an enrichment of the language is that ✷-p becomes too weak to ensure persistence. Let us state this formally, defining for this purpose a somewhat too general notion: Definition 3.1. A preframe is a triple F := hW, , ❁i, where  is a partial order, and ❁ is a binary relation. A premodel based on F is K := hF, V i, where V is a valuation mapping propositional variables to -upward closed sets. The forcing relation K, k φ is defined in the standard way for the intuitionistic connectives and using equation (4) for J. 11 It can be easily shown (see, e.g., [Zho03; IDZ05]) that the condition equivalent to persistence becomes precisely J-p , that is: Fact 3.2. For a preframe K := hW, , ❁i, J-p above corresponds to the following condition: for any two sets U, V upward closed wrt , the set U J V := {k ∈ W | ∀ℓ ❂ k, if ℓ ∈ U, then ℓ ∈ V } is upward closed wrt . We will thus take J-p to be the minimal condition in what follows. Definition 3.3. A (J-)frame is a preframe satisfying J-p . We can define in a standard way what it means for a formula to be valid or refuted in a class of models. As we have already suggested, for LJ the brilliancy condition does not remain “mostly harmless” in the sense described above for L✷ : Fact 3.4. [Zho03] The following conditions are equivalent for a J-frame: • validity of (φ ∧ ψ) J χ → φ J (ψ → χ); • validity of ψ J χ → ⊤ J (ψ → χ); • validity of brilliancy . One easily sees the converse implication φ J (ψ → χ) → (φ ∧ ψ) J χ and, consequently, its special instance (where φ is equal to ⊤) ✷(ψ → χ) → ψ J χ to be valid on any J-frame; see Lemma 4.1 for a syntactic derivation. Let us take stock. In order to restore definability of J in terms of ✷, i.e., validity of (2) above, one needs to impose the brilliancy condition. In general, ✷(φ → ψ) implies φ J ψ, but not necessarily the other way around. Of course, in classical Kripke frames,  is a discrete order, which trivializes all conditions discussed above and all distinctions between them. As we will see in Corollary 4.8, the boolean deconstruction of J can be also derived syntactically. We will return to Kripke semantics in § 6 below. 4. Axiomatizations 4.1. A fistful of logics In this section, we present a Hilbert-style study of LJ -logics. Discussion of arithmetically oriented principles was originated by Visser [Vis81; Vis82; Vis85; Vis94] and developed further by Iemhoff and coauthors [Iem01b; Iem03; IDZ05], who also studied the basic theory of J-frames. IPC and CPC denote, respectively, the intuitionistic propositional calculus and its classical counterpart. 12 4.1.1. Logics in L✷ Before we start discussing J-logics in § 4.1.2, Table 4.1 presents some axioms involving only ✷, which is a definable connective in LJ . Table 4.1: List of principles for ✷. Here, the names of systems in the right column refer to languages restricted to connectives appearing in the axiomatization, i.e., not involving J. Later in the text, we will also use some of these principles as axioms over iA, i.e., the minimal “normal” system for J (cf. Table 4.2), where ✷ is a defined connective. N✷ ⊢ φ ⇒ ⊢ ✷φ CPC := IPC + peirce K✷ ✷(φ → ψ) → ✷φ → ✷ψ i-K✷ := IPC + N✷ + K✷ i-GL✷ := i-K✷ + L✷ 4✷ ✷φ → ✷✷φ c-GL✷ := CPC + i-GL✷ C4✷ ✷✷φ → ✷φ i-S✷ := i-K✷ + S✷ i-SL✷ := i-K✷ + SL✷ L✷ ✷(✷φ → φ) → ✷φ i-PLL✷ := i-S✷ + C4✷ S✷ φ → ✷φ i-mHC✷ := i-S✷ + CB✷ i-KM✷ := i-SL✷ + CB✷ SL✷ (✷φ → φ) → φ i-KM.lin✷ := i-KM✷ + Lin✷ Lei ✷(φ ∨ ψ) → ✷(φ ∨ ✷ψ) 4✷ is known to be derivable in i-GL✷ . In the provability logic literature: CB✷ ✷φ → (ψ → φ) ∨ ψ CB′✷ ✷(ψ → φ) → (ψ → φ) ∨ ψ • N✷ is known as L1, Lin✷ ✷(φ → ψ) ∨ ✷(ψ → φ) • K✷ is known as L2, peirce ((φ → ψ) → φ) → φ • 4✷ is known as L3, em φ ∨ ¬φ • L✷ is known as L4. • The axioms of i-GL✷ (intuitionistic Löb logic) and c-GL✷ (classical Löb logic) are well-known. The logic c-GL✷ is arithmetically complete for all classical Σ01 -sound theories extending Elementary Arithmetic EA. The logic i-GL✷ is arithmetically valid in all arithmetical theories extending i-EA. We discuss these matters further in § 5.3. • The principle Lei is known as Leivant’s Principle. The principle is, in a sense, a shadow of the disjunction property. The disjunction property of an arithmetical theory T cannot be verified in T itself. Leivant’s Principle is arithmetically valid in a substantial class of arithmetical theories that includes Heyting Arithmetic HA. We discuss Leivant’s Principle in § 5.3. • S✷ axiomatizes strong modalities (cf. § 7), but arises also in some arithmetically motivated logics (§ 5.4.4). Strong Löb logic is obtained by adding S✷ to i-GL✷ —or, alternatively, by using SL✷ instead of L✷ as an axiom. 13 • The principle C4✷ classically corresponds to a semantic condition known as density (cf. Figure 6.2). From another point of view, this axiom arises naturally in the Curry-Howard logic of monads (§ 7). It is a typical “nonLöb-like” axiom: in combination with L✷ , we could derive ✷⊥. • CB✷ comes from the intuitionistic system i-KM✷ of Kuznetsov and Muravitsky and its later weakening to i-mHC✷ by Esakia and the Tbilisi group (see [Lit14] for more information and references); its equivalent variant CB′✷ (see Lemma 4.16a) was discussed [Vis82] in connection with PA∗ (§ 5.4.4). In our setting, it is interesting to contrast it with CBa in Table 4.2 (Figure 6.2 and Example 6.11). See also § 8.2 for the arithmetical perspective on the contrast between i-mHC✷ and i-mHCa . The name CB used here comes from Litak [Lit14], where it was used to suggest the Cantor-Bendixson derivative. • Lin✷ is a typical axiom valid on total orders. In Fact 4.18 and Example 6.12, we compare and contrast this axiom with its J-counterpart. 4.1.2. Logics in LJ Table 4.2 displays potential axioms for J central for this paper. Most of them come with an explicit arithmetical interpretation. Typically, the “primed” variants of axioms will be their equivalent reformulations (§ 4.2). • Iemhoff [Iem01b; Iem03; IDZ05] identified system iA as the logic of all (finite) frames satisfying the J-p condition; this and other completeness results are discussed in § 6. However, Di is an axiom which is not exactly trivial from an arithmetical point of view. It does hold in the preservativity logic of Heyting Arithmetic but it fails in the preservativity logic of Peano Arithmetic (§ 5.4 and Appendix B). The non-triviality of Di, and the potential interest in a disjunction-free system (§ 7) are the reasons why we isolated iA− as a subsystem. • The principles 4a , Wa , Ma are arithmetically valid for the preservativity interpretation of J. This means that they are in the logic i-PreL− which is arithmetically valid in all arithmetical theories we consider in this paper (§ 5.4). The principle La , weaker than Wa , is mainly of technical interest. • If we interpret φ J ψ as ¬ ψ ✄ ¬ φ, then the principle Pa is the distinctive principle of the interpretability logic of finitely axiomatized extensions of EA+ aka I∆0 + Supexp. The modality ✄ stands for interpretability over a theory. This modality is explained in § C.20 The specific result mentioned here is discussed in detail in § C.3. 20 On a side note, some CS readers may be familiar with the use of triangle-like notation like ✄ for unary modalities in the context of guarded (co)recursion discussed in § 7.2. The tradition of using such notation for binary operators and connectives such as arithmetical interpretability is much longer and we believe this convention to be more natural. 14 Table 4.2: List of principles for J and logics considered in this paper. Everywhere below, when we write i-X ? , the superscript “?” can be either “−” or nothing, depending whether or not Di is used. Na ⊢ φ → ψ ⇒ ⊢ φ J ψ Tr φ J ψ → ψ J χ → φ J χ Ka φ J ψ → φ J χ → φ J (ψ ∧ χ) iA0 := IPC + Na + Tr, K′a φ J ψ → (φ ∧ χ) J (ψ ∧ χ) iA− := iA0 + Ka , K′′ a φ J ψ → φ J (ψ → χ) → φ J χ iA := iA− + Di, K′′′ φ J (ψ → χ) → (φ ∧ ψ) J χ a BL ✷(φ → ψ) → φ J ψ i-GLa ? := iA? + La , LB φ J ψ → ✷φ → ✷ψ i-GWa ? := iA? + Wa , Di φ J χ → ψ J χ → (φ ∨ ψ) J χ i-PreL? := i-GWa ? + Ma . Di′ φ J ψ → (φ ∨ χ) J (ψ ∨ χ) For each logic i-X ? , i-SX ? denotes its extension with S✷ , in particular Pa φ J ψ → ✷(φ J ψ) 4a φ J ✷φ i-SA := iA + S✷ . La (✷φ → φ) J φ Set also: i-PLAA := i-SA + C4a , Wa (φ ∧ ✷ψ) J ψ → φ J ψ i-mHCa := i-SA + CBa , Wa′ φ J ψ → (✷ψ → φ) J ψ i-KMa := i-mHCa + L✷ , Ma φ J ψ → (✷χ → φ) J (✷χ → ψ) i-KM.lina := i-KMa + Lina , M′a (φ ∧ ✷χ) J ψ → φ J (✷χ → ψ) Sa (φ → ψ) → φ J ψ For each logic i-X , i-BoxX denotes its extension with Box, e.g., S′a φ J ψ → φ → ✷ψ i-BoxA := iA + Box, Box φ J ψ → ✷(φ → ψ) Box′ (χ ∧ φ) J ψ → χ J (φ → ψ) i-BoxGLa := i-GLa + Box. Box′′ φ J ψ → (χ → φ) J (χ → ψ) Note that i-BoxGLa is just a notational variant of i-GL✷ . Note also that notation i-BoxX − would be redundant, see Lemma 4.4c. A fortiori, the same applies to extensions of i-mHCa by Lemma 4.16c. Similarly, i-PLAA− would be redundant by Lemma 4.17g. In all these systems, Di can be derived from the remaining axioms. Furthermore, as we will show in Lemma 4.19, i-KM.lina and i-KM.lin✷ are notational variants of the same system. CBa φ J ψ → (φ → ψ) ∨ φ Lina φ J ψ ∨ ψ J φ Appa (φ ∧ (φ J ψ)) J ψ C4a ✷φ J φ Hug (φ → ✷ψ) → (φ J ψ) 15 • Sa and S′a are J-variants axiomatize the same logic as S✷ (Lemma 4.10). In general, this is rarely the case with J-generalizations of ✷-axioms; often the J-version is stronger, but Lina illustrates such a rule is not universal. • We have already seen Box in § 3 above; its equivalence with Box′ and Box′′ is established in Lemma 4.4. The conjunction of this axiom with BL, derivable in iA (Lemma 4.1c), collapses J. Note that CBa makes Box derivable (Lemma 4.16), unlike CB✷ (Example 6.11). • The last group of J-principles—i.e., Appa , C4a and Hug—which should be contrasted with C4✷ , will play a prominent rôle in § 7.1 on monads, idioms and arrows in functional programming. For similar reasons as C4✷ , they are of drastically “anti-Löb” character, a fact made explicit by their semantic correspondents displayed in Figure 6.2 in § 6. It is worth mentioning that Appa was in fact adopted by Lewis as an axiom even in his weakest system S1, cf. Remark 7.3. 4.2. An armful of derivations In this subsection we put the Hilbert-systems proposed above to actual use. We begin with a discussion of minimal axiom systems, with and without Ka or Di. Later on, we move to those inspired by concrete applications. We are not giving the details of these derivations here; some are available in existing references (and we give references in several cases), some are left for the reader as an exercise, and some will be published in future work [LV]. For a calculus X defined by a list of axioms and rules, write X ⊢ φ to denote deducibility from all substitution instances of axioms/rules in X plus Modus Ponens. Whenever we have that for any φ, Y ⊢ φ implies X ⊢ φ, we write X ⊢ Y. For X ⊢ φ → ψ, iA0 ⊢ φ → ψ, iA− ⊢ φ → ψ or iA ⊢ φ → ψ (see Table 4.2 below), write, respectively, φ ⊢X ψ, φ ⊢0 ψ, φ ⊢− ψ and φ ⊢∨ ψ. In other words, we use ⊢− (and ⊢X − ) for derivability without instances of non-IPC schemes involving disjunction (Di or equivalently Di′ ) and ⊢0 for a still more restrictive case when deduction relies on monotonicity only. Correspondingly, interderivability (equivalence) is denoted using, respectively ⊣⊢, ⊣⊢X , ⊣⊢0 , ⊣⊢− and ⊣⊢∨ . Also, let us abbreviate X ⊢ φ J ψ, iA0 ⊢ φ J ψ, iA− ⊢ φ J ψ or iA ⊢ φ J ψ as, respectively, φ |JX ψ, φ |J0 ψ, φ |J− ψ and φ |J∨ ψ. Note that even the weakest of these relations, i.e., |J0 is transitive and contains ⊢0 ; in fact, this is precisely essence of the minimal deduction system iA0 . Finally, for deductions in ✷-only language, using i-K✷ as the minimal system, one can use similar conventions as above with ✷ as subscript (e.g, “⊢✷ ” and “⊣⊢✷ ”). 4.2.1. Axiomatizations for minimal systems Lemma 4.1. a. The principles Ka , K′a , K′′a and K′′′ a are equivalent over iA0 . 16 b. The principles Di and Di′ are equivalent over iA0 . c. iA0 ⊢ BL and iA0 ⊢ LB. As noted in existing references (cf., e.g., [Iem01b, Chapter 3] or [IDZ05, Theorem 2.5]), there is some freedom in the choice of minimal rules: Fact 4.2. iA− ⊣⊢ IPC + N✷ + K✷ + Tr + Ka . Open Question 4.3. Even in the absence of intuitionistic , the negationfree logic obtained by replacing → with J is a subintuitionistic logic [Cor87; Doš93; CJ01; CJ05]. Is there a good a presentation of the minimal logic of J in terms of fusion or dovetailing/fibring of IPC and this minimal subintuitionistic logic, rather analogous to the logic of bunched implications BI [OP99; Pym02; POY04]? Note that the analogy with BI is limited, e.g., both local and global consequence relations of J in the absence of → cease to be protoalgebraic [CJ01; CJ05]. 4.2.2. Collapsing and decomposing J In Fact 3.4, we have observed that there are two syntactically similar conditions one can use to enforce brilliancy. Now we can prove syntactically their equivalence, which explains why we used the seemingly weaker one as Box: Lemma 4.4. a. i-BoxA− ⊣⊢ iA + Box′ , i.e., Box and Box′ are equivalent over iA− . b. i-BoxA− ⊣⊢ iA + Box′′ , i.e., Box and Box′′ are equivalent over iA− . c. i-BoxA− ⊢ Di and consequently i-BoxX ⊣⊢ i-BoxX − for any X . Remark 4.5. We presented one possible way to translate a J-logic i-X into a ✷-logic, to wit to take φ J ψ as an abbreviation for ✷(φ → ψ). This translation relates i-X to its extension i-BoxX , which is term-equivalent to a ✷-logic. Another way, studied in detail by Iemhoff and coauthors [Iem01b; Iem03; IDZ05], takes the validity of LB as a starting point and translates φ J ψ as ✷φ → ✷ψ. A third interpretation of J in terms of ✷ relating i-PLL✷ and i-PLAA is discussed in Remark 7.2; it builds on a i-PLAA-decomposition of J in terms of → provided by Lemma 4.17f. For more on reductions of J to unary modalities see [LV]. We have suggested that the degeneration of J in the presence of classical laws can be derived syntactically. In fact, this can be obtained as a consequence of an equivalence derivable over the intuitionistic base but, atypically, using disjunction with its Di axiom in an essential way: Lemma 4.6. We have: ψ J χ ⊣⊢∨ (ψ ∨ ¬ψ) J (ψ → χ). 17 Nevertheless, as (ψ ∨ ¬ψ) J (ψ → χ) is parametric in the antecedent of strict implication, it does not seem a satisfying reduction of J to →. Let us also note in passing that if one adds J to Johansson’s minimal logic instead of IPC, even this transformation does not work anymore. Moreover, there is no one-variable formula φ(p) in the disjunction-free fragment of the intuitionistic signature s.t. p J q ⊣⊢− φ(p) J (p → q) and CPC ⊢ φ(p), cf. Example 6.10. Open Question 4.7. In general, we stick to extensions of L, but let us make a digression concerning a language without all standard connectives. Suppose we define [φ]ψ as (φ ∨ ¬ φ) J ψ. As we saw above, φ J ψ is equivalent with [φ](φ → ψ). Is there an elegant axiomatization for the minimal fragment of the language with [·](·)? It seems richer than the disjunction-free fragment of LJ . Corollary 4.8. iA + em ⊢ i-BoxA. Remark 4.9. This is one of very few places in this section where we need full iA rather than iA− , i.e., where Di is used in an essential way. This happens for a very good reason: it is not possible to derive i-BoxA from iA− + em. One can see this, e.g., by considering the interpretation of φ J ψ as ✷φ → ✷ψ. In Appendix B we will explain that the logic ILM of Π01 -conservativity and interpretability corresponds to c-PreL := i-PreL− + em. This provides a proof that even c-PreL does not extend i-BoxA. The proof may use either the arithmetical interpretation or the Veltman semantics used for ILM. We will discuss collapsing and decomposing further in a later paper [LV]; see also remarks preceding Theorem 6.6 below. 4.2.3. Derivations between arithmetical principles We turn our attention to derivations between principles of central importance, especially from the point of view of arithmetical interpretations. Lemma 4.10. We have: a. i-SA− ⊣⊢ iA− + Sa , i.e., over iA− , the principles S✷ and Sa are equivalent. b. i-SA− ⊣⊢ iA− + S′a , i.e., over iA− , the principles S✷ and S′a are equivalent. c. In the presence of Sa , Na is derivable using just Modus Ponens. Hence, axiomatizations of “strength” in terms of ✷ and in terms of J yield the same logic over iA− . As we are going to see below, this is a relatively rare phenomenon. Still, many well-known modal derivations can be easily translated into the J-setting, e.g., a derivation of 4 from the Löb axiom: − − Lemma 4.11. i-GL− a ⊣⊢ iA + L✷ + 4a . It follows that, over iA + 4a , the principles L✷ and La are interderivable. Lemma 4.12. 18 a. iA− + 4a ⊢ 4✷ . c. i-SA− ⊢ Pa . b. i-SA− ⊢ 4a . d. i-SA− + L✷ ⊢ i-GL− a . Lemma 4.13. [IDZ05, Cor. 2.6 and 2.7] Wa and Wa′ are equivalent over iA− 0. Similarly, Ma and M′a are equivalent over iA− . 0 Lemma 4.14. We have: − a. i-GW− a ⊢ i-GLa . − − b. i-GL− a + Ma ⊢ Wa . In other words, i-PreL ⊣⊢ i-GLa + Ma . c. i-GL− a + P a ⊢ Wa . d. iA− + Ma ⊢ (✷φ J ψ) → ✷(✷φ → ψ). Examples 6.7, 6.8 and 6.9 illustrate that clauses (a) and (b) cannot be reversed. Lemma 4.15. iA + 4a ⊢ Lei. This implies that the logics i-GLa , i-GWa and i-PreL are not conservative over i-GL✷ . Both i-GLa and i-GWa are conservative over i-GL✷ + Lei [IDZ05]. 