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Mathematical Semantics of Intuitionistic Logic 2 MATHEMATICAL SEMANTICS OF INTUITIONISTIC LOGIC SERGEY A. MELIKHOV Abstract. This work is a mathematician’s attempt to understand intuitionistic logic. It can be read in two ways: as a research paper interspersed with lengthy digressions into rethinking of standard material; or as an elementary (but highly unconventional) introduction to first-order intuitionistic logic. For the latter purpose, no training in formal logic is required, but a modest literacy in mathematics, such as topological spaces and posets, is assumed. The main theme of this work is the search for a formal semantics adequate to Kol- mogorov’s informal interpretation of intuitionistic logic (whose simplest part is more or less the same as the so-called BHK interpretation). This search goes beyond the usual model theory, based on Tarski’s notion of semantic consequence, and beyond the usual formalism of first-order logic, based on schemata. Thus we study formal semantics of a simplified version of Paulson’s meta-logic, used in the Isabelle prover. By interpreting the meta-logical connectives and quantifiers constructively, we get a generalized model theory, which covers, in particular, realizability-type interpretations of intuitionistic logic. On the other hand, by analyzing Kolmogorov’s notion of semantic consequence (which is an alternative to Tarski’s standard notion), we get an alternative model theory. By using an extension of the meta-logic, we further get a generalized alternative model theory, which suffices to formalize Kolmogorov’s semantics. On the other hand, we also formulate a modification of Kolmogorov’s interpreta- tion, which is compatible with the usual, Tarski-style model theory. Namely, it can be formalized by means of sheaf-valued models, which turn out to be a special case of Palmgren’s categorical models; intuitionistic logic is complete with respect to this semantics. Contents 1. Introduction 3 1.1. Formal semantics 3 1.2. Informal semantics 4 Disclaimers 5 arXiv:1504.03380v3 [math.LO] 1 May 2017 Acknowledgements 5 2. Motivation: homotopies between proofs 6 2.1. Hilbert’s 24th Problem 6 2.2. The collapse of non-constructive proofs 8 2.3. About constructive proofs 10 3. What is intuitionistic logic? 10 3.1. Semantics of classical logic 10 Supported by Russian Foundation for Basic Research Grant No. 15-01-06302. 1 MATHEMATICAL SEMANTICS OF INTUITIONISTIC LOGIC 2 3.2. Platonism vs. Verificationism 11 3.3. The issue of understanding 12 3.4. Solutions of problems 14 3.5. The Hilbert–Brouwer controversy 15 3.6. Problems vs. conjectures 16 3.7. The BHK interpretation 18 3.8. Understanding the connectives 20 3.9. Something is missing here 24 3.10. Classical logic revisited 26 3.11. Clarified BHK interpretation 33 3.12. The principle of decidability and its relatives 36 3.13. Medvedev–Skvortsov problems 37 3.14. Some intuitionistic validities 38 4. What is a logic, formally? 42 4.1. Introduction 43 4.2. Typed expressions 46 4.3. Languages and structures 53 4.4. Meta-logic 63 4.5. Logics 73 4.6. Intuitionistic logic (syntax) 80 4.7. Models 87 4.8. Classical logic (semantics) 92 5. Topology, translations and principles 101 5.1. Deriving a model from BHK 101 5.2. Tarski models 103 5.3. Topological completeness 107 5.4. Jankov’s principle 112 5.5. Markov’s principle 116 5.6. -Translation 120 ¬¬ 5.7. Provability translation 123 5.8. Independence of connectives and quantifiers 129 6. Sheaf-valued models 130 6.1. Sheaves 130 6.2. Operations on sheaves 134 6.3. Sheaf-valued models 138 7. Semantics of the meta-logic 145 7.1. Generalized models 145 7.2. Alternative semantics 151 7.3. Generalized alternative semantics 157 References 168 MATHEMATICAL SEMANTICS OF INTUITIONISTIC LOGIC 3 1. Introduction 1.1. Formal semantics Much of this long treatise is devoted to understanding in rigorous mathematical terms A. Kolmogorov’s 1932 short note “On the interpretation of intuitionistic logic” [108]. What the logical community has grasped of this text is contained in the so-called BHK interpretation, often considered to be “the intended interpretation” of intuitionistic logic and very extensively analyzed in the literature (see §§3.7–3.8 below). However, as ob- served in the classic textbook by Troelstra and van Dalen [202], “the BHK-interpretation in itself has no ‘explanatory power’ ” since “on a very ‘classical’ interpretation of [the informal notions of] construction and mapping,” its six clauses “justify the principles of two-valued (classical) logic.” But as discussed in §§3.9–3.13 below, this arguably does not apply to Kolmogorov’s original interpretation. The problem with Kolmogorov’s original interpretation is that it is incompatible with the usual model theory, based on Tarski’s notion of semantic consequence; worse yet, it is an interpretation of something that does not exist in the usual formalism of first-order logic, based on schemata. Instead of this usual formalism we use a simplified (and, eventually, slightly extended) version of L. Paulson’s 1989 meta-logic [157], which is the basis of the Isabelle prover (see §§4.1–4.6). (In fact, it builds not on the standard modern formalism but rather on the early tradition of first-order logic, as in the textbooks by Hilbert–Bernays [90] and Hilbert–Ackermann [89].) The meta-logical connectives and quantifiers can be used to express, for instance, the horizontal bar and comma in inference rules (in fact, the entire deductive system of a first-order logic is a single meta-formula); derivability of a formula or a rule; implication between principles (in the usual sense, which has nothing to do with ordinary implication between formulas or schemata); variables fixed in a derivation (in the sense of Kleene), etc. In this approach, principles and rules, and even implications between them become finite entities within the formal language (without any meta-variables even in situations where the standard formalism would have side conditions such as “where x is not free in F ” or “where t is free for x in F (x)”) and so one can speak of their interpretation by mathematical objects. Formal semantics of the meta-logic of first-order logics seems to have never been developed, however; thus we have to undertake this task (see §§4.7–4.8 and §7). In usual model theory, regarded as a particular semantics of the meta-logic, the meta-logical connectives and quantifiers are interpreted classically (and moreover in the two-valued way) even though they are constructive. By interpreting them constructively, we get a generalized model theory (see §7.1). It suffices to formalize, in particular, realizability- type interpretations of intuitionistic logic, which do not fit