4.2.4. More derivations Derivations discussed in the remainder of this section are mostly of importance in § 7, although, e.g., Lemma 4.16a will be also relevant in § 5.4.4: Lemma 4.16. We have: a. i-K✷ − + CB✷ ⊣⊢✷ i-K✷ − + CB′✷ b. iA− + CBa ⊢ CB✷ . c. i-mHCa − ⊢ Box and consequently i-mHCa − ⊢ Di. d. i-mHCa ⊢ Ma . e. i-KMa ⊢ Wa . Clause (c) implies that the notation “i-mHCa − ” is redundant. Example 6.11 below illustrates that clause (b) cannot be reversed. Lemma 4.17. We have: a. iA− + C4a ⊢ C4✷ . e. i-PLAA− ⊢ Hug. b. iA− + Appa ⊢ C4a . f. φ J ψ ⊣⊢i-PLAA− φ → ✷ψ − c. iA + Hug ⊢ C4a . g. i-PLAA− ⊢ Di. d. i-PLAA− ⊢ Appa . 19 Example 6.13 illustrates irreversibility of several clauses in this lemma. So far, principles involving J tended to be stronger than their relatives formulated in L✷ . It is indeed quite often but not always the case. For example, in the case of “semi-linearity” axioms, the situation is reversed: Fact 4.18. We have: • iA− + Lin✷ ⊢ Lina . • i-BoxA + Lina ⊢ Lin✷ , in particular i-mHCa + Lina ⊢ Lin✷ . It is hard to make Lina and Lin✷ coincide in the absence of Box, cf. Example 6.12. Nevertheless, here is an important exception, used in § 7.2 below: Lemma 4.19. We have: a. (✷φ → ✷ψ) → ✷(φ → ψ) ⊢0 Box. b. i-KM.lin✷ ⊢ (✷φ → ✷ψ) → ✷(φ → ψ). c. i-KM.lin✷ ⊣⊢ i-KM.lina . That is, not only both systems are notational variants of each other, but the Box axiom can be derived from i-KM.lin✷ . 5. Arithmetical interpretations: provability and preservativity « In § 6 we continue the discussion of the modal side of our calculi. But now, we cannot postpone any further the presentation of our original motivation for studying constructive J and a number of its axiom systems in terms of Σ01 preservativity for an arithmetical theory T : • A JT B if, for all Σ01 -sentences S, if T ⊢ S → A, then T ⊢ S → B. In order to provide a framework for such interpretations of modal connectives, we introduce the notion of a schematic logic. This notion can be given a very general treatment. However, for the purposes of this paper, we will restrict ourselves to the case of arithmetical theories, studying propositional logics of theories (§ 5.2), provability logics (§ 5.3) and our true target: preservativity logics (§ 5.4). For an instructive contrast, we provide some extra information about logics for Π01 -conservativity and interpretability in Appendices B and C. 5.1. Schematic logics An arithmetical theory T is, for the purposes of this paper, an extension of i-EA, the intuitionistic version of Elementary Arithmetic, in the arithmetical language.21 We demand that the axiom set of T is given by a ∆0 (exp)-formula. 21 The classical theory EA is I∆ + Exp. This theory consists of the basic axioms for zero, 0 successor, addition, multiplication plus ∆0 -induction plus the axiom that states that exponentiation is total. The theory i-EA is the same theory only with constructive logic as underlying logic. The theory proves the decidability of ∆0 (exp)-formulas. Some basic information about constructive arithmetic can be found in [TD88; Tro73; Dra88]. 20 Let L⊚0 ,...,⊚k−1 be the language extending L with operators ⊚0 , . . . , ⊚k−1 , where ⊚i has arity ni . Let a function F be given that assigns to every ⊚i an arithmetical formula A(v0 , . . . , vni −1 ), where all free variables are among the variables shown. We write ⊚i,F (B0 , . . . , Bni −1 ) for F (⊚i )(pB0 q, . . . , pBni −1 q). Here pCq is the numeral of the Gödel number of C. Suppose f is a mapping from the propositional atoms to arithmetical sentences. We define (φ)fF as follows: • (p)fF := f (p) • (·)fF commutes with the propositional connectives • (⊚i (φ0 , . . . , φni −1 ))fF := ⊚i,F ((φ0 )fF , . . . , (φni −1 )fF ) Let T be an arithmetical theory. We say that a modal formula in our given signature is T -valid w.r.t. F if, for all assignments f of arithmetical sentences to the propositional atoms, we have T ⊢ (φ)fF . We write ΛT,F for the set of modal formulas that are T -valid w.r.t. F . Of course, we will focus exclusively on “natural” F yielding well-behaved ΛT,F with interesting properties. 5.2. Propositional logics of a theory Let us first consider the case where our finite set of modal operators is empty. If T is consistent and classical, then ΛT := ΛT,∅ is, trivially, precisely CPC and if T is Heyting Arithmetic (HA), then ΛT has the de Jongh property: ΛT = IPC. There are theories for which ΛT is an intermediate logic strictly between IPC and CPC. De Jongh, Verbrugge & Visser [JVV11] show that whenever Θ is an intermediate logic with the finite frame property (cf. § 6) and U is the result of extending HA with all axioms of Θ as schemes, ΛU = Θ. For some theories like Markov’s Arithmetic MA = HA + MP + ECT0 , where MP is the Markov’s Principle ([TD88, 4.5, p.203], [Tro73, 1.11.5, p.93]): (∀x(Ax ∨ ¬Ax) ∧ ¬¬∃xAx) → ∃xAx. and ECT0 is the Extended Church’s Thesis (cf. Appendix A), the characterization of the set of valid principles is an open problem connected to the question of the propositional logic of realizability. See e.g. [Pli09, § 13]. For more on intuitionistic schematic logics see [Smo73; Vis99; Pli09; JVV11; AM14]. 5.3. Provability Logic Next we consider the extension of propositional logic with a unary modal operator ✷. It allows numerous interesting arithmetical interpretations, but at this point we focus on the interpretation of ✷ as provability. Consider any arithmetical theory T . We assume that T comes equipped with a ∆0 (exp)predicate αT that gives the codes of its axiom set. Let provability in T be arithmetized by provT . We note that T really occurs in the guise of αT . We set F0,T (✷) := provT (v0 ). Let Λ∗T := ΛT,F0,T . Intuitionistic Löb’s Logic i-GL✷ 21 is contained in all Λ∗T , where T is an arithmetical theory in the sense of this paper. This insight is due to Löb [Löb55].22 Remark 5.1. In many treatments of intuitionistic modal logic the interdefinability of ✷ and ✸ fails and ✷ and ✸ both are treated as primitive operations. This is not so in the context of provability logic for constructive theories and its extensions. Here the connective ✸ is always defined as ¬✷¬. Thus, ✷, which signals the existence of a proof, is the positive notion and ✸ is the negative less informative notion. We note that ¬✸¬ is equivalent to ¬¬✷¬¬ which, in the context of theories like HA, is certainly weaker than ✷. Remark 5.2. One of the first global insights into schematic theories is due to George Gargov [Gar84]: they inherit the disjunction property from the underlying arithmetic theory. Thus, if an extension of i-EA has the disjunction property, then so has its provability logic.23 The theory c-GL✷ is obtained by extending i-GL✷ with classical logic. If T is a Σ10 -sound classical theory, then Λ∗T = c-GL✷ . This insight is due to Solovay [Sol76]. In contrast, the logic i-GL✷ is not complete for HA. The system for preservativity logic i-PreL+V discussed in § 5.4 derives many more arithmetically valid principles for the provability logic of HA underivable in i-GL✷ , e.g., • ✷¬¬ ✷φ → ✷✷φ. • ✷(¬¬ ✷φ → ✷φ) → ✷✷φ • ✷(φ ∨ ψ) → ✷(φ ∨ ✷ψ). (Lei) We note that the first principle is a consequence of classical c-GL✷ , but the second and third are not. This illustrates that Λ∗(·) is not monotonic. To make this understandable, the reader may note that we both change the theory and the interpretation of the modal operator. Note that substituting ✷⊥ for φ and ¬ ✷⊥ for ψ in Lei yields ⊢ ✷(✷⊥ ∨ ¬ ✷⊥) → ✷(✷⊥ ∨ ✷¬ ✷⊥) → ✷(✷⊥ ∨ ✷⊥) → ✷✷⊥ Hence, adding Lei to c-GL✷ yields ✷✷⊥, i.e., Leivant’s principle is ‘weakly inconsistent’ with classical logic over i-GL✷ . 22 Three remarks are in order. The fact that Löb’s Principle follows from Löb’s work was noted by Leon Henkin who was the referee of Löb’s paper. Secondly, Löb’s proof of Löb’s Principle is fully constructive and goes through even in constructive versions of S12 . Thirdly, Kripke’s proof of Löb’s Principle from the Second Incompleteness Theorem is not constructive —even if we give the Second Incompleteness Theorem the form: if a theory proves its own consistency then it is inconsistent. 23 Interestingly, Gargov’s argument itself uses classical logic. 22 We write Λ ⊞ Λ′ for the closure of Λ ∪ Λ′ under modus ponens. Our insight above yields: c-GL✷ + ✷✷⊥ ⊆ Λ∗HA ⊞ Λ∗PA . In fact, by Theorem 5.3, we have: Λ∗HA ⊞ Λ∗PA = c-GL✷ + ✷✷⊥. Theorem 5.3 (Silly Upperbound). We have: (✷✷⊥ → ¬¬ ✷⊥) 6∈ Λ∗T iff Λ∗T ⊆ c-GL✷ + ✷✷⊥. Our proof presupposes knowledge of the proof of Solovay’s Theorem. The proof can be skipped since nothing but the Silly Upperbound rests on it. Proof. “→” Suppose φ ∈ Λ∗T and c-GL✷ + ✷✷⊥ 0 φ. Then there is a counter Kripke model of depth 2 to φ, say with nodes 0,. . . , n − 1 and root 0. We have i ❁ j iff i = 0 and j > 0. Let T + be T plus ✷T ✷T ⊥ plus sentential excluded third. We work in T + . We define a Solovay function in the usual way: • h0 := 0 • h(p + 1) := ( i h(p) if h(p) ❁ i and proof T (p, pℓ 6= iq) otherwise Here ℓ is the limit of h. W We note that, since ✷T ✷T ⊥ we have ✷T 0<j<n ∃x hx = j. This tells us that inside the box, we can indeed prove that the limit exists. Moreover we have excluded third for sentences of the form ℓ = i. Outside the box we can also prove the existence of the limit by sentential excluded third. Using these two observations we can execute the usual Solovay argument. This gives us ✷T ⊥ and we may conclude that T ⊢ ✷T ✷T ⊥ → ¬¬✷T ⊥. “←” Suppose (✷✷⊥ → ¬¬ ✷⊥) ∈ Λ∗T and Λ∗T ⊆ c-GL✷ + ✷✷⊥. Then it would follow that c-GL✷ ⊢ ✷✷⊥ → ✷⊥. Quod non. ✷ Note that (✷✷⊥ → ¬¬ ✷⊥) 6∈ Λ∗T if T is one of HA, HA + MP, HA + ECT0 . The situation is different for T = HA∗ (cf. § 5.4.4). We formulate the main question of constructive provability logic. Open Question 5.4. What is the provability logic of HA? Is it decidable? We note that the logic is prima facie Π02 . The basic information about classical provability logic can be found in [Smo85; BS91; Boo93; Lin96; Jd98; Šve00; AB04; HV14]. For information about intuitionistic provability logic, see e.g. [Vis94; Iem01b; Iem01a; Vis08; AM14]. 5.4. Preservativity Logic As stated above, Σ01 -preservativity [Vis85; Vis94; Vis02; Iem03; IDZ05] for a theory T is defined as follows: • A JT B if, for all Σ01 -sentences S, if T ⊢ S → A, then T ⊢ S → B. 23 In contrast to Π01 -conservativity and interpretability (see Appendices B and C), defining Σ01 -preservativity does not require an inter-theory notion T J U . We give a characterization of Σ01 -preservativity that is analogous to the OreyHájek characterization for interpretability over PA. Suppose T extends HA. We write ✷T,n for the arithmetization of provability from the axioms of T with Gödel number ≤ n. The theory T is, HA-verifiably, essentially reflexive: for all n and A, we have T ⊢ ✷T,n A → A. Here we allow parameters in the formulation of the reflection principle.24 Theorem 5.5. Suppose T is an extension of HA. Then, A JT B iff, for all n, T ⊢ ✷T,n A → B. This result is verifiable in i-EA. Proof. “→” Suppose A JT B. It follows that (a) if T ⊢ ✷T,n A → A, then (b) T ⊢ ✷T,n A → B. Now note that (a) follows from essential reflexivity. “←” Suppose (c) for all n, T ⊢ ✷T,n A → B and (d) T ⊢ S → A. From (d), we have, for some m, that T ⊢ ✷T,m (S → A). We choose m so large that the finite axiomatization of i-EA has Gödel number ≤ m. By i-EA verifiable Σ01 completeness of extensions of i-EA, T ⊢ S → ✷T,m A. Hence, by (c), T ⊢ S → B. The verifiability in i-EA can be seen by inspection of the above proof. ✷ 5.4.1. i-PreL− and i-PreL We note that ⊤ JHA A is i-EA-provably equivalent to ✷HA A. This means that in our study of Σ01 -preservativity logic of arithmetical theories, we can treat ✷ as a defined connective and focus on LJ . Let F1,T (J) := pT (v0 , v1 ), where pT is a good arithmetization of the relation A JT B. We define Λ◦T := ΛT,F1,T . The principles of i-PreL− are arithmetically valid for all arithmetical theories in our sense, to wit all ∆0 (exp)-axiomatized theories in the arithmetical language extending i-EA. However, i-PreL = i-PreL− + Di is valid for a more restricted group. We need the notion of closure under q-realizability: T ⊢ A implies there is an n such that T ⊢ n·ε↓ ∧ n · ε qe A (cf. Appendix A for notation). A sufficient condition for Di to hold is that T is not only closed under q-realizability, but it also verifies this fact. Theorem 5.6. Suppose T is T -provably closed under q-realizability. Then Di is arithmetically valid in T , i.e., Di is in Λ◦T . 24 This result is folklore. We could not locate a fully worked-out proof in the literature. Some ingredients can be found in [Tro73, Part I, §5], but the treatment of these ingredients contains some gaps. The proof looks as follows. The theory HA verifies cut-elimination for predicate logic. Consider any n. Reason in T . Suppose ✷T,n A. Let p be a cut-free witness of ✷T,n A. All formulas occurring in p will have complexity ≤ m, for some standard m. Here our complexity measure is depth of logical connectives. We can develop a partial satisfaction predicate for formulas of complexity ≤ m that HA-verifiably satisfies the commutation conditions. The standard axioms of T that have Gödel number ≤ n are true (in the sense of our satisfaction predicate), since the Tarski bi-conditionals are derivable. By induction, we can show that all m-derivations from true axioms yield true conclusions. So, a fortiori, we have A. 24 Proof. We write J for JT and ✷ for ✷T . Suppose T is T -provably closed under q-realizability. We reason in T . Suppose (a) A J C and (b) B J C. Suppose ✷(S → (A ∨ B)), where S is a Σ01 -sentence. By q-realizability and the fact that Σ01 -sentences are self-realizing, we can find a recursive index e of a 0-ary, 0,1-valued function, such that (c) ✷(S → e·ε ↓), (d) ✷((e · ε = 0 ∧ S) → A) and (e) ✷((e · ε = 1 ∧ S) → B). From (a) and (d), we get: (f) ✷((e · ε = 0 ∧ S) → C). From (b) and (e) we get (g) ✷((e · ε = 1 ∧ S) → C). From (c), (f) and (g), we obtain ✷(S → C). ✷ The following salient theories T are T -provably (even i-EA-provably) closed under q-realizability: HA, HA + MP, HA + ECT0 , MA = HA + MP + ECT0 and HA∗ (see § 5.4.4), hence i-PreL is arithmetically valid in them. Open Question 5.7. It would be interesting to have a more perspicuous condition for the satisfaction of Di than closure under q-realizability. Moreover, in many cases we can also prove Di using the de Jongh translation. Are there separating examples where either q-realizability works and the de Jongh translation does not or where the de Jongh translation works but q-realizability does not? 5.4.2. The Preservativity Logic of HA The logic i-PreL is incomplete for HA [Vis94; Iem03]. Define: • (χ)(σ) := σ for σ ::= ⊤ | ⊥ | (⊤ J φ) | (σ ∨ σ), where φ ranges over the full language. • (χ)(φ ∧ ψ) := ((χ)(φ) ∧ (χ)(ψ)), • (χ)(φ) := (χ → φ) in all other cases. The following principle is arithmetically valid over HA. V V For χ := i<n (φi → ψi ),Wwe have: ⊢ (χ → (φn ∨ φn+1 )) J j<n+2 (χ)(φj ). An example of a consequence of V is as follows. Consider any non-modal propositional formula φ(p) with at most p free. Suppose that φ(p) is not constructively valid. Then, the principle φ(✷ψ) J (✷ψ ∨ ¬ ✷ψ) is arithmetically valid over HA. Remark 5.8. We have the following salient result about the admissible rules of HA. Suppose φ and ψ are non-modal propositional formulas. Define: • φ ∼ HA ψ if for all arithmetical substitutions σ we have: HA ⊢ σ(φ) ⇒ HA ⊢ σ(ψ). The following are equivalent: (i) φ ∼ HA ψ, (ii) i-PreL + V ⊢ φ J ψ, (iii) i-PreL + V ⊢ ✷φ → ✷ψ. See [Iem03] in combination with [Vis94]. 25 Is i-PreL + V the preservativity logic of HA? We do not think so. The second author has discovered a valid scheme that does not appear to be derivable from i-PreL + V. To save space, we postpone a detailed discussion to future work. Open Question 5.9. Here is a list of more open problems. I. Is i-PreL− the preservativity logic of all extensions of i-EA? In other words, is i-PreL− the intersection of all Λ◦T , where T is an arithmetical extension of i-EA? II. Is i-PreL− the preservativity logic of all extensions of HA? III. Is there an extension T of i-EA such that Λ◦T = i-PreL− ? IV. Is there an extension T of HA such that Λ◦T = i-PreL− ? V. Is there an extension T of i-EA such that Λ◦T = i-PreL? VI. Is there an extension T of HA such that Λ◦T = i-PreL? VII. What is the preservativity logic of HA? VIII. What is the preservativity logic of HA + MP? IX. What is the preservativity logic of HA + ECT0 ? The questions VII, VIII, IX are obviously quite difficult.a As far as we know nobody has seriously worked on questions I–VI. a They could be easier than the question what the provability logic of HA, HA + MP or HA + ECT0 is. Sometimes theories in a richer language are easier to manage. 5.4.3. The Preservativity Logic of classical theories We know a lot about the preservativity logic of classical theories, since JT can be intertranslated with Π01 -conservativity ◮T in the classical case. As a consequence we can translate what we know about the logic of Π01 -conservativity to a result about preservativity logic. Let c-PreL := i-PreL− + em. Theorem 5.10. Suppose that T is Σ01 -sound classical theory that extends IΠ− 1 + Exp. Then, the preservativity logic of T is precisely c-PreL. This result is a translation of Theorem 12 of [BV05], which is a strengthening of the main result of [HM90], the latter in turn being an adaptation of [Sha88] and [Ber90]. For details see Appendix B. 5.4.4. HA∗ and PA∗ The Completeness Principle for a theory T is defined as CPT A → ✷T A. 26 Here A is allowed to contain parameters. Consider any theory T such that T is HA + CPT . Such a theory is easily constructed by the Fixed Point Lemma. One can show that, if HA verifies that T is HA + CPT , then T is unique modulo provable equivalence. Thus, the following definition is justified: HA∗ is the unique theory such that, HA-verifiably, HA∗ is HA + CPHA∗ . The theory HA∗ was introduced and studied in [Vis82]. We have a second way of access to HA∗ via a variant of Gödel’s translation of IPC in S4. We define: • Ag := A if A is atomic. • (·)g commutes with ∧, ∨ and ∃. • (B → C)g := ((B g → C g ) ∧ ✷HA (B g → C g )). • (∀x B)g := (∀x B g ∧ ✷HA ∀x B g ). We have HA∗ ⊢ A iff HA ⊢ Ag . Using the translation (·)g on can show that HA∗ is conservative over HA with respect to formulas that have only Σ1 -formulas as antecedents of implications. The theory HA∗ is the theory in which the incompleteness phenomena lie most closely to the logical surface. We have the strong form of Löb’s Principle HA∗ ⊢ (✷HA∗ A → A) → A. Note that HA∗ ⊢ ¬¬ ✷HA∗ ⊥ is a special case. We are inclined to read this principle as: inconsistency can never be excluded. If we extend PA to U = PA + CPU , we end up with the inglorious U ⊢ ✷U ⊥. However, HA∗ is conservative over HA for a wide class of formulas. So, the Completeness Principle is an example of a kind of extension that makes no real sense in the classical case. The theory HA∗ can be used to provide easy proofs of the independence of KLS (Kreisel-Lacombe-Schoenfield) and MS (Myhill-Shepherdson) from HA [Vis82], simplifying the original ones by Beeson [Bee75] while preserving their basic idea. De Jongh and Visser showed that every prime recursively enumerable Heyting algebra on finitely many generators can be embedded in the Heyting algebra of HA∗ . See [JV96]. Their proof is an adaptation of a proof by Shavrukov [Sha93] in the simplified form due to Zambella [Zam94] concerning the embeddability of Magari algebras in the Magari algebra of Peano Arithmetic. A consequence of the De Jongh-Visser result is the fact that the admissible propositional rules for HA∗ are precisely the derivable rules. In contrast, the admissible propositional rules for HA are the same as the admissible rules for IPC: this is the maximal set of admissible rules that is possible for a theory with the de Jongh property. Thus among theories with the de Jongh property both the minimal possible set of admissible rules and the maximal one are exemplified. See also [Vis99]. We want to show that HA∗ is HA-verifiably closed under q-realizability. The easiest route is via the notion of self-q-realizability. A formula A(~x) (with all 27 free variables shown) is self-q-realizing if there is a number sA such that HA ⊢ A(~x) → (sA · (~x)) qe A(~x), cf. Appendix A for notation. A substantial class of i-EA-verifiably self-q-realizing formulas is the class of auto-q formulas given as follows. Let S range over all Σ01 -formulas, let A range over all formulas and let v range over all variables: • B ::= S | (B ∧ B) | ∀v B | (A → B) We note that the class of auto-q formulas substantially extends the almost negative formulas that are self-r-realising. The instances of the completeness scheme have the form ∀~x (A(~x) → S(~x)), where S is Σ01 . Thus, these instances are auto-q. It follows that HA∗ is HAverifiably closed under q-realizability. Thus, i-PreL∗ := iA + SL✷ + Ma is contained in the preservativity logic of HA∗ , to wit Λ◦HA∗ . There are examples of valid principles that are most probably not in i-PreL∗ . We do not know whether this has any traces in the provability logic of HA∗ . As will be explained in Remark C.3, there is a certain analogy between HA + CT0 ! and HA∗ . We turn to the theory PA∗ , axiomatized by the set α of all sentences A such that PA ⊢ Ag . One can easily show that α is closed under deduction and that PA∗ satisfies CPHA∗ .25 The theory PA∗ verifies the Trace Principle : TP ✷PA∗ ∀x (Ax → Bx) → (∃x Ax ∨ ∀x (Ax → Bx)). This principle is equivalent to ✷PA∗ ∀x Bx → (∃x Ax ∨ ∀x (Ax → Bx)). The presence of the trace principle has as a modal consequence the principle CB✷ ✷φ → ((ψ → φ) ∨ ψ) In [Vis82], it is shown that the logic i-KM✷ is precisely the provability logic of PA∗ .26 We remind the reader that: i-KM✷ = i-SL✷ + CB✷ = i-GL✷ + S✷ + CB✷ . The preservativity logic of PA∗ contains i-PreL− and Sa , but neither Di nor CBa (§ 8.2). 6. Kripke completeness and correspondence ? Apart from being our original motivation to study J, the arithmetical interpretation can occasionally complement the deductive systems proposed in § 4 25 We have demanded that the axiom set of a theory is ∆ (exp). The axioms of PA∗ do not 0 satisfy this demand. So, the official axiom set should be a suitable ∆0 (exp)-set manufactured from α using a version of Craig’s trick. 26 In [Vis82] the equivalent form CB′ is used, cf. Lemma 4.16. ✷ 28 by providing a route to disprove certain judgements of the form X ⊢ φ, i.e., to show non-derivability from suitable sets of axioms (namely, those valid in some arithmetical interpretations): Example 6.1. Interpreting φ J ψ as ✷(φ → ψ) over HA yields iA + L✷ + Ma . This interpretation refutes 4a , La and a fortiori Wa . It follows that 4a is really needed in Lemma 4.14b above to derive Wa . To disprove more such judgements, we need to return to relational insights of § 3 and provide Kripke completeness and correspondence results. Most of this section is based on work we will discuss in a parallel publication [LV]. 6.1. Notions of completeness Given a logic i-X , set Fram(i-X ) := {F | for any V, hF, V i i-X }. Say that i-X is (weakly) complete for (or with respect to) a class of frames K if it is • sound wrt K, i.e., K ⊆ Fram(i-X ) and • any α s.t i-X 6⊢ α can be refuted in a model based on a frame from K. We say that a condition (which may be expressed in a natural language or in a formalized metalanguage like first- or second-order logic) corresponds to a given J-logic i-X if it defines precisely Fram(i-X ). In particular, when a condition is a correspondent of iA + φ, we say it corresponds to φ and correspondingly (pun unintended) use notation Fram(φ). A logic i-X can be complete for much smaller a class than Fram(i-X ) but if it is complete for some class of frames, it is also complete for Fram(i-X ); we can thus take this as a definition what it means to be (weakly) complete without additional qualifications. Incomplete logics, i.e., those which have some non-theorems which cannot be refuted in Fram(i-X ), are sometimes even encountered among those with an arithmetical interpretation, c.f. systems known as GLB and GLP [Jap88; Boo93; HL16], though most “naturally” defined logics tend to be complete. Remark 6.2. Let us recall an important difference between completeness and correspondence V Twhen it comes to combinations (conjunctions) of axioms. Clearly, Fram( Γ) = Fram(γ), so whenever α is a correspondent of φ and β is a corγ∈Γ respondent of ψ, α ∧ β is a correspondent of φ ∧ ψ. Nothing like this needs to hold for completeness, even for a finitely axiomatizable logic. Completeness of iA + φ and iA + ψ for frames defined, respectively, by α and β does not automatically imply that iA + φ + ψ is complete for α ∧ β—or, indeed, for any class of frames whatsoever. This is why in Figure 6.2, Theorems 6.4 and 6.6 below we do not mention correspondence conditions for logics axiomatized by conjunctions/combinations of axioms, but completeness results for such logics need to be stated explicitly. The notion of completeness can be refined further in two orthogonal directions. One of them is the finite model property (fmp, also known as the finite frame 29 Figure 6.2: Correspondence conditions. In this figure, and elsewhere in this paper, ❀ stands for ❁ and → stands for . Some names of principles are taken from Iemhoff and coauthors [Iem01b; Iem03; IDZ05; Zho03], others come from our work to be published separately [LV]. and subset X ⊆ W , set X ↑R := {y ∈ W | ∃x ∈ X.xRy}; in particular, write x ↑R for {x} ↑R . Box k❁ℓm⇒k❁m brilliant m ~> > O /ℓ >~ k 4✷ semi-transitive / mO xO k❁ℓ❁m⇒ ∃x.k ❁ x  m O O k 4a gathering ( /ℓ k 6m ❁-Noetherian (conversely well-founded) and semi-transitive L✷ Wa k❁ℓ❁m⇒ℓm /ℓ supergathering on finite frames: k❁ℓ❁m⇒ ∃x ❂ k.(ℓ ≺ x  m) x ? ? ? / mO ` 6= /ℓ k k ❁ ℓ  m ⇒ ∃x ❂ k.(ℓ  x  m & x ↑❁· ⊆ m ↑❁· ) Ma Montagna S✷ strong k❁ℓ⇒kℓ k CBa ❁-dominated k≺ℓ⇒k❁ℓ k CB✷ weakly ❁-dominated ' 7ℓ 'ℓ 7 6= ? ℓO k ≺ ℓ ⇒ ∃m ❂ k.m  ℓ 6= /o k /o / m Lina weakly semi-linear k ❁ ℓ & k ❁ m ⇒ (m  ℓ OR ℓ  m) Lin✷ strongly semi-linear k ❁ ℓ  ℓ′ & k ❁ m  m′ ⇒ (m′  ℓ′ OR ℓ′  m′ ) ℓO C4✷ semi-dense Hug O pre-reflexive k ❁ ℓ ⇒ ∃x ❂ ℓ.x  ℓ semi-nucleic k ❁ ℓ ⇒ ∃m  k. ∃m′ ❂ m. ℓ  m.m′  ℓ almost reflexive k❁ℓ⇒ℓ❁ℓ /o / y ' > mO /o /ℓ k 30 /o /ℓg k k Appa O k ❁ ℓ ⇒ ∃x  ℓ.∃y ❂ k.y ❁ x k C4a xO o /ℓ{ /o / x property) which simply means completeness wrt a class of finite frames. While the fmp is a much stronger property than weak completeness, it is still rather standard among most “natural” logics. It is not quite the case, however, with another refinement of interest: the notion of strong completeness, i.e., completeness for deductions from infinite sets of premises. This notion can be defined in two different ways using either • the relation Γ ⊢i-X φ defined as “φ is deducible from Γ using all theorems of i-X and Modus Ponens” or • the relation Γ ⊢J i-X φ defined as “φ is deducible from Γ using all theorems of i-X , Modus Ponens and the rule Na ”. A given J-logic i-X is then • strongly locally complete if whenever Γ 6⊢i-X φ, there exists F ∈ Fram(i-X ), a valuation V and a point k in F s.t. F, V, k Γ and F, V, k 6 φ. • strongly globally complete if whenever Γ 6⊢J i-X φ, there exists F ∈ Fram(i-X ), a valuation V s.t. F, V Γ yet for some point k in F, F, V, k 6 φ. As discovered by Frank Wolter [Wol93], these two notions coincide for Kripke semantics of ordinary modal logics. While Wolter was not working with extensions of iA, his reasoning extends to our setting: Theorem 6.3. A J-logic i-X is strongly locally complete iff it is strongly globally complete. Strong completeness is typically achieved as a corollary of stronger results, such as canonicity, which in turn, as first observed by Fine [Fin75] (see also Gehrke et al. [GHV06] for a general treatment), can be obtained as a corollary of elementarity: that is, being complete wrt a first-order definable class of frames. It is not hard to see intuitively the reason for this connection: for a (weakly) complete logic at least, a failure of strong completeness implies a failure of compactness of the Kripke consequence relation, whereas being elementarily definable guarantees compactness of this relation. A suitable notion of canonicity for J-logics has been proposed and studied in the literature [Iem01b; Iem03; Zho03]; in fact, clauses regarding strong completeness in Theorem 6.4 below are corollaries of such canonicity results. 6.2. Completeness and correspondence results Figure 6.2 lists various completeness/correspondence conditions for LJ -principles. Löb-like axioms tend to have counterparts which are not of first-order character, but numerous others can in fact be expressed in first-order logic. Let us turn these claims into proper theorems. First, let us summarize results which are available in the existing literature, or can be relatively easily derived: Theorem 6.4. 31 a. iA is strongly complete (wrt the class of all frames) and enjoys the finite model property [Iem01b, Prop. 4.1.1], [Iem03, Prop. 7], [Zho03, Th. 2.1.10]. b. i-BoxA = iA + Box corresponds to the class of brilliant frames, is strongly complete and enjoys the finite model property. c. i-SA = iA + S✷ corresponds to the class of strong frames, is strongly complete and enjoys the finite model property. d. iA+4✷ corresponds to the class of semi-transitive frames, is strongly complete and enjoys the finite model property. e. iA + 4a corresponds to the class of gathering frames, is strongly complete and enjoys the finite model property [Iem01b, Prop. 4.2.1], [Iem03, Prop. 8]. f. iA + L✷ corresponds to the class of Noetherian semi-transitive frames and enjoys the finite model property [Iem01b, Prop. 4.3.2], [Zho03, Th. 2.2.7]. g. i-GLa = iA + L✷ + 4a (cf. Lemma 4.11) corresponds to the class of Noetherian gathering frames [Iem03, Lem. 9], [IDZ05, Lem. 3.10] and enjoys the finite model property. h. i-GWa = iA + Wa corresponds to the “supergathering” property of Figure 6.2 on the class of finite frames [Zho03, Lem. 3.5.1], [IDZ05, Th. 3.31]. i. iA+Ma corresponds to the class of Montagna frames of Figure 6.2, is strongly complete [Iem03, Prop. 11] and enjoys the finite model property [IDZ05, Lem. 3.21], [Zho03, Th. 3.3.5]. We could not find in the literature an explicit statement of the finite model property of i-PreL = i-GWa + Ma . Moreover, an astute reader probably noticed that we do not claim strong completeness for all logics appearing in the statement of this theorem. The reason is obvious: it is very well-known that variants of the Löb axiom clash with strong completeness and, a fortiori, with canonicity. Boolos and Sambin [BS91] credit Fine and Rautenberg with this observation, which can be now found in any standard monograph on modal logic. This can be extended in several directions, e.g., to logics with weaker axioms (cf. Amerbauer [Ame96]) or to failure of broader notions of strong completeness [Lit05]; see Litak [Lit07, § 3] for more on both counts. In the context of logics for (relative) interpretability (cf. Appendix C), problems with canonicity and strong completeness have been pointed out, e.g., by de Jongh and Veltman [JV90]. Let us adapt such arguments to our setting: Theorem 6.5. i-X is not strongly complete whenever • it is contained between iA + L✷ and c-GL✷ + Lin✷ or • it is contained between iA + L✷ and i-KM.lina . In particular, i-GLa , i-GWa , i-PreL or i-KMa fail to be strongly complete. 32 Proof sketch. We can work in the standard modal language containing just ✷ rather J (in fact, ✷ and → are the only connectives really used). We can also use the freedom offered by Theorem 6.3 and choose to disprove global completeness. Consider now Γ := {✷pi+1 → pi | i ∈ ω} and note that that in any model where Γ is globally satisfied but p0 fails, there must exists an infinite ❁-ascending chain, which allows us to refute Noetherianity, hence refuting L✷ . However, taking ❁ to be an ordered sum of ω with its copy with reverse ordering ω ∗ ,  to be either (for the first clause) discrete or (for the second clause) the reflexive version of ❁ and setting V (pi ) := (i + 1) ↑ produces a model where Γ is globally valid, p0 fails and all theorems of i-X hold under V (despite being refutable in the underlying frame). ✷ Theorem 6.4 above does not cover correspondence and completeness claims for all axioms and frame conditions displayed in Figure 6.2, especially those not directly related to preservativity and provability principles. As it turns out, there is a technique of transferring generic results available for (bi)modal logics over CPC into the intuitionistic setting. For ✷-logics, it has been developed in a series of papers by Wolter and Zakharyaschev [WZ97; WZ98]. We are going to present details of generalization of this technique to J-logics in a separate paper [LV]. For now, let us just list some consequences regarding strong completeness and canonicity (we leave the finite model property out of the picture here): Theorem 6.6 ([LV]). a. iA + CBa correspond to the class of ❁-dominated frames of Figure 6.2 and is strongly complete. b. iA + CB✷ correspond to the class of weakly ❁-dominated frames of Figure 6.2 and is strongly complete. c. i-mHCa is strongly complete (wrt the class of strong ❁-dominated frames) . d. iA + Lina correspond to the class of weakly semilinear frames of Figure 6.2 and is strongly complete. e. iA + Lin✷ correspond to the class of strongly semilinear frames of Figure 6.2 and is strongly complete. f. iA + C4✷ correspond to the class of semi-dense frames of Figure 6.2 and is strongly complete. g. iA + C4a correspond to the class of pre-reflexive frames of Figure 6.2 and is strongly complete. h. iA + Appa correspond to the class of almost reflexive frames of Figure 6.2 and is strongly complete. i. i-PLAA is strongly complete (wrt the class of strong almost reflexive frames) . 33 6.3. Non-derivations Having a developed semantics, we are now in a position to provide more examples of non-derivations between formulas and non-containments between logics. Example 6.7. Consider the formula φ0 := ✷⊥ J ⊥ → ✷⊥. It is easy to see that this formula is in the closed fragment of i-GWa . This means that φ0 is variable-free and provable in i-GWa . We show that φ0 is not in the closed fragment of i-GLa . By Theorem 6.4g, i-GLa is determined by Noetherian gathering frames. Consider the following (Noetherian gathering) model: HcV /b a Clearly, a All points are ❁-irreflexive. We set b  c, a ❁ b ❁ c and the valuation is empty, i.e., V (a) = V (b) = V (c) = ∅. Note that CBa holds in this model. ✷⊥ J ⊥, but a 1 ✷⊥. Example 6.8. Consider the formula φ1 := ✷⊥ J p → ✷(✷⊥ → p). Lemma 4.14d implies that this formula is provable in i-PreL. We prove that i-GWa 0 φ1 by considering the following model satisfying the condition for finite frames for i-GWa as stated in Theorem 6.4h (and Figure 6.2): d: I U p :c All points are ❁-irreflexive. We set b ↑ ⊇ {c, d}, a ❁ b, b ❁ d, a ❁ d and the valuation is V (d) = p and empty otherwise. /b a It is now easy to see that a ✷⊥ J p, but a 1 ✷(✷⊥ → p). Example 6.9. We can improve Example 6.8 by providing a separating closed formula. Consider the formula φ2 := ✷✷⊥ J ¬¬ ✷⊥ → ✷(✷✷⊥ → ¬¬ ✷⊥). Again, Lemma 4.14d implies that this formula is provable in i-PreL. We prove that i-GWa 0 φ2 by considering the following model satisfying the condition for finite frames for i-GWa as stated in Theorem 6.4h (and Figure 6.2): E fO U _ ; dZ i a It is now easy to see that a 5 eO V o <c All points are ❁-irreflexive. As usual, we do not draw the transitive and reflexive closure of → (which, recall, stands for the poset order ). The valuation is empty (and irrelevant anyway). /b ✷✷⊥ J ¬¬ ✷⊥, but a 1 ✷(✷✷⊥ → ¬¬ ✷⊥). 34 Example 6.10. Recall that following Lemma 4.6, we noted that in the disjunctionfree setting, there is no one-variable formula φ(p) s.t. p J q ⊣⊢− φ(p) J (p → q) and CPC ⊢ φ(p). This follows from the fact that φ3 := p J q → (¬¬p → p) J (p → q) is not a theorem of iA: c O p :d All points are ❁-irreflexive. We set q to be false everywhere. It is easy to check the antecendent of φ3 holds at a, but the consequent fails. It is worth noting that i-PreL holds in this model. /b a We can complement this observation by another one: it is not possible to improve the iA-equivalence of Theorem 4.6 by taking a one-variable intuitionistic formula stronger that p ∨ ¬p as the antecedent of J replacing em, as φ4 := p J q → (¬¬p ∨ ¬p) J (p → q) is not a theorem of iA either: c O p All points are ❁-irreflexive. We set q to be false everywhere. Again the antecendent of φ4 holds at a, but the consequent fails; moreover, i-PreL holds in this model. /b a Example 6.11. Here are diagrams illustrating that CBa is not a theorem of i-mHC✷ ; that is, strong frames which are only weakly ❁-dominated. We use the convention that ◦ stand for a ❁-reflexive loop and • for lack thereof. •O ? •O a • ◦ ' 7• & 4 9. . . Arithmetical interpretation provides another interesting way of distinguishing between CB✷ and CBa : § 5.4.4 noted that CB✷ is in the preservativity logic of PA∗ , whereas as stated in Theorem 8.10, CBa does not belong to this system (the only problem is that neither does Di). Example 6.12. So far, we were seeing examples showing that principles for J are often properly stronger than their relatives formulated in terms of ✷ only. Recall that when introducing Fact 4.18, we indicated it is not always the case, as witnessed by semi-linearity axioms. Here is a simple frame for i-GWa +Lina where Lin✷ fails (for both claims one can use Theorem 6.6 and Figure 6.2, but they are straightforward to verify anyway). We are following the same conventions regarding ❁-reflexive and ❁-irreflexive points as in the preceding example: ? G•W • ?• /• 35 Example 6.13. In order to separate C4✷ , C4a and Appa , we provide an example of a semi-dense frame which is not pre-reflexive (on the left) and a pre-reflexive one which are not almost reflexive (on the right): •O o ◦O • /◦ ? •W •  ◦ Again, even without using completeness results of Theorem 6.6, one can easily verify everything by hand (including finding suitable valuations). Example 6.14. In § 7.1, we will use the fact that i-PLAA does not contain Box. As made clear by Theorem 6.6 and Figure 6.2, for this purpose we need a strong almost reflexive frame which is not brilliant: ( /• • 6◦ 7. Strength: arrows, monads, idioms and guards | We have already seen that arithmetical interpretation of modalities provides good motivation for studying intuitionistic logics with strict implication, including those with the strength axiom. This is a very good motivation indeed, but by no means the only one. Such formalisms have continuously reappeared in several recent lines of research, especially in theoretical computer science. 7.1. Notions of computation and arrows Surprisingly, the functional programming community discovered a variant of constructive strict implication at roughly the same time as it appeared in the context of preservativity. More specifically, “(classical) arrows” in the terminology of John Hughes [Hug00] (see also [LWY11]) are in our terminology strong Lewis arrows. Interestingly enough, their unary cousins knows as “idioms” or “applicative functors” [MP08] were discovered later in this community, though a special subclass of applicative functors—to wit, monads corresponding to i-PLL✷ modalities [BBP98; FM97; Kob97]—has been enjoying continuous attention since the seminal paper of Moggi [Mog91]. A particularly convenient basis for our discussion contrasting arrows, idioms and monads is provided by Lindley et al. [LWY11], which we take as the main reference for this subsection. The connection between intuitionistic logics and functional programming is provided by the Curry-Howard correspondence, also known as the Curry-Howard isomorphism or proposition-as-types paradigm (cf. [SU06]).27 While the details are outside of the scope of this paper, the shortest outline is that 27 As pointed out by Sørensen and Urzyczyn, “The Brouwer - Heyting - Kolmogorov Schönfinkel - Curry - Meredith - Kleene - Feys - Gödel - Läuchli - Kreisel - Tait - Lawvere - Howard - de Bruijn - Scott - Martin-Löf - Girard - Reynolds - Stenlund - Constable Coquand - Huet - . . . - isomorphism might be a more appropriate name, still not including 36 • (intuitionistic) formulas correspond to types, • logical connectives correspond to type operators/constructors, • logical axioms correspond to inhabited types and hence deciding theoremhood corresponds to the type inhabitation problem, • logical proofs—e.g., in a variant of a natural deduction system or in a Hilbert-style system—are encoded by proof terms—in a variant of lambda calculus or of combinatory logic—understood as a (functional) programming language and hence • proof normalization corresponds to reduction of these terms, understood as representing computation. In particular, ordinary intuitionistic implication φ → ψ corresponds to forming the function space of programs (proofs) which take data from (proofs for) φ as their input and produce members of (proofs for) ψ as their output. The introduction rule for → corresponds to λ-abstraction and its elimination rule (i.e., ordinary Modus Ponens) corresponds to function application. Nevertheless, one may ask: are “computations” exactly co-extensional with “members of function space”? In the words of Ross Paterson Many programs and libraries involve components that are function-like, in that they take inputs and produce outputs, but are not simple functions from inputs to outputs. . . [S]uch “notions of computation” defin[e] a common interface, called “arrows”. [Pat03, p. 201] What are the laws such a notion of computation is supposed to satisfy? The inhabitation laws of the calculus of “classic arrows” [LWY11, Fig. 4] in a disjunction-free language are given by the following axioms:28 Sa (φ → ψ) → φ J ψ, Tr φ J ψ → ψ J χ → φ J χ, K′a φ J ψ → (φ ∧ χ) J (ψ ∧ χ). Thanks to Lemmas 4.1 and 4.10, we know it is just an axiomatization for i-SA− ! Open Question 7.1. As Lindley et al. [LWY11] work in a type theory without the co-product operator (i.e., the Curry-Howard counterpart of disjunction), the issue of validity of Di simply does not arise. Nevertheless, given the problematic status of Di in preservativity logics of some theories all the contributors.” [SU06, p. viii] Indeed, the Curry-Howard isomorphism provides the most commonly accepted specification of the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic connectives. We could thus only half-jokingly argue that this subsection is yet another place in our paper where Lewis meets Brouwer. 28 Lindley et al. [LWY11] call these axioms arr, >>> and first, respectively. They also use ❀ in place of J. 37 (cf. Open Question 5.7), it seems a valid question whether Di should be a law imposed on all notions of computation—and if not, how to characterize those where it holds. It is an inhabited type for both arrows with apply (monads) and static arrows (idioms), as follows from the discussion below and, correspondingly, Lemmas 4.17g and 4.16c. What is the status of the Box law then (or any of its equivalent forms)? As it turns out, the Curry-Howard interpretation provides another rationale for considering (strong) Lewis arrows not determined by an unary ✷. Lindley et al. [LWY11] call arrows satisfying Box static arrows and show that such arrows correspond to the “idioms” or “applicative functors” of McBride and Paterson [MP08]. Indeed, the inhabitation laws of the calculus for idioms [LWY11, Fig. 3] are exactly those of i-S✷ . This is, however, only a special subclass of computations encoded by arrows: namely those computations “in which commands are oblivious to input” [LWY11]. Lindley [Lin14] rephrases this claim to the effect that idioms are distinguished by their static approach to data flow. However, as said above, just a special subclass of applicative functors is by far the most important from a programming point of view: that of (strong) monads. This subclass of idioms whose type system satisfies in addition the inhabitation law corresponding to C4✷ (and, obviously, a number of equalities between proof terms, which are not of concern to us here) provides the most popular framework for effectful computations. In other words, the Curry-Howard counterpart (the logic of type inhabitation) of the calculus for (strong) monads proposed by Moggi under the name of computational metalanguage [Mog91] is i-PLL✷ : propositional lax logic [BBP98; FM97; Kob97]. Monads can be shown [Hug00; LWY11] to be in 1-1 correspondence with higher-order arrows or classical arrows with apply. To wit, these are arrows satisfying the law: Appa (φ ∧ (φ J ψ)) J ψ. Thus, by Lemma 4.17, the logic of type inhabitation for this subclass of arrows is precisely i-PLAA (propositional logic of arrows with apply). Lindley et al. present a two-context natural deduction system for both i-SA− and i-PLAA, whose proof-term assignment is based on a distinction between terms and commands and argue that higher-order arrows are “promiscuous (in the broader sense of undiscriminating)”, as the “apply” construct corresponding to Appa bridges this distinction carefully maintained in the calculus for i-SA− (which can be thus called meticulous). Another perspective is offered by Lindley [Lin14]: higher-order arrows are distinguished by their dynamic approach not only to data flow, but also to control flow. Remark 7.2. The correspondence between monads and arrows with apply should not be conflated with the one between idioms (whose logic of type inhabitation is i-SA− ) and static arrows, whose logic of type inhabitation is i-BoxSA: i.e., a system where φ J ψ is definable as ✷(φ → ψ). In contrast, Box is obviously not valid in i-PLAA (cf. Example 6.14) and the ✷-only fragment of 38 i-PLAA + Box is a ✷-logic stronger than i-PLL✷ ; e.g., we have that i-PLAA + Box ⊢ ✷(✷φ → φ) and one can easily check that i-PLL✷ 6⊢ ✷(✷φ → φ). Instead, i-PLAA is embedded into i-PLL✷ by interpreting φ J ψ as φ → ✷ψ, cf. [LWY11, § 6]. In fact, we can derive this fact syntactically from Lemma 4.17f above! Remark 7.3. To finish this subsection on another theme from Lewis, note that Symbolic Logic [LL32] had this to say about Appa (appearing therein as postulate 11.7 in the main text and in the famous Appendix II as B7): It might be supposed that this principle would be implicit in any set of assumptions for a calculus of deductive inference. As a matter of fact, 11.7 cannot be deduced from other postulates. [LL32, p. 125] The last sentence29 is pertinent indeed: Appa is the only axiom of the smallest system Lewis was interested in, i.e., S1, which is not a theorem of iA− ! 7.2. Modalities for guarded (co)recursion Another area of recent computer science where strong intuitionistic modalities have found numerous applications is the study of guarded (co)recursion: as an important tool to ensure productivity in (co)programming with coinductive types [KB11a; KB11b; KBH12; AM13; Møg14; BM15; CBGB15] and, on the metalevel, in semantic reasoning about programs involving higher-order store or a combination of impredicative quantification with recursive types [DAB11; BMSS12; BBM14; SB14; SBB15; Jun+15; BGCMB16]. The logics of type inhabitation of these systems are mostly extensions of i-SL✷ , involving either first- or higher-order quantifiers (corresponding to dependent, polymorphic or impredicative types) or additional entities like clock variables [AM13; Møg14; BM15; BGCMB16], or (a constructive analogue of) the universal modality [CBGB15]. Nakano [Nak00; Nak01] proposed using the axioms of i-SL✷ for approximation modality crediting Sambin-de Jongh-style results on elimination of fixpoints as one of his motivations (see [Lit14, § 3] for a detailed discussion of this point); more recent discussion of Nakano-style systems can be found in Abel and Vezzosi [AV14] and Severi [Sev17]. The idea of using such modalities also in the metalanguage for reasoning about semantics of programs has been popularized by Appel et al. [AMRV07], who were nevertheless working with the axiom L✷ rather than SL✷ seen in most later references. As the above overview makes clear, the area has grown too large to allow an adequate summary in this paper. See [Lit14] for more information and [ML17] for an overview of models of guarded (co)recursion, i.e., from our point of view, categorical models for proof systems for fragments of such logics. Our question here is whether the Lewis arrow naturally occurs in this context. 29 It is proved later on p. 495 of [LL32] using a matrix proposed by Parry. 39 In fact, starting from the original paper of Nakano [Nak00] and even more so in references like Abel and Vezzosi [AV14], the introduction/elimination/inference rules governing the behaviour of such an“approximation” or “delay” modality are often formulated combining ✷ and →. This point is perhaps most explicitly addressed by Clouston and Goré [CG15], a reference highly relevant from our point of view, as it does use J (denoted therein as ։), claiming moreover: The main technical novelty of our sequent calculus is that we leverage the fact that the intuitionistic accessibility relation is the reflexive closure of the modal relation, by decomposing implication into a static (classical) component and a dynamic ‘irreflexive implication’ J that looks forward along the modal relation. In fact, this irreflexive implication obviates the need for ✷ entirely, as ✷φ is easily seen to be equivalent to ⊤ J φ. Semantically, the converse of this applies also, as φ J ψ is semantically equivalent to ✷(φ → ψ), but the J connective is a necessary part of our calculus. We maintain ✷ as a first-class connective in deference to the computer science applications and logic traditions from which we draw, but we note that formulae of the form ✷(φ → ψ) are common in the literature—see Nakano’s (→ E) rule [Nak00], and even more directly the ⊛ constructor of [BM13]. We therefore suspect that treating J as a firstclass connective could be a conceptually fruitful side-benefit of this work ([CG15], in a notation adjusted to this paper). Clouston and Goré [CG15] provide a sequent calculus for a logic called here i-KM.lina . The focus on this logic is motivated by Litak’s observation [Lit14] that i-KM.lin✷ is the propositional fragment of the Mitchell-Bènabou logic of the topos of trees proposed as a model of guarded (co)recursion by Birkedal and coauthors [BMSS12] and used ever since [Møg14; CBGB15; Sev17]. Let us note here that Lemma 4.19 implies that any semantics for i-KM.lina must make Box valid: in other words, i-KM.lina can be just seen as another syntactic presentation of i-KM.lin✷ . However, Lemma 4.19 requires all the axioms of i-KM.lin✷ and when studying broader classes of models of guarded (co)recursion [ML17], more flexibility in adding J is possible. Open Question 7.4. Are there natural applications of J-logics not including the Box axiom in terms of guarded (co)recursion? And, more broadly, do arithmetically relevant principles discussed in this paper have a computational interpretation? Let us add that, while Gentzen-style systems are not our main interest here, the above quote from Clouston and Goré [CG15] hints at another motivation for studying constructive J. Namely, even in the setups which make it a definable connective, it can still prove a more convenient primitive from a proof-theoretical point of view than ✷ is. 7.3. Intuitionistic epistemic logic Finally, let us briefly mention yet another recent area of research where strong intuitionistic modalities made a surprising appearance: in the work of Artemov 40 and Protopopescu on intuitionistic epistemic logic [AP16], presented also in this collection.30 These authors work with unary ✷ and call S✷ the principle of “coreflection”. The minimal system denoted by these authors as IEL− corresponds to i-S✷ in our notation, their IEL is obtained by adding31 ¬✷⊥ and IEL+ arises by adding C4✷ —i.e., is an extension of i-PLL✷ . A proof-theoretic justification for these systems is presented in terms of the Brouwer-Heyting-Kolmogorov interpretation. This seems to provide a natural connection with references discussed in § 7.1—but, curiously, none of them seems to be mentioned by Artemov and Protopopescu, neither the extensive literature on i-PLL✷ , nor the rôle of i-S✷ as the logic of applicative functors (idioms, prenuclei . . . ). We leave an epistemic interpretation of strong arrows and extensions of i-SA as a promising subject for future study. » 8. Applications of preservativity Having briefly overviewed other motivations for studying constructive J, let us return to our main one. Preservativity has many applications. A number of these applications can be found in [Vis85] and [Vis94]. We describe one of the main results of those papers in § 8.1. In § 8.2, we show how one can capture the invalidity of the law of excluded middle in terms of preservativity. We illustrate how this result imposes a constraint on possible preservativity logics of theories. 8.1. NNIL The NNIL-formulas (No Nestings of Implications to the Left [Vis85; Vis94]) are defined as follows: • φ ::= ⊥ | ⊤ | p | (φ ∧ φ) | (φ ∨ φ) | (p → φ) It is easy to see that there are only finitely many nonequivalent NNILformulas on finitely many variables. Let p~ be the propositional variables of φ and define φ⋆ as the disjunction of representatives of all IPC-equivalence classes of NNIL-formulas ψ in the variables p~ such that IPC ⊢ ψ → φ. Using the Interpolation Theorem, we see that, for any NNIL-formula χ, we have IPC ⊢ χ → φ if and only if IPC ⊢ χ → φ⋆ . So, φ⋆ is the best NNIL-approximation from below of φ. In more fancy terms, (·)⋆ is the right adjoint of the embedding functor of the preorder category of the NNIL-formulas into the preorder category of all propositional formulas, both preorders being IPC-provable implication. Theorem 8.1 ([Vis85; Vis94]). For any function f from the propositional variables to Σ01 -sentences, φf JHA (φ⋆ )f . Hence, if HA ⊢ φf , then HA ⊢ (φ⋆ )f . 30 For other approaches to intuitionistic epistemic logic cf. also Williamson [Wil92] or Proietti [Pro12] and for a more dynamic take, see Kurz and Palmigiano [KP13]. 31 Litak [Lit14] denotes ¬✷⊥ as (nv)—non-verum. 41 The original aim of [Vis85] was to show: if HA ⊢ φf , then HA ⊢ (φ⋆ )f . However, it turned out that the inductive assumption requires the stronger statement involving preservativity. Thus, preservativity was discovered as a tool for induction loading. Theorem 8.1 can be reformulated in terms of admissible consequence. We define: • φ ∼ HA,Σ01 ψ if for any Σ01 -substitution f , HA ⊢ ψ f whenever HA ⊢ φf . Thus, φ ∼ HA,Σ01 ψ means that φ/ψ is an admissible rule for Σ01 -substitutions over HA. Theorem 8.1 now simply says: φ ∼ HA,Σ01 φ⋆ . It is optimal in the sense that, whenever φ ∼ HA,Σ01 ψ, we have IPC ⊢ φ⋆ → ψ [Vis94]. Thus, φ ∼ HA,Σ01 ψ iff φ⋆ ⊢IPC ψ. If we view ∼ HA,Σ01 and ⊢IPC are pre-ordering categories, this says that (·)⋆ is the left adjoint of the embedding functor of ∼ HA,Σ01 in ⊢IPC . The NNIL-formulas play an important rôle in: the characterization of the provability logic of HA for Σ01 -substitutions by Ardeshir and Mojtahedi [AM14], the study of infon logic [CG13] and several other contexts [Ren89; VBJL95; Yan08]. 8.2. On the falsity of Tertium non Datur In intuitionistic propositional logic, we have the principle ¬ ¬ (φ ∨ ¬ φ). As a consequence, there is no direct logical expression of the constructive insight of the invalidity of the law of excluded middle.32 The connective (·) J ⊥ is a weaker form of negation, say ∼. Can we have, provably in i-EA, that ∼HA (A ∨ ¬ A), for some suitable A? We will show that, for a wide range of theories T , we can indeed find such a sentence A, including T being HA, HA + MP or HA + ECT0 , HA∗ . We write: • T ≤ U if i-EA verifies that T is a subtheory of U . Suppose i-EA verifies Di for U , i.e. suppose that Di is in ΛF1,U ,i-EA . We note that over i-EA we have (✷U ⊥ ∨ ¬ ✷U ⊥) JU ✷U ⊥. This is in the desired direction since we can consider ✷U ⊥ as a weak form of falsity. However, we cannot get the desired result as long as we stay with Σ01 -sentences. Theorem 8.2. Consider any consistent theory U . There is, verifiably in i-EA+ ✸U ⊤, no Σ1 -sentence S such that ∼U (S ∨ ¬ S). Proof. We work in i-EA + ✸U ⊤. Consider a Σ1 -sentence S. Suppose we have ∼U (S ∨ ¬ S). It follows that (S → (S ∨ ¬ S)) JU (S → ⊥). Thus, ⊤ JU ¬ S, ¬ S JU (S ∨ ¬ S) and (S ∨ ¬ S) JU ⊥. Ergo, ✷U ⊥. Quod non. ✷ 32 We can consistently add ¬ ∀x (A(x) ∨ ¬ A(x)) to constructive arithmetic for certain A. E.g., HA plus a weak version of Church’s Thesis (cf. Appendix A) proves ¬ ∀x (x · x ↓∨x · x ↑). 42 To prepare the construction of the promised sentence, we first consider theories V with HA ≤ V . Recall that ✷V,x A stands for (arithmetized) provability from the axioms of V with Gödel number ≤ x. • Feferman provability for V is defined by: △V A := ∃x (✷V,x A ∧ ✸V,x ⊤). We have:33 Fe1 V ⊢ A ⇒ V ⊢ △V A. Fe2 i-EA ⊢ △V (A → B) → (△V A → △V B). Fe3 i-EA ⊢ S → △V S, for Σ01 -sentences S. We note that it follows that i-EA ⊢ ✷V B → △V ✷V B. Fe4 i-EA ⊢ △V B → ✷V B. Fe5 i-EA ⊢ ✸V ⊤ → (△V A ↔ ✷V A). Fe6 i-EA ⊢ ▽V ⊤, where ▽ is ¬△¬. We note that classically Fe4 follows form Fe5. Shavrukov [Sha94] provides a complete axiomatization for the bimodal logic of ordinary provability and Feferman provability for PA. Open Question 8.3. Shavrukov employs a different interpretation of ✷PA,x , to wit provability in IΣx . It would be interesting to find a better analogue of the version of the Feferman predicate employed by Shavrukov for the case of (extensions of) HA. Moreover, the principles given above provide a part of the principles given by Shavrukov for the classical case. We do not get all Shavrukov’s principles in the constructive case. It would be interesting to study how close we can get to his system. Note that, supposing that V is consistent, we cannot get that, for all A, we have V ⊢ △V A → △V △V A. Otherwise, we could reproduce the reasoning for Gödel’s Second Incompleteness Theorem. This leads immediately to a contradiction with Fe6. We remind the reader that the theory V is V -verifiably essentially reflexive. This means that both truly and V -provably, we have: for all n and all A, we have V ⊢ ✷V,n A → A.34 Theorem 8.4. Suppose HA ≤ T . We have i-EA ⊢ A JV △V A. 33 We do not present the principles for triangle as a schematic logic. This is because of the occurrence of a variable over Σ01 -sentences. We would need a many-sorted propositional theory. Of course this is perfectly doable. We just did not develop it in this paper. 34 We have this even for formulas A, when we employ the usual convention for free variables under the box. 43 Proof. Reason in i-EA. Consider any x. We have, by essential reflexivity, ✷V (✷V,x A → (✷V,x A ∧ ✸V,x ⊤)). Hence, ✷V (✷V,x A → △V A). Ergo, by Theorem 5.5, A JV △V A. ✷ Consider the Gödel sentence GV of Feferman provability for V . We have then i-EA ⊢ GV ↔ ¬ △V GV . Whenever the intended theory is clear from the context, we write G for GV . Theorem 8.5. Suppose HA ≤ V . Then, i-EA ⊢ G JV ⊥ and i-EA ⊢ ¬ G JV ⊥. Proof. We reason in i-EA. We have G JV △V G, and hence, G JV ¬ G. Since also G JV G, it follows that G JV ⊥. We have, ¬ G JV △V G and ¬ G JV △V ¬ G. Hence, ¬ G JV △V ⊥. So, by Fe6, we have ¬ G JV ⊥. ✷ Theorem 8.6. Suppose HA ≤ T and that T0 verifies Di for T , i.e. that Di is in ΛF1,U ,T0 . Then, we have T0 ⊢∼T (GT ∨ ¬ GT ). This follows immediately from Theorem 8.5. The reason why T0 appears in the formulation is that we want the result both for T0 = i-EA and for T0 = T . Open Question 8.7. Can we extend Theorem 8.6 to cases where we do not have HA ≤ T ? We can now show that the preservativity logic of PA∗ does not contain Di and CBa . We first prove a purely modal result that delivers both cases. We can achieve it in two ways. Theorem 8.8. A. i-GWa + CB✷ ⊢ (p J ⊥ ∧ ¬ p J ⊥) → ✷⊥. B. i-GLa + CB✷ ⊢ (p J ⊥ ∧ ¬ p J ⊥) → ✷⊥. Proof. (A): We reason in i-GWa + CB✷ + (p J ⊥ ∧ ¬ p J ⊥). By Di, we have (a) (p ∨ ¬ p) J ⊥. On the other hand, we have, by CB✷ , that ✷⊥ → (p ∨ ¬ p). By Na , we have (b) ✷⊥ J (p ∨ ¬ p). Combining (a) and (b), we find ✷⊥ J ⊥ and, hence, by Wa , we obtain ✷⊥. (B): We reason in i-GLa + CB✷ + (p J ⊥ ∧ ¬ p J ⊥). By Di, we have (a) (p ∨ ¬ p) J ⊥. The principle CB✷ gives us ✷⊥ → (p ∨ ¬ p). It follows, by Na , that ✷✷⊥ → ✷(p ∨ ¬ p). Ergo, we have ✷✷⊥ → ✷⊥. We now apply the extended Löb’s Rule, using that our assumption (p J ⊥ ∧ ¬ p J ⊥) is self-necessitating, to conclude that ✷⊥. ✷ As an immediate consequence of Theorems 8.5, 8.6 and 8.8, we have: 44 Theorem 8.9. Suppose HA ≤ T and T is Σ01 -sound. Then, we cannot have both Di and CB✷ in Λ◦T . Theorem 8.10. Neither Di nor CBa are in Λ◦PA∗ . Proof. Since PA∗ is Σ01 -sound and validates CB✷ , by Theorem 8.9, it cannot validate Di. Suppose now PA∗ validates CBa . Then Λ◦PA∗ extends i-mHCa − = iA− + S✷ + CBa . It follows, by Lemma 4.16(c), that Λ◦PA∗ contains Di. Quod non, as we just saw. ✷ Another salient consequence of Theorems 8.5, 8.6 and 8.8 is the following result. Theorem 8.11. For no T ≥ HA, we have: Λ◦T = i-PreL + CB✷ . Proof. Suppose HA ≤ T . Clearly, if (i-PreL + CB✷ ) ⊆ Λ◦T , it follows that T ⊢ ✷T ⊥. But then ✷⊥ ∈ Λ◦T . On the other hand, by a simple Kripke model argument, we can show that i-PreL + CB✷ 0 ✷⊥. ✷ Thus, not every extension of i-PreL− can be obtained as the preservativity logic of a T ≥ HA. We finish this subsection by giving a better condition under which CBa cannot be in the preservativity logic of a theory. This condition will again imply that CBa is not in Λ◦PA∗ . Theorem 8.12. Suppose HA ≤ T , T has the disjunction property and T is consistent. Then, Λ◦T does not contain CBa . Proof. Suppose HA ≤ T , T has the disjunction property and T is consistent. Moreover, suppose Λ◦T contains CBa . We will derive a contradiction. Let G := GT . Since T ⊢ G J ⊥, it follows, by CBa , that T ⊢ G ∨ ¬ G. Hence, by the disjunction property, we find T ⊢ G or T ⊢ ¬ G. Hence ⊤ JT G and ⊤ JT ¬ G. Ergo, T ⊢ ⊥. ✷ We note that PA trivially satisfies CBa . Moreover, HA ≤ PA and PA is (hopefully) consistent. However, PA does not have the disjunction property. 9. Conclusions We are not nearly done, but our space is running out: if we did not stop now, we would have to turn this paper into a monograph. We hope to have convinced the reader that constructive J provides a fascinating subject of research wherever it appears—be it computer science, philosophy or, especially, metatheory of arithmetic. This last context is particularly rife in challenges, despite decades of diligent research in the area. Let us highlight again several lists of unsolved problems regarding arithmetical interpretations: Open Questions 5.4, 5.7, 5.9, 8.3, 8.7 and (in Appendix C below) C.4 and C.11. This, however, is not the only area where interesting open questions abound. As a simple example, consider the study of axiomatization and proof systems 45 for various fragments of LJ (e.g., Open Questions 4.3 and 4.7). Moreover, we have only briefly touched on the question of computational significance of J. Extending category-, proof- and type-theoretic frameworks for “strong arrows” in computer science (§ 7 and references therein) and providing CurryHoward/computational interpretations of different axioms in Table 4.2 (cf. in particular Open Questions 7.1 and 7.4) would seem a natural research direction. A century after the publication of Lewis’ first papers on J and the Survey, the full potential of the strict implication connective still remains to be exploited. It could have been otherwise if Lewis followed his evident interest in non-boolean logics (cf. §2.2). Another decision which in hindsight proved premature was to insist on principles like Appa in even the weakest variant of his system (cf. Remark 7.3), which effectively rules out some of the most fruitful provabilitymotivated applications of J. With these conceptual blocks out of the way and having the advantage of an additional century worth of research on constructive logic, we have no excuse not to carry the torch further. Acknowledgements. We would like to thank: Wesley H. Holliday and Merlin Götlinger for proof-reading parts of an earlier draft of this paper; Mark van Atten for pointing out references on the work of Orlov; the referees for their comments; and, especially, the editors for their successful effort to make the entire process as smooth as possible with a paper of this size. We are also grateful to Erwin R. Catesbeiana for his reflections on the reflection principle. References [AV14] [Abr93] [Ame96] [AMRV07] [AM14] [AP16] [AB04] A. Abel and A. Vezzosi. “A Formalized Proof of Strong Normalization for Guarded Recursive Types”. In: Proceedings of APLAS. Vol. 8858. LNCS. Springer International Publishing, 2014, pp. 140–158. url: http: //dx.doi.org/10.1007/978-3-319-12736-1_8 (cit. on pp. 39–40). S. Abramsky. “Computational interpretations of linear logic”. In: Theoretical Computer Science 111.1–2 (1993), pp. 3–57. url: http://www. sciencedirect . com / science / article / pii / 030439759390181R (cit. on p. 8). M. Amerbauer. “Cut-Free Tableau Calculi for Some Propositional Normal Modal Logics”. In: Studia Logica 57.2/3 (1996), pp. 359–372. url: http://www.jstor.org/stable/20015881 (cit. on p. 32). A. W. Appel, P.-A. Melliès, C. D. Richards, and J. Vouillon. “A very modal model of a modern, major, general type system”. In: Proceedings of POPL. ACM SIGPLAN-SIGACT. 2007, pp. 109–122 (cit. on p. 39). M. Ardeshir and S. Mojtahedi. “The Σ1 -Provability Logic of HA”. In: arXiv preprint arXiv:1409.5699 (2014) (cit. on pp. 21, 23, 42). S. Artemov and T. Protopopescu. “Intuitionistic Epistemic Logic”. In: The Review of Symbolic Logic 9.2 (2016), pp. 266–298 (cit. on p. 41). S. Artemov and L. Beklemishev. “Provability Logic”. In: Handbook of Philosophical Logic, 2nd ed. Vol. 13. Springer, 2004, pp. 229–403 (cit. on pp. 23, 60). 46 [AM13] [Bar53] [Bar46] [Baz03] [Bec30] [Bee75] [BV05] [BBP98] [Ber90] [BJ13] [Bie94] [BM13] [BMSS12] [BGCMB16] [BBM14] [BM15] R. Atkey and C. McBride. “Productive coprogramming with guarded recursion”. In: International Conference on Functional Programming, (ICFP). ACM SIGPLAN. 2013, pp. 197–208 (cit. on p. 39). R. Barcan Marcus. “Strict Implication, Deducibility and the Deduction Theorem”. In: The Journal of Symbolic Logic 18.3 (1953), pp. 234–236. url: http://www.jstor.org/stable/2267407 (cit. on p. 7). R. C. Barcan. “The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication”. In: The Journal of Symbolic Logic 11.4 (1946), pp. 115–118. url: http://www.jstor.org/stable/ 2268309 (cit. on p. 7). V. A. Bazhanov. “The Scholar and the “Wolfhound Era”: The Fate of Ivan E. Orlov’s Ideas in Logic, Philosophy, and Science”. In: Science in Context 16.4 (2003), pp. 535–550 (cit. on p. 7). O. Becker. Zur Logik der Modalitäten. Jahrbuch für Philosophie und phänomenologische Forschung. Halle, 1930 (cit. on p. 7). M. Beeson. “The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations”. In: The Journal of Symbolic Logic 40 (1975), pp. 321–346 (cit. on p. 27). L. Beklemishev and A. Visser. “On the Limit Existence Principles in Elementary Arithmetic and Σ0n -consequences of Theories”. In: Annals of Pure and Applied Logic 136.1–2 (2005), pp. 56–74 (cit. on pp. 26, 57). P. N. Benton, G. M. Bierman, and V. de Paiva. “Computational Types from a Logical Perspective”. In: J. Funct. Programming 8.2 (1998), pp. 177–193 (cit. on pp. 36, 38). A. Berarducci. “The interpretability logic of Peano arithmetic”. In: The Journal of Symbolic Logic 55 (1990), pp. 1059–1089 (cit. on pp. 26, 59). G. Bezhanishvili and R. Jansana. “Esakia Style Duality for Implicative Semilattices”. In: Appl. Categor. Struct. 21 (2013). 10.1007/s10485-0119265-0, pp. 181–208. url: http://dx.doi.org/10.1007/s10485-0119265-0 (cit. on p. 11). G. Bierman. On intuitionistic linear logic. Tech. rep. UCAM-CL-TR346. University of Cambridge, Computer Laboratory, 1994 (cit. on p. 8). L. Birkedal and R. E. Møgelberg. “Intensional Type Theory with Guarded Recursive Types qua Fixed Points on Universes”. In: Proceedings of LiCS. ACM/IEEE. 2013, pp. 213–222 (cit. on p. 40). L. Birkedal, R. E. Møgelberg, J. Schwinghammer, and K. Støvring. “First Steps in Synthetic Guarded Domain Theory: Step-Indexing in the Topos of Trees”. In: LMCS 8 (4 2012), pp. 1–45 (cit. on pp. 39–40). A. Bizjak, H. Grathwohl, R. Clouston, R. Møgelberg, and L. Birkedal. “Guarded Dependent Type Theory with Coinductive Types”. In: Proceedings of FoSSaCS. 2016 (cit. on p. 39). A. Bizjak, L. Birkedal, and M. Miculan. “A Model of Countable Nondeterminism in Guarded Type Theory”. In: Proceedings of RTA-TLCA. Vol. 8560. LNCS. Springer International Publishing, 2014, pp. 108–123. url: http://dx.doi.org/10.1007/978- 3- 319- 08918- 8_8 (cit. on p. 39). A. Bizjak and R. E. Møgelberg. “A Model of Guarded Recursion With Clock Synchronisation”. In: ENTCS 319 (2015), pp. 83–101. url: http: //dx.doi.org/10.1016/j.entcs.2015.12.007 (cit. on p. 39). 47 [Boo93] [BS91] [BD84] [CJ01] [CJ05] [CBGB15] [CG15] [Cor87] [CG13] [Doš93] [Doš92] [Dra88] [DAB11] [Dun72] [FM97] [Fin86] G. Boolos. The Logic of Provability. Cambridge University Press, 1993 (cit. on pp. 23, 29). G. Boolos and G. Sambin. “Provability: the emergence of a mathematical modality”. In: Studia Logica 50 (1991), pp. 1–23 (cit. on pp. 23, 32). M. Božić and K. Došen. “Models for Normal Intuitionistic Modal Logics”. In: Studia Logica 43.3 (1984), pp. 217–245. url: http : / / www . jstor.org/stable/20015164 (cit. on pp. 10–11). S. Celani and R. Jansana. “A Closer Look at Some Subintuitionistic Logics”. In: Notre Dame J. Formal Logic 42.4 (2001), pp. 225–255. url: http://dx.doi.org/10.1305/ndjfl/1063372244 (cit. on pp. 9, 17). S. Celani and R. Jansana. “Bounded distributive lattices with strict implication”. In: Mathematical Logic Quarterly 51.3 (2005), pp. 219– 246. url: http://dx.doi.org/10.1002/malq.200410022 (cit. on pp. 9, 17). R. Clouston, A. Bizjak, H. B. Grathwohl, and L. Birkedal. “Programming and Reasoning with Guarded Recursion for Coinductive Types”. In: Proceedings of FoSSaCS. Vol. 9034. LNCS. Springer, 2015, pp. 407– 421. url: http : / / dx . doi . org / 10 . 1007 / 978 - 3 - 662 - 46678 - 0 _ 26 (cit. on pp. 39–40). R. Clouston and R. Goré. “Sequent Calculus in the Topos of Trees”. In: Proceedings of FoSSaCS. Vol. 9034. LNCS. Springer, 2015, pp. 133–147 (cit. on p. 40). G. Corsi. “Weak Logics with Strict Implication”. In: Mathematical Logic Quarterly 33.5 (1987), pp. 389–406. url: http : / / dx . doi . org / 10 . 1002/malq.19870330503 (cit. on pp. 9, 17). C. Cotrini and Y. Gurevich. “Transitive primal infon logic”. In: The Review of Symbolic Logic 6.02 (2013), pp. 281–304 (cit. on p. 42). K. Došen. “Modal Translations in K and D”. In: Diamonds and Defaults: Studies in Pure and Applied Intensional Logic. Springer Netherlands, 1993, pp. 103–127 (cit. on pp. 9, 17). K. Došen. “The first axiomatization of relevant logic”. In: Journal of Philosophical Logic 21.4 (1992), pp. 339–356. url: https://doi.org/ 10.1007/BF00260740 (cit. on pp. 7–8). A. G. Dragalin. Mathematical Intuitionism: Introduction to Proof Theory. Vol. 67. Translations of Mathematical Monographs. Translated by E. Mendelson, ed. by B. Silver. American Mathematical Society, 1988 (cit. on pp. 20, 56). D. Dreyer, A. Ahmed, and L. Birkedal. “Logical Step-Indexed Logical Relations”. In: LMCS 7.2 (2011). url: http://dx.doi.org/10.2168/ LMCS-7(2:16)2011 (cit. on p. 39). J. M. Dunn. “A modification of Parry’s analytic implication.” In: Notre Dame J. Formal Logic 13.2 (1972), pp. 195–205. url: http://dx.doi. org/10.1305/ndjfl/1093894715 (cit. on p. 8). M. Fairtlough and M. Mendler. “Propositional Lax Logic”. In: Inform. and Comput. 137.1 (1997), pp. 1–33 (cit. on pp. 36, 38). K. Fine. “Analytic implication.” In: Notre Dame J. Formal Logic 27.2 (1986), pp. 169–179. url: http : / / dx . doi . org / 10 . 1305 / ndjfl / 1093636609 (cit. on p. 8). 48 [Fin75] [Gar84] [GHV06] [Gir87] [Göd86] [Gol81] [HM90] [HV14] [HHST96] [HL16] [Hug00] [Iem01a] [Iem03] [IDZ05] [Iem01b] [Jap88] [Jd98] K. Fine. “Some Connections between Elementary and Modal Logic”. In: Proceedings of the Third Scandinavian Logic Symposium. North-Holland Publishing Company, 1975 (cit. on p. 31). G. K. Gargov. “A note on the provability logics of certain extensions of Heyting’s Arithmetic”. In: Mathematical Logic, proceedings of the conference on Mathematical Logic, dedicated to the memory of A. A. Markov (1903–1979), Sofia, September 22–23, 1980. Bulgarian Academy of Sciences, 1984, pp. 20–26 (cit. on p. 22). M. Gehrke, J. Harding, and Y. Venema. “MacNeille completions and canonical extensions”. In: Transactions of the American Mathematical Society 358 (2006), pp. 573–590 (cit. on p. 31). J.-Y. Girard. “Linear Logic”. In: Theoretical Computer Science 50 (1987), pp. 1–102 (cit. on p. 8). K. Gödel. Kurt Gödel: Publications 1929–1936. Vol. 1. Collected Works. ed. by S. Feferman et al. OUP USA, 1986 (cit. on pp. 7–8). R. I. Goldblatt. “Grothendieck Topology as Geometric Modality”. In: MLQ 27.31–35 (1981), pp. 495–529. url: http : / / dx . doi . org / 10 . 1002/malq.19810273104 (cit. on p. 11). P. Hájek and F. Montagna. “The Logic of Π1 -conservativity”. In: Archiv für Mathematische Logik und Grundlagenforschung 30 (1990), pp. 113– 123 (cit. on p. 26). V. Halbach and A. Visser. “The Henkin sentence”. In: The Life and Work of Leon Henkin. Springer, 2014, pp. 249–263 (cit. on p. 23). Logic: from foundations to applications. Clarendon Press, Oxford, 1996 (cit. on p. 50). W. H. Holliday and T. Litak. Complete Additivity and Modal Incompleteness. eScholarship system of UC Berkeley, Working Papers series. Accepted by “The Review of Symbolic Logic”. 2016. url: http://www. escholarship.org/uc/item/8pp4d94t (cit. on p. 29). J. Hughes. “Generalising monads to arrows”. In: Science of Computer Programming 37.1-3 (2000), pp. 67–111. url: http://dx.doi.org/10. 1016/S0167-6423(99)00023-4 (cit. on pp. 36, 38). R. Iemhoff. “A Modal Analysis of Some Principles of the Provability Logic of Heyting Arithmetic”. In: Proceedings of AiML’98. Vol. 2. 2001 (cit. on pp. 11, 23). R. Iemhoff. “Preservativity logic: An analogue of interpretability logic for constructive theories”. In: Mathematical Logic Quarterly 49.3 (2003), pp. 230–249 (cit. on pp. 2–3, 11–12, 14, 17, 23, 25, 30–32). R. Iemhoff, D. De Jongh, and C. Zhou. “Properties of intuitionistic provability and preservativity logics”. In: Logic Journal of IGPL 13.6 (2005), pp. 615–636 (cit. on pp. 2–3, 11–12, 14, 17, 19, 23, 30, 32). R. Iemhoff. “Provability Logic and Admissible Rules”. PhD thesis. University of Amsterdam, 2001 (cit. on pp. 3, 11–12, 14, 17, 23, 30–32). G. K. Japaridze. “The polymodal logic of provability”. In: Intensional Logics and the Logical Structure of Theories: Proceedings of the Fourth Soviet-Finnish Symposium on Logic, Telavi, May 1985. Metsniereba, 1988, pp. 16–48 (cit. on p. 29). G. Japaridze and D. de Jongh. “The logic of provability”. In: Handbook of proof theory. North-Holland Publishing Co., 1998, pp. 475–546 (cit. on pp. 23, 60). 49 [JVV11] [JV90] [JV96] [Jun+15] [Kob97] [Köh81] [KB11a] [KB11b] [KBH12] [KP13] [Lew13] [Lew15] [Lew32] [Lew12] [Lew20] D. de Jongh, R. Verbrugge, and A. Visser. “Intermediate Logics and the de Jongh property”. In: Archive for Mathematical Logic 50.1–2 (2011), pp. 197–213. url: http://dx.doi.org/10.1007/s00153-010-0209-4 (cit. on p. 21). D. de Jongh and F. Veltman. “Provability logics for relative interpretability”. In: [Pet90]. 1990, pp. 31–42 (cit. on p. 32). D. de Jongh and A. Visser. “Embeddings of Heyting algebras”. In: [HHST96]. 1996, pp. 187–213 (cit. on p. 27). R. Jung et al. “Iris: Monoids and Invariants as an Orthogonal Basis for Concurrent Reasoning”. In: Proceedings of POPL. ACM, 2015, pp. 637– 650. url: http://doi.acm.org/10.1145/2676726.2676980 (cit. on p. 39). S. Kobayashi. “Monad as modality”. In: Theoretical Computer Science 175.1 (1997), pp. 29–74. url: http://www.sciencedirect.com/ science/article/pii/S0304397596001697 (cit. on pp. 36, 38). P. Köhler. “Brouwerian Semilattices”. In: Trans. Amer. Math. Soc. 268.1 (1981), pp. 103–126. url: http : / / www . jstor . org / stable / 1998339 (cit. on p. 11). N. R. Krishnaswami and N. Benton. “A semantic model for graphical user interfaces”. In: Proceedings of ICFP. ACM SIGPLAN. ACM, 2011, pp. 45–57 (cit. on p. 39). N. R. Krishnaswami and N. Benton. “Ultrametric Semantics of Reactive Programs”. In: Proceedings of LiCS. IEEE. 2011, pp. 257–266 (cit. on p. 39). N. R. Krishnaswami, N. Benton, and J. Hoffmann. “Higher-order functional reactive programming in bounded space”. In: Proceedings of POPL. ACM, 2012, pp. 45–58. url: http://doi.acm.org/10.1145/2103656. 2103665 (cit. on p. 39). A. Kurz and A. Palmigiano. “Epistemic Updates on Algebras”. In: Logical Methods in Computer Science 9.4 (2013). url: https://doi.org/ 10.2168/LMCS-9(4:17)2013 (cit. on p. 41). C. I. Lewis. “A New Algebra of Implications and Some Consequences”. In: The Journal of Philosophy, Psychology and Scientific Methods 10.16 (1913), pp. 428–438. url: http://www.jstor.org/stable/2012900 (cit. on p. 65). C. I. Lewis. “A Too Brief Set of Postulates for the Algebra of Logic”. In: The Journal of Philosophy, Psychology and Scientific Methods 12.19 (1915), pp. 523–525. url: http://www.jstor.org/stable/2012996 (cit. on p. 65). C. I. Lewis. “Alternative Systems of Logic”. In: The Monist 42.4 (1932), pp. 481–507. eprint: http://monist.oxfordjournals.org/content/ 42 / 4 / 481 . full . pdf. url: http : / / monist . oxfordjournals . org / content/42/4/481 (cit. on pp. 8–9). C. I. Lewis. “Implication and the Algebra of Logic”. In: Mind 21.84 (1912), pp. 522–531. url: http://www.jstor.org/stable/2249157 (cit. on pp. 5, 65). C. I. Lewis. “Strict Implication–An Emendation”. In: The Journal of Philosophy, Psychology and Scientific Methods 17.11 (1920), pp. 300– 302. url: http : / / www . jstor . org / stable / 2940598 (cit. on pp. 6, 65–66). 50 [Lew14] [LL14] [Lew18] [Lew30] [LL32] [Lin14] [LWY11] [Lin96] [Lit05] [Lit14] [Lit07] [LV] [Löb55] [Mar14] [Mar04] [MP08] [McC88] [McC91] C. I. Lewis. “The Matrix Algebra for Implications”. In: The Journal of Philosophy, Psychology and Scientific Methods 11.22 (1914), pp. 589– 600. url: http://www.jstor.org/stable/2012652 (cit. on pp. 6, 9, 65). C. I. Lewis and C. H. Langford. “A Note on Strict Implication (1935)”. In: History and Philosophy of Logic 35.1 (2014), pp. 44–49 (cit. on p. 7). C. Lewis. A Survey of Symbolic Logic. University of California Press, 1918 (cit. on pp. 2–3, 6, 9, 65). C. Lewis. “Logic and Pragmatism”. In: Contemporary American Philosophy: Personal Statements. Vol. 2. Library of philosophy, ed. by J. H. Muirhead. G. Allen & Unwin Limited, 1930. url: https://books. google.de/books?id=3VYNAAAAIAAJ (cit. on p. 65). C. Lewis and C. Langford. Symbolic Logic. Dover, 1932 (cit. on pp. 2, 6–8, 39, 66). S. Lindley. “Algebraic effects and effect handlers for idioms and arrows”. In: Proceedings of WGP. ACM, 2014, pp. 47–58 (cit. on p. 38). S. Lindley, P. Wadler, and J. Yallop. “Idioms are Oblivious, Arrows are Meticulous, Monads are Promiscuous”. In: ENTCS 229.5 (2011). Proceedings of MSFP, pp. 97–117. url: http://www.sciencedirect. com/science/article/pii/S1571066111000557 (cit. on pp. 36–39). P. Lindström. “Provability logic – a short introduction”. In: Theoria 62.1-2 (1996), pp. 19–61 (cit. on p. 23). T. Litak. “An algebraic approach to incompleteness in modal logic”. PhD thesis. Japan Advanced Institute of Science and Technology, 2005 (cit. on p. 32). T. Litak. “Constructive modalities with provability smack”. In: Leo Esakia on duality in modal and intuitionistic logics. Vol. 4. Outstanding Contributions to Logic. Springer, 2014, pp. 179–208. url: https : / / arxiv.org/abs/1708.05607 (cit. on pp. 10, 14, 39–41). T. Litak. “The non-reflexive counterpart of Grz”. In: Bulletin of the Section of Logic 36.3–4 (2007). A special issue In Honorem Hiroakira Ono edited by Piotr Lukowski, pp. 195–208. url: http://www.filozof. uni.lodz.pl/bulletin/pdf/36_34_10.pdf (cit. on p. 32). T. Litak and A. Visser. “Lewis arrow fell off the wall: decompositions of constructive strict implication”. in preparation (cit. on pp. 16–18, 29–30, 33). M. Löb. “Solution of a problem of Leon Henkin”. In: Journal of Symbolic Logic 20 (1955), pp. 115–118 (cit. on p. 22). E. Mares. “Editor’s Introduction to C.I. Lewis and C.H. Langford “A Note on Strict Implication””. In: History and Philosophy of Logic 35.1 (2014), pp. 38–43 (cit. on p. 7). E. D. Mares. Relevant Logic: A Philosophical Interpretation. Cambridge University Press, 2004 (cit. on p. 8). C. McBride and R. Paterson. “Applicative programming with effects”. In: J. Funct. Programming 18.1 (2008), pp. 1–13 (cit. on pp. 36, 38). D. McCarty. “Constructive validity is nonarithmetic”. In: The Journal of Symbolic Logic 53 (1988), pp. 1036–1041 (cit. on p. 2). D. McCarty. “Incompleteness in intuitionistic metamathematics”. In: Notre Dame Journal of Formal Logic 32 (1991), pp. 323–358 (cit. on p. 2). 51 [ML17] [MR13] [Møg14] [Mog91] [Mur05] [Nak00] [Nak01] [OP99] [Par70] [Pat03] [Pet90] [Pit15] [Pli09] [Pro12] [POY04] [Pym02] [Ren89] S. Milius and T. Litak. “Guard Your Daggers and Traces: Properties of Guarded (Co-)recursion”. In: Fundamenta Informaticae 150 (2017). special issue FiCS’13 edited by David Baelde, Arnaud Carayol, Ralph Matthes and Igor Walukiewicz, pp. 407–449. url: http://arxiv.org/ abs/1603.05214 (cit. on pp. 39–40). I. Moerdijk and G. Reyes. Models for smooth infinitesimal analysis. Springer Science & Business Media, 2013 (cit. on p. 2). R. E. Møgelberg. “A Type Theory for Productive Coprogramming via Guarded Recursion”. In: Proceedings of CSL-LiCS. ACM, 2014, 71:1– 71:10. url: http://doi.acm.org/10.1145/2603088.2603132 (cit. on pp. 39–40). E. Moggi. “Notions of computation and monads”. In: Inform. and Comput. 93 (1 1991), pp. 55–92. url: http://portal.acm.org/citation. cfm?id=116981.116984 (cit. on pp. 36, 38). M. G. Murphey. C. I. Lewis: The Last Great Pragmatist. SUNY Series in Philosophy. SUNY Press, 2005 (cit. on pp. 5–7, 65). H. Nakano. “A Modality for Recursion”. In: Proceedings of LiCS. IEEE. 2000, pp. 255–266 (cit. on pp. 39–40). H. Nakano. “Fixed-Point Logic with the Approximation Modality and Its Kripke Completeness”. In: Proceedings of TACS. Vol. 2215. LNCS. Springer, 2001, pp. 165–182 (cit. on p. 39). P. W. O’Hearn and D. J. Pym. “The Logic of Bunched Implications”. In: B. Symb. Log. 5.2 (1999), pp. 215–244. url: http://www.jstor. org/stable/421090 (cit. on pp. 8, 17). W. T. Parry. “In memoriam: Clarence Irving Lewis (1883–1964).” In: Notre Dame J. Formal Logic 11.2 (1970), pp. 129–140. url: http : //dx.doi.org/10.1305/ndjfl/1093893933 (cit. on pp. 6–7). R. Paterson. “Arrows and Computation”. In: The Fun of Programming. Palgrave, 2003, pp. 201–222. url: http : / / www . soi . city . ac . uk / ~ross/papers/fop.html (cit. on p. 37). Mathematical logic, Proceedings of the Heyting 1988 summer school in Varna, Bulgaria. Plenum Press, Boston, 1990 (cit. on p. 50). Proceedings of FoSSaCS. Vol. 9034. LNCS. Springer, 2015. V. Plisko. “A survey of propositional realizability logic”. In: The Bulletin of Symbolic Logic 15.01 (2009), pp. 1–42 (cit. on p. 21). C. Proietti. “Intuitionistic Epistemic Logic, Kripke Models and Fitch’s Paradox”. In: Journal of Philosophical Logic 41.5 (2012), pp. 877–900. url: http://dx.doi.org/10.1007/s10992-011-9207-1 (cit. on p. 41). D. J. Pym, P. W. O’Hearn, and H. Yang. “Possible worlds and resources: the semantics of BI”. In: Theoretical Computer Science 315.1 (2004). Mathematical Foundations of Programming Semantics, pp. 257– 305. url: http://www.sciencedirect.com/science/article/pii/ S0304397503006248 (cit. on pp. 8, 17). D. Pym. The Semantics and Proof Theory of the Logic of Bunched Implications. Vol. 26. Appl. Log. Ser. Kluwer Academic Publishers, 2002 (cit. on pp. 8, 17). G. Renardel de Lavalette. “Interpolation in fragments of intuitionistic propositional logic”. In: Journal of Symbolic Logic 54.04 (1989), pp. 1419–1430 (cit. on p. 42). 52 [Sev17] [Sha94] [Sha93] [Sha88] [SBB15] [Sim94] [Smo73] [Smo85] [Sol76] [SU06] [Sot84] [Šve00] [SB14] [Tro92] [Tro73] [TD88] [van07] P. Severi. “A Light Modality for Recursion”. In: Proceedings of FOSSACS 2017. Springer Berlin Heidelberg, 2017, pp. 499–516. url: http: //dx.doi.org/10.1007/978-3-662-54458-7_29 (cit. on pp. 39–40). V. Shavrukov. “A smart child of Peano’s”. In: Notre Dame Journal of Formal Logic 35 (1994), pp. 161–185 (cit. on p. 43). V. Shavrukov. “Subalgebras of diagonalizable algebras of theories containing arithmetic”. In: Dissertationes mathematicae (Rozprawy matematyczne) CCCXXIII (1993) (cit. on p. 27). V. Shavrukov. The logic of relative interpretability over Peano arithmetic (in Russian). Tech. rep. Report No.5. Stekhlov Mathematical Institute, Moscow, 1988 (cit. on pp. 26, 59). F. Sieczkowski, A. Bizjak, and L. Birkedal. “ModuRes: A Coq Library for Modular Reasoning About Concurrent Higher-Order Imperative Programming Languages”. In: Proceedings of ITP. Vol. 9236. LNCS. Springer, 2015, pp. 375–390. url: http://dx.doi.org/10.1007/9783-319-22102-1_25 (cit. on p. 39). A. K. Simpson. “The Proof Theory and Semantics of Intuitionistic Modal Logic”. PhD thesis. University of Edinburgh, 1994. url: http: //homepages.inf.ed.ac.uk/als/Research/thesis.ps.gz (cit. on p. 10). C. Smoryński. “Applications of Kripke Models”. In: Metamathematical Investigations of Intuitionistic Arithmetic and Analysis. Springer Lecture Notes 344. Springer, 1973, pp. 324–391 (cit. on p. 21). C. Smoryński. Self-Reference and Modal Logic. Universitext. Springer, 1985 (cit. on p. 23). R. Solovay. “Provability interpretations of modal logic”. In: Israel Journal of Mathematics 25 (1976), pp. 287–304 (cit. on p. 22). M. H. Sørensen and P. Urzyczyn. Lectures on the Curry-Howard Isomorphism. Vol. 149. Stud. Logic Found. Math. Elsevier Science Inc., 2006 (cit. on pp. 36–37). V. Sotirov. “Modal theories with intuitionistic logic”. In: Mathematical Logic, Proc. Conf. Math. Logic Dedicated to the Memory of A. A. Markov (1903 - 1979), Sofia, September 22 - 23, 1980. 1984, pp. 139– 171 (cit. on p. 11). V. Švejdar. “On Provability Logic”. In: Nordic Journal of Philosophical Logic 4.2 (2000), pp. 95–116 (cit. on p. 23). K. Svendsen and L. Birkedal. “Impredicative Concurrent Abstract Predicates”. In: Proceedings of ESOP. Vol. 8410. LNCS. Springer, 2014, pp. 149–168. url: http://dx.doi.org/10.1007/978-3-642-548338_9 (cit. on p. 39). A. S. Troelstra. Lectures on Linear Logic. CSLI Lecture Notes 29, Center for the Study of Language and Information, 1992 (cit. on p. 8). A. Troelstra. Metamathematical investigations of intuitionistic arithmetic and analysis. Springer Lecture Notes 344. Springer Verlag, 1973 (cit. on pp. 20–21, 24, 55–56, 63). A. Troelstra and D. van Dalen. Constructivism in Mathematics, vol 1. Vol. 121. Studies in Logic and the Foundations of Mathematics. North Holland, 1988 (cit. on pp. 20–21, 55–56). M. van Atten. “The hypothetical judgement in the history of intuitionistic logic”. In: Logic, Methodology, and Philosophy of Science XIII 53 [van90a] [van90b] [Vis98] [Vis81] [Vis06a] [Vis08] [Vis85] [Vis90] [Vis82] [Vis06b] [Vis94] [Vis99] [Vis02] [Vis14] [VBJL95] (LMPS XIII). College Publications, 2007, p. 662. url: https://halshs. archives-ouvertes.fr/halshs-00791548 (cit. on p. 8). J. van Oosten. “Lifschitz’ Realizability”. In: The Journal of Symbolic Logic 55.02 (1990), pp. 805–821 (cit. on p. 61). W. P. van Stigt. Brouwer’s intuitionism. North-Holland, 1990 (cit. on p. 4). A. Visser. “An Overview of Interpretability Logic”. In: Advances in Modal Logic. Vol. 1, 87. CSLI Lecture Notes. Center for the Study of Language and Information, 1998, pp. 307–359 (cit. on p. 60). A. Visser. Aspects of diagonalization and provability. Ph.D. Thesis, Department of Philosophy, Utrecht University, 1981 (cit. on p. 12). A. Visser. “Categories of Theories and Interpretations”. In: Logic in Tehran. Proceedings of the workshop and conference on Logic, Algebra and Arithmetic, held October 18–22, 2003. Vol. 26. Lecture Notes in Logic. ASL, A.K. Peters, Ltd., 2006, pp. 284–341 (cit. on p. 59). A. Visser. “Closed Fragments of Provability Logics of Constructive Theories”. In: Journal of Symbolic Logic 73.3 (2008), pp. 1081–1096 (cit. on p. 23). A. Visser. Evaluation, provably deductive equivalence in Heyting’s Arithmetic of substitution instances of propositional formulas. Logic Group Preprint Series 4. Faculty of Humanities, Philosophy, Utrecht University, 1985 (cit. on pp. 2, 12, 23, 41–42). A. Visser. “Interpretability logic”. In: Mathematical logic, Proceedings of the Heyting 1988 summer school in Varna, Bulgaria. Plenum Press, Boston, 1990, pp. 175–209 (cit. on p. 60). A. Visser. “On the completeness principle: A study of provability in Heyting’s arithmetic and extensions”. In: Annals of Mathematical Logic 22.3 (1982), pp. 263–295 (cit. on pp. 12, 14, 27–28). A. Visser. “Predicate logics of constructive arithmetical theories”. In: Journal of Symbolic Logic 71.4 (2006), pp. 1311–1326 (cit. on pp. 60– 61). A. Visser. Propositional combinations of Σ-sentences in Heyting’s Arithmetic. Logic Group Preprint Series 117. Faculty of Humanities, Philosophy, Utrecht University, 1994 (cit. on pp. 2, 12, 23, 25, 41–42, 58). A. Visser. “Rules and Arithmetics”. In: Notre Dame Journal of Formal Logic 40.1 (1999), pp. 116–140 (cit. on pp. 21, 27). A. Visser. “Substitutions of Σ01 -sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic”. In: Annals of Pure and Applied Logic 114 (2002), pp. 227–271 (cit. on p. 23). A. Visser. “Why the theory R is special”. In: Foundational Adventures. Essays in honour of Harvey Friedman. Originally published online by Templeton Press in 2012. See http://foundationaladventures.com/. College Publications, 2014, pp. 7–23 (cit. on p. 59). A. Visser, J. van Benthem, D. de Jongh, and G. R. de Lavalette. “NNIL, a Study in Intuitionistic Propositional Logic”. In: Modal Logic and Process Algebra, a Bisimulation Perspective. CSLI Lecture Notes, no. 53. Center for the Study of Language and Information, 1995, pp. 289–326 (cit. on p. 42). 54 [Wil92] [Wol93] [WZ98] [WZ97] [Yan08] [Zam94] [Zho03] T. Williamson. “On Intuitionistic Modal Epistemic Logic”. In: Journal of Philosophical Logic 21.1 (1992), pp. 63–89. url: http://www.jstor. org/stable/30226465 (cit. on p. 41). F. Wolter. “Lattices of Modal Logics”. PhD thesis. Fachbereich Mathematik, Freien Universität Berlin, 1993 (cit. on p. 31). F. Wolter and M. Zakharyaschev. “Intuitionistic Modal Logics as fragments of Classical Bimodal Logics”. In: Logic at Work, Essays in honour of Helena Rasiowa. Springer–Verlag, 1998, pp. 168–186 (cit. on pp. 11, 33). F. Wolter and M. Zakharyaschev. “On the relation between intuitionistic and classical modal logics”. In: Algebra and Logic 36 (1997), pp. 121– 125 (cit. on pp. 11, 33). F. Yang. “Intuitionistic subframe formulas, NNIL-formulas and n-universal models”. MA thesis. Universiteit van Amsterdam, 2008 (cit. on p. 42). D. Zambella. “Shavrukov’s theorem on the subalgebras of diagonalizable algebras for theories containing I∆0 + EXP”. In: Notre Dame Journal of Formal Logic 35 (1994), pp. 147–157 (cit. on p. 27). C. Zhou. “Some Intuitionistic Provability and Preservativity Logics (and their interrelations)”. MA thesis. ILLC, University of Amsterdam, 2003 (cit. on pp. 12, 30–32). A. A recap of realizability – We need Kleene’s T-predicate: T(e, x, p) means p is a halting computation for the partial recursive function with index e on input x. We write U(p) = y for: the result of computation p is y. We employ the usual assumptions that for at most one p we have T(e, x, p) and that U(p) ≤ p. Define: • e · x = y if ∃p (T(e, x, p) ∧ U(p) = y). • p : (e · x = y) if T(e, x, p) ∧ U(p) = y. • e ·z x = y if ∃p ≤ z p : (e · x = y) or (∀q ≤ z ¬T(e, x, q) ∧ y = 0). • e · x ↓ for (the partial recursive function with index) e being defined on x and e · x ↑ otherwise. Sometimes we will need Kleene application for functions of several arguments. In such cases, we will write x · (~y ). The tuple (~y ) is tacitly identified with a number, in particular we use ε for the (code of the) empty sequence. We have several variants of the (intuitionistic) Church’s Thesis: CT0 ∀x ∃y Axy → ∃e ∀x ∃y (e·x = y ∧Axy). This is the standard arithmetical form of the Thesis, with only numerical quantifiers appearing (modulo a version of the choice principle), rather than an universally quantified function symbol [Tro73, 1.11.7,p.95], [TD88, 4.3,p.193]. CT0 ! ∀x ∃!y Axy → ∃e ∀x ∃y (e·x = y∧Axy). This slightly weakened form will play a central rôle in Appendix C.4.2, where more references are provided. 55 ECT0 is the extended Church’s Thesis [TD88, 4.4,p.199], [Tro73, 3.2.14,p.195]: ∀x (Bx → ∃y Axy) → ∃e ∀x (Bx → ∃y (e · x = y ∧ Axy)) where B ranges over almost negative formulas: • B ::= S | (B ∧ B) | (B → B) | ∀v B and S ranges over Σ01 -formulas. Almost negative formulas will play an important rôle in Appendix C.4.3. From § 5.4.1 on, we have been using the notion of q-realizability [Tro73, § 3.2.3, p. 189], a variant of the usual Kleene realizability: xe qA := A (A atomic) xe q (A ∧ B) := (j1 x)e q A ∧ (j2 x)e qB xe q (A ∨ B) := (j1 x = 0 → (j2 x)e q A) ∨ (j1 x 6= 0 → (j2 x)e q B) xe q (A → B) := (A → B) ∧ ∀v(ve q A → ∃z(x · v = z ∧ ze q B)) xe q (∃vAv) := (j2 x)e q A(j1 x) xe q (∀vAv) := ∀v(Av ∧ ∃z(x · v = z ∧ ze q Av)) where j1 , j2 are the inverses of a chosen pairing function. Note that, unlike Troelstra [Tro73, § 3.2.3], we choose to plug additional conjuncts into clauses for → and ∀, rather than for ∨ and ∃. Apart for Troelstra [Tro73] and Troelstra and van Dalen [TD88], another reference on realizability in HA we recommend is Dragalin [Dra88]. B. Π01 -conservativity – In this appendix, we discuss both classical and constructive interpretability logic. An arithmetical theory U is Π01 -conservative over a theory T or T ◮ U if, for all Π01 -sentences P , we have, if U ⊢ P , then T ⊢ P .35,36 We write A ◮T B for (T + A) ◮ (T + B). We expand the language of propositional logic with the unary ✷ and the binary ✄. Consider any theory T . We set F2,T (✷) := provT (v0 ) and F2,T (✄) := piconT (v0 , v1 ). Par abus de langage, we write ◮T for ✄F2,T , thus introducing an innocent ambiguity. We write Λ•T for ΛT,F2,T . We note that a Π01 -sentence is constructively equivalent to the negation of Σ01 sentence. This implies that A → P is equivalent to ¬¬ A → P . Thus, we 35 The use of the notation ◮ is just local in this paper. Often one uses ✄Π0 . 1 of the general case, where we also consider non-arithmetical theories, reveals that Π1 -conservativity is ‘really’ a relation between interpretations of a basic arithmetical theory in various theories. 36 Reflection 56 find that ¬¬ A and A are mutually Π01 -conservative over T . This means that ✷T can only be defined from ◮T for theories in which ✷T A and ✷T ¬¬ A are provably equivalent for all A. Hence, in general provability cannot be defined from Π01 -conservativity over constructive theories. B.1. The classical case Consider a classical theory T . We have T -verifiably that A JT B iff ¬ B ◮T ¬ A, and A ◮T B iff ¬ B JT ¬ A. Thus, over T , Σ01 -preservativity and Π01 conservativity are intertranslatable. This tells us that the Σ01 -preservativity logic of T can be found via a transformation of the Π01 -conservativity logic of T . The logic ILM consists of c-GL✷ plus the following principles. J1 ✷(φ → ψ) → φ ✄ ψ BL ✷(φ → ψ) → φ J ψ J2 (φ ✄ ψ ∧ ψ ✄ χ) → φ ✄ χ Tr (φ J ψ ∧ ψ J χ) → φ J χ J3 (φ ✄ χ ∧ ψ ✄ χ) → (φ ∨ ψ) ✄ χ Ka (φ J ψ ∧ φ J χ) → φ J (ψ ∧ χ) J4 φ ✄ ψ → (✸φ → ✸ψ) LB φ J ψ → (✷φ → ✷ψ) J5 ✸φ ✄ φ 4a φ J ✷φ M φ ✄ ψ → (φ ∧ ✷χ) ✄ (ψ ∧ ✷χ) Ma φ J ψ → (✷χ → φ) J (✷χ → ψ) The list of principles for preservativity given above is equivalent to c-PreL := i-PreL− + em. See Lemma 4.1, Fact 4.2, Lemmas 4.11 and 4.14. Theorem 12 of [BV05] yields that the Π01 -conservativity logic of T is ILM whenever T is an extension of IΠ− 1 + Exp. This class of theories contains such salient theories as IΣ1 and PA. Thus, we have justified Theorem 5.10, which tells us that Λ◦T = c-PreL if T is a Σ01 -sound extension of IΠ− 1 + Exp. We note that the principle corresponding to La would have been: (†) (φ ◮ ψ ∧ φ ◮ χ) → φ ◮ (ψ ∧ χ). Let T be a Σ10 -sound theory with PA ≤ T . Consider the sentence G := GT from § 8.2. Suppose T satisfies (†). We have, in T , both ⊤ ◮T G and ⊤ ◮T ¬ G. It follows that we have ⊤ ◮T ⊥, i.e. ✷T ⊥. However, this contradicts Σ10 soundness. B.2. The constructive case In this subsection we zoom in on the case of HA. Here the situation for Π01 conservativity is quite different. We still have, HA-verifiably, A ◮HA B iff ¬ B JHA ¬ A. However, we do not have the equivalence of A JHA B and ¬ B ◮HA ¬ A. The equivalence fails in both directions. 57 We have (¬¬ ✷HA ⊥ → ✷HA ⊥) J ✷HA ⊥ [Vis94], but we do not have ¬ ✷HA ⊥ ◮HA ¬ (¬¬ ✷HA ⊥ → ✷HA ⊥), as this is equivalent to ✷HA ¬¬ ✷HA ⊥. In the other direction, trivially, we do have ¬ (✷HA ⊥ ∨ ¬ ✷HA ) ◮HA ⊥. But as shown in § 5.3, ⊤ J (✷HA ⊥ ∨ ¬ ✷HA ⊥) fails. It is easily seen that the logic Λ•HA contains i-ILM, the theory axiomatized by i-GL✷ + J1-5 + M. However, it contains more. As noted above, we have the principle ⊢ ¬¬ φ ✄ φ. C. Interpretability – In this appendix, we discuss both classical and constructive interpretability logic. C.1. Basics NB: The definitions of this subsection work for all theories in finite signature. So in this subsection the theory need not be arithmetical and the axiom set can be just any set of axioms regardless of the complexity. As is well known, purely relational signatures can simulate signatures with terms via a term-unraveling procedure. Thus, we can justify defining interpretations only for relational languages. A one-dimensional translation τ between relational signatures Ξ and Θ provides a domain formula δτ (v0 ) of signature Θ and assigns to each n-ary Ξ-predicate a Θ-formula Pτ (v0 , . . . , vn−1 ). Here the variables of δτ and Pτ are among those shown. We define a translation A 7→ Aτ from Ξ-formulas to Θ-formulas as follows: • P τ (x0 , . . . , xn−1 ) := Pτ (x0 , . . . , xn−1 ) (in case an xi is not free for vi in Pτ (v0 , . . . , vn−1 ), we employ the mechanism of renaming bound variables.) • (·)τ commutes with the propositional connectives. • (∀x B)τ := ∀x (δτ (x) → B τ ), (∃x B)τ := ∃x (δτ (x) ∧ B τ ). Nota bene: we also allow identity to be translated to a different formula. We can define the more complex notion of many-dimensional translation with parameters. In the many-dimensional case a sequence of objects of the interpreting theory represents an object in the interpreted theory. In the case with parameters allow a sequence of extra free variables, the parameters, to occur in the domain formula and in the translations of the predicate symbols. Suppose T has signature Θ and U has signature Ξ. We define: • An interpretation K : U → T is a triple hU, τ, T i, such that, for all Ξsentences A, if U ⊢ A, then T ⊢ Aτ . • T ✄ U if there is an interpretation K : U → T . • A ✄T B if (T + A) ✄ (T + B). 58 If we allow parameters, we add a parameter-domain αK to the specification of K. We demand that K : U → T iff, T proves that αK is non-empty and that, for all Ξ-sentences A, if U ⊢ A, then T ⊢ ∀w ~ (αK (w) ~ → Aτ,w~ ). We write δK for δτK and PK for PτK . For more information about the definition of an interpretation, see e.g. [Vis06a] and [Vis14]. In the case of extensions of i-EA as the interpreting theory one can show that, for our purposes, allowing many-dimensional interpretations makes no difference. We can eliminate the higher dimensions using Cantor pairing. In case we have extensions of PA as the interpreting theory, allowing parameters makes no difference. We can eliminate parameters using the Orey-Hájek Characterization that guarantees an interpretation without parameters whenever there is an interpretation. In case we are not considering extensions of PA, it is in most cases unknown whether allowing parameters has an effect on the interpretability logic. If the interpreting theory is an extension of PA we can always eliminate domain relativization and we can always replace an interpretation by an identity preserving equivalent. In case the interpreted theory has PA-provably infinitely many arguments, we even can do both at the same time. If the interpreting theory is classical and does define one element in the interpreted theory, we can eliminate the domain relativization by setting all elements outside the original domain equal to the definable element. If we allow parameters we can eliminate the domain relativization always as long as the interpreting theory is classical. C.2. Interpretability Logic introduced The relation ✄T can be arithmetized, say by intT . We expand the language of propositional logic with the unary ✷ and the binary ✄. Consider any theory T with a ∆0 (exp)-axiomatization. We set F3,T (✷) := provT (v0 ) and F3,T (✄) := intT (v0 , v1 ). Par abus de langage, we write ✄T for ✄F3,T , thus introducing an e T for ΛT,F . innocent ambiguity. We write Λ 3,T In the classical case ✷T A is equivalent to ¬ A ✄T ⊥. Thus, classically, we also have the option to expand only with ✄ and treat ✷ as a defined symbol. This equivalence can fail intuitionistically. One can see this, e.g., by taking T := HA and A := (✷HA ⊥ ∨ ¬ ✷HA ⊥). At present it is unknown whether ✷HA A is HA-provably equivalent to ⊤ ✄HA A, so we cannot exclude that there would be a definition of the ✷ in terms of interpretability over HA. C.3. Classical Interpretability Logic Over PA arithmetic interpretability and Π01 -conservativity coincide. Thus, the e PA = ILM. The arithmetical completeness of ILM for interpretability over PA Λ was proven by Berarducci [Ber90] and Shavrukov [Sha88] proved that this result 59 also holds for all Σ10 -sound extensions of PA. 37 The reader is referred to [Jd98; Vis98; AB04] for more information about classical interpretability logic. We know two further arithmetically complete interpretability logics. The first is ILP. This is the logic of Σ10 -sound finitely axiomatized extensions of EA+ , also known as I∆0 + Supexp. If we take the contraposed preservativity-style version of ILP, we obtain the logic i-GW− a + Pa + em [Vis90]. C.4. Constructive Interpretability Logic In this subsection we treat constructive interpretability logic with the interpretability logic of HA as our main focus. We need some preliminary material to get the discussion off the ground. C.4.1. i-Isomorphism The materials of the subsubsection work for any theories of finite signature. We will need the notion of i-isomorphism between interpretations. Two interpretations K, M : U → T are i -isomorphic if there is an i-isomorphism G between K and M . A T -formula G is an i-isomorphism between K and M if the theory T verifies that ‘G is a bijection between δK and δM that preserves the predicate symbols of U ’. For example if P is unary, we ask: T ⊢ ∀u∀v ((δK (u) ∧ δM (v) ∧ G(u, v)) → (P K (u) ↔ P M (v))). Let T be any extension of HA. Suppose K : i-EA → T . We also have the identical interpretation E : i-EA → T that translates all predicate symbols to themselves. E.g. AE (v0 , v1 , v2 ) := A(v0 , v1 , v2 ), where A is the relation representing addition. Then, by a special case of the Dedekind-Pudlák Theorem, there exists a formula F such that T proves that F is an initial embedding of E in K. Now it is easy to see that E is i-isomorphic to K iff T proves that F is surjective. Thus, there is a single fixed statement, say CK , that expresses that E is i-isomorphic to K.38 C.4.2. CT0 ! In this subsection, we present some basic facts about CT0 ! (cf. Appendix A), which we will use to derive a new interpretability principle over HA. The theorem below is proven in [Vis06b]. For completeness’ sake, we repeat the proof here. The proof is an adaptation of the proof of Tennenbaum’s Theorem. Such proofs were used before to prove the categoricity of i-EA in constructive meta-theories under the assumption of Church’s Thesis and Markov’s Principle. By taking some extra care we can avoid the assumption of Markov’s Principle. 37 If we leave, for a moment, the context of arithmetical theories, we can say that the result holds for all classical essentially reflexive sequential theories (with respect to some interpretation of arithmetic). 38 We need minor modifications of the formulation in case we have parameters. 60 Theorem C.1. The theory i-EA verifies the following. Suppose T extends HA+ CT0 ! and K : T ✄ i-EA. Then, T ⊢ CK . Proof. We give the proof for the case without parameters. We need minor modifications to add parameters. Suppose T extends HA + CT0 ! and K : T ✄ i-EA. We note that i-EA proves that λeλx.(e ·z x) is total. Let sig(x) = 1 if x > 0 and sig(x) = 0 if x = 0. Let F be the initial embedding of E in K. We work in T . Fix an element z of δK . We define the operation ∗ as follows. • e ∗z x = y if ∃e′ ∃x′ ∃y ′ (F (e, e′ ) ∧ F (x, x′ ) ∧ F (y, y ′ ) ∧ (sig(e′ ·z x′ ) = y ′ )K ). It is easy to see that Hz := λe.(1 − e ∗z e) is a total 0,1-valued function. CT0 !, there is a recursive function that computes Hz , say with index h. p : (h · h = i). Suppose F (h, h′ ) and F (i, i′ ) and F (p, p′ ). We have Hz (h) and, hence, h ∗z h = 1 − i. This means that (sig(h′ ·z h′ ) = 1 − i′ )K . On other hand, since F is an initial embedding, we find (p′ : (h′ · h′ = i′ ))K . By Let =i the We reason inside K. In case p′ ≤ z, we have that h′ · h′ = h′ ·z h′ . Hence, i′ = sig(1 − i′ ). Quod non. Hence z < p′ . We leave K. Since F is an initial embedding, we can find a z ∗ < p such that F (z ∗ , z). Since z was an arbitrary element of δK , we may conclude that F is surjective. ✷ It follows that the interpretability logic of extensions T of HA + CT0 ! contains the following principle: • φ ✄ ψ → ✷(φ → ψ). Remark C.2. The Tarski biconditionals TB for the arithmetical language are all sentences of the form True(pAq) ↔ A. It is clear that every arithmetical theory locally interprets itself plus TB. In the classical case it follows that PA ✄ (PA+TB). However, we cannot have HA✄(HA+TB). If we had HA✄(HA+TB), then we would have K : (HA + CT0 !) ✄ (HA + TB), for some K. However, since the reduct of K to the arithmetical language is i-isomorphic to E, this would enable us to define truth for the arithmetical language in HA + CT0 !. By Tarski’s Theorem on the undefinability of truth, we would find that HA + CT0 ! is inconsistent. Quod non. For some further information about CT0 !, see [van90a]. Remark C.3. With respect to interpretability, there is a certain analogy between HA + CT0 ! and HA∗ . In [Vis06b], the following result is proved. Let τ be translation from the arithmetical language to itself. Consider the theory T := HA∗ +(i-EA)τ . Clearly, τ carries an interpretation of i-EA in T . Let Fτ be the standard embedding of the T -numbers into the τ -numbers. We have: HA∗ + (i-EA)τ ⊢ ∀y (δτ (y) → (∃x Fτ (x, y) ∨ ✷HA ⊥)).39 39 In case τ has parameters a slight adaptation of the formulation is needed. 61 It is easy to see that we cannot generally eliminate the ✷HA ⊥ from the result since PA + ✷PA ⊥ is an extension of HA∗ . The theory PA + ✷PA ⊥ has many non-trivial interpretations of i-EA. It has not been explored whether the result described here throws any shadows on the interpretability logic of HA. C.4.3. The Interpretability Logic of HA The interpretability logic of HA has not yet been studied. It seems to us that there are some good reasons for this neglect, the first being that the more basic problem of the provability logic of HA is still wide open. Unlike the case of the logic of Σ01 -preservativity, there are no indications that the study of the logic of interpretability will help in the study of provability logic. Interpretability itself is intuitionistically significant, e.g., the usual translations of elementary syntax in arithmetic work equally well classically and intuitionistically. But—and here is our second reason—the usefulness of interpretations to compare arithmetical theories is much diminished. For example, the ¬¬-translation does not commute with disjunction, and, thus, fails to carry an interpretation. The demand of commutation with disjunction and existential quantification is much more restrictive intuitionistically than classically. Still, studying the differences between the interpretability logic of HA and that of PA highlights how the classical principles depend on the chosen logic. Also, the relevant methods are quite interesting. Finally, a good friend makes an appearance here: Tennenbaum’s Theorem plays a significant rôle. Which of the axioms of ILM remain in the interpretability logic of HA? The principles of i-GL✷ and the principles J1,2,4 and M are valid over HA. However, J5 fails since, e.g., its instance ✸✷⊥ ✄ ✷⊥ fails.40 The status of J3 is unknown. We note that the classical argument for J3 does yield following weakened version. • (φ ✄ χ ∧ ψ ✄ χ) → ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)) ✄ χ We define the modal Σ01 -formulas as follows: • σ ::= ⊤ | ⊥ | ✷φ | (σ ∨ σ) The following valid principle was noted by Lev Beklemishev in conversation. • (σ ✄ χ ∧ σ ′ ✄ χ) → (σ ∨ σ ′ ) ✄ χ, with σ and σ ′ being modal Σ01 . Open Question C.4. Let A0 := ∀S ∈ Σ01 (TrueΣ01 S ∨ ¬ TrueΣ01 S) and A1 := ∀S ∈ Σ01 (✷HA S → TrueΣ01 S). As ✸HA A1 implies, by the Double Negation Translation, ✸PA A1 , we have i-EA-verifiably (A0 ∧✸HA A1 )✄HA A1 . We can do then the Henkin construction for PA + A1 using the decidability for Σ01 -sentences. We also have trivially A1 ✄HA A1 . But do we have: 40 The fact that ✸✷⊥ ✄ ✷⊥ is not valid for HA follows, for example, from Theorem C.8 in combination with what we already know about the provability logic of HA. 62 ((A0 ∧ ✸HA A1 ) ∨ A1 ) ✄HA A1 ? Similarly, we have for any B that (A0 ∧ ✸HA ⊤) ✄HA (B ∨ ¬ B). But do we have for all B that ((A0 ∧ ✸HA ⊤) ∨ B) ✄HA (B ∨ ¬ B) ? Is the classically invalid principle ⊢ (φ ✄ ψ ∧ φ ✄ χ) → φ ✄ (ψ ∧ χ) still invalid over HA? We do not know that for φ = ⊤. However, the usual construction of Orey sentences for PA can be adapted to give a sentence O such that A ✄HA O and A ✄HA ¬ O, where A is the universal closure of an instance of Tertium non Datur that is sufficient to make the classical argument work. Theorem C.1 throws a shadow downward on HA. We need to define the Γ0 formulas to describe it. Let S range over Σ01 -formulas and let A range over almost negative formulas, as defined in Appendix A: • B ::= S | (B ∧ B) | (B ∨ B) | (A → B) | ∀x B | ∃x B Anne Troelstra shows in [Tro73, §3.6.6] that HA + ECT0 is Γ0 -conservative over HA. A fortiori, HA + CT0 ! is Γ0 -conservative over HA. Inspection of the proof shows that this fact is verifiable in i-EA. We have: Theorem V the following. Suppose C is in Γ0 . We V C.5. The theory i-EA verifies have: if i<n (Ai ✄HA Bi ) and HA ⊢ i<n (Ai → Bi ) → C, then HA ⊢ C. V V Proof. Suppose C is in Γ0 and V (Ai → Bi ) → C. V i<n (Ai ✄HA Bi ) and HA ⊢ i<n It follows that HA + CT0 ! ⊢ i<n (Ai → Bi ) and HA + CT0 ! ⊢ i<n (Ai → Bi ) → C. Hence HA + CT0 ! ⊢ C. Since C is in Γ0 , it follows that HA ⊢ C. ✷ Corollary C.6. The theory i-EA verifies the following. Suppose A is almost negative and B is in Γ0 . Suppose further that A ✄HA B. Then, HA ⊢ A → B. Corollary C.7. The theory i-EA verifies the following: if ⊤ ✄HA O and ⊤ ✄HA ¬ O, then HA ⊢ ⊥. Thus, if HA is consistent, it has no Orey-sentences. We give counterparts of the above classes in the modal language, beginning with the almost negative ones.41 Let φ range over all formulas and • σ ::= ⊥ | ⊤ | ✷φ | (σ ∨ σ) • ψ ::= σ | (ψ ∧ ψ) | (ψ → ψ) 41 By the Orey-Hájek characterization, A ✄ 0 PA B is a Π2 -relation. (It was shown to be complete Π02 independently by Per Lindström and Robert Solovay.) No such reduction is known for the relation A ✄HA B. This relation is prima facie Σ03 and might, for all we know, be Σ03 -hard. We note that Π02 is almost negative but Σ03 is not. So we cannot take φ ✄ ψ as a modal almost negative formula. This does not exclude that further insight might allow us to include it at a later stage. 63 We define the Γ0 -formulas of the bi-modal language as follows. Let φ range over all formulas and let ψ range over the almost negative formulas. • χ ::= ⊥ | ⊤ | ✷φ | (φ ✄ φ) | (χ ∧ χ) | (χ ∨ χ) | (ψ → χ). Theorem C.8. Let χ be in Γ0 . The following principle is in the interpretability logic of HA: ^ ^ ( (φi ✄ ψi ) ∧ ✷( (φi → ψi ) → χ)) → ✷χ. i<n i<n Example C.9. The principle ⊢ (¬¬ ✷⊥ → ✷⊥) ✄ ¬¬✷⊥ → ✷¬¬ ✷⊥ is valid over HA. Since HA is HA-verifiably closed under the primitive recursive Markov’s Rule, it follows that ⊢ ((¬¬ ✷⊥ → ✷⊥) ✄ ¬¬✷⊥) → ✷✷⊥ is valid over HA. Remark C.10. We note that the seemingly stronger principle ^ ^ (φi → ψi ) ✄ χ) → ✷χ. ( (φi ✄ ψi ) ∧ i<n i<n in fact follows from Theorem C.8. Open Question C.11. Is there an interpretation of i-EA in HA that is not i-isomorphic to E? There are many strengthenings of our question. We can demand HAverifiability of the self-interpretation. We could ask whether there is an A such that ⊤ ✄HA A, but HA 0 A. Etcetera. If one combines the proof of Theorem C.1 with q-realizability, one obtains the following. In case the domain and the parameter domain of an interpretation K of i-EA in HA are auto-qa , then K is i-isomorphic with E. Thus, a non-trivial interpretation i-EA in HA should either have a sufficiently complex domain or a sufficiently complex parameter domain.b Note also that ⊤ ✄HA A both implies that ✷HA+CT0 ! A and that ⊤ ✄PA A, which puts some severe constraints on the possible A. a See § 5.4.4 for the notion of auto-q. apologize for the classical reasoning. However, since the relevant predicates are decidable, it can be constructively justified. b We C.4.4. Interpretability and Π01 -Conservativity We have seen that interpretability and Π01 -conservativity coincide over PA. Over other classical theories, interpretability and Π1 -conservativity part ways. For example, they come apart over Primitive Recursive Arithmetic PRA: we have ⊤ ◮PRA IΣ1 , but not ⊤ ✄PRA IΣ1 . Over HA, interpretability and Π1 -conservativity likewise separate ways. We still find that, HA-verifiably, ✄HA is a sub-relation of ◮HA . However, for example, 64 we have ✸✷HA ⊥ ◮HA ✷HA ⊥, but not ✸✷HA ⊥ ✄HA ✷HA ⊥. Also, ¬¬ ✷HA ⊥ ◮HA ✷HA ⊥, but not ¬¬ ✷HA ⊥ ✄HA ✷HA ⊥.42 = D. The problem of the Survey We are returning here to the issue briefly mentioned in the main text: the collapse of J to → in Lewis’ original system [Lew14; Lew18] discovered by Post and addressed by Lewis in a subsequent note [Lew20]. This episode is instructive in illustrating how Lewis’ own thinking about J was often sabotaged by a combination of several factors, including: • an insistence on boolean laws for “material” connectives, including in particular classical, involutive laws for negation; • especially in the 1910’s, a certain carelessness in accepting deductive laws for “intensional” connectives, especially those involving contraposition. The second problem was pretty much admitted by Lewis himself: In developing the system, I had worked for a month to avoid this principle, which later turned out to be false. Then, finding no reason to think it false, I sacrificed economy and put it in ([Lew30], via [Mur05, p.92]). In hindsight, these problems are unsurprising, especially given the publication date of the Survey. Not only were non-boolean systems in the prenatal stage, but also semantics of propositional logics was poorly understood at the time. Symbolic Logic published in 1932 was already in a much better position, mostly thanks to efforts of Mordechaj Wajsberg and William Parry, who provided several crucial algebraic (counter)models used in Appendix II to establish independence results for axioms between S1 and S5. No such assistance was available to Lewis when writing the earlier Survey and consequently, when deciding whether or not to adopt a specific axiom for J, he would mostly rely just on his philosophical intuitions, much like other authors in that period.43 From our point of view, it is of particular interest to isolate the actual rôle played by classical logic with its involutive negation, the axiom Box and redefinition of ✷ as (1) in the collapse of the system of the Survey. 42 These results follow from Theorem C.8 in combination with facts about provability logic. in this respect his remark [Lew20]: “Mr. Post’s example which demonstrates the falsity of 2.21 is not here reproduced, since it involves the use of a diagram and would require considerable explanation.” A “diagram” is presumably a finite matrix/algebra (which could indicate a largely overlooked inspiration Lewis’ work provided for Post in developing nonclassical logical “matrices”, a.k.a. algebras or truth-tables!). In Appendix II to Symbolic Logic, the counterexamples of Parry and Wajsberg were called “groups”. It is worth mentioning that early Lewis’ papers tended to have titles like Implication and the Algebra of Logic [Lew12], A New Algebra of Implications and Some Consequences [Lew13], The Matrix Algebra for Implications [Lew14] or A Too Brief Set of Postulates for the Algebra of Logic [Lew15], but this should not mislead us: the word “algebra” (or “matrix”) is not taken here in the sense of modern model theory or universal algebra. 43 Cf. 65 The problematic axiom is the converse of the one which was latter baptised A8 in Appendix II to Symbolic Logic : A8 (φ J ψ) J (¬✸ψ J ¬✸φ). In the Survey, this axiom was postulated as a strict bi-implication, i.e., (φ J ψ) L (¬✸ψ J ¬✸φ). In our setting, with ✷ rather than ✸ as the primitive and with φ not being equivalent to ¬¬φ, the missing half can be rendered as 2.21 (✷ψ J ✷φ) J (¬φ J ¬ψ), “2.21” being Lewis’ name for this axiom [Lew20]. Of course, there are other conceivable variants, for example: 2.21′ (✷¬φ J ✷¬ψ) J (ψ J φ). As it turns out, however, 2.21 is exactly what we need to reproduce Post’s derivation over iA (together with a sub-boolean axiom Auxp introduced below). To present further details, let us also recall that Lewis uses Modus Ponens for J, i.e., φ ψφJψ as the main inference rule. This in itself is telling: in iA, φ jointly with φ J ψ entails only ✷ψ. The rule ✷φ φ is admissible, but not derivable, unless one postulates as an axiom explicitly ✷φ → φ, something that Lewis’ insistence on formulating all the axioms with J as the principal connective prevented him from doing; ✷φ J φ is not quite the same thing.44 In the setup with Modus Ponens for J as the central rule and L as the “real” equivalence or identity, instead of deducing φ ↔ ✷φ in the extension of our iA with Lewis’ axioms we need to show both ✷φ J φ (which is already a theorem for Lewis, cf. the discussion of Appa and Remark 7.3 above) and 4a φ J ✷φ, deriving 4a in turn requiring only finding another theorem χ s.t. χ J (φ J ✷φ) is also a theorem; in other words, to derive still weaker ✷4a ✷(φ J ✷φ). This in turn can be done if one has both 2.21 and a law which is a mild consequence of excluded middle, namely 44 One can see here yet another instance of Lewis’ peculiar paradox, pointed out by Ruth Barcan Marcus: despite his insistence that “the relation of strict implication expresses precisely that relation which holds when valid deduction is possible” and that “the system of Strict Implication may be said to provide that canon and critique of deductive inference” [LL32, p. 247], his own systems tend to run into problems with the relationship between →, J, entailment and deducibility (relevance logicians would point it out too, cf. Footnote 15, but their own systems have their own share of similar problems). 66 Auxp (¬¬p J ¬(✷p → ✷¬p)) J (p J ✷p). Note that to derive Auxp, it is enough to have as an axiom scheme, e.g., (¬¬✷p ∧ ¬✷¬p) → ✷p; this is why we call Auxp a mild consequence of boolean laws. Note also that in presence of 2.21, we have that Auxp2 ✷(✷p → ✷¬p) J ✷¬p. To get this formula, substitute ⊥ for φ and p for ψ in 2.21, use BL and the fact that ✷p → ✷¬p ⊣⊢− ✷p → ✷⊥. Now we can redo in our setting the Post derivation as quoted by Lewis. Substituting ✷p → ✷¬p for ψ and ¬p for φ in 2.21 yields ((✷p → ✷¬p) J ✷¬p) J (¬¬p J ¬(✷p → ✷¬p)). The antecedent of this strict implication is precisely Auxp2 and the consequent is the antecedent of Auxp. Remark D.1. Of course, there are simpler ways of collapsing the system of the Survey when full boolean logic and all Lewis axioms are assumed. Note that using classical logic and Box (which is an axiom for Lewis, and as we established in Corollary 4.8 can anyway be derived in iA + CPC), we can replace 2.21 with ✷(✷ψ → ✷φ) → ✷(ψ → φ). Classically, this axiom in turn can be replaced with ✸ψ → ✸(✷ψ ∧ ✸ψ). Now, if ✷φ → φ (i.e., reflexivity) is also an axiom or a theorem (which, as shown above, should be indeed the case in a modern representation of Lewis’ original system, with ✷φ φ as an admissible or derivable rule), we can derive ✷φ ↔ φ, trivializing the modal operator. 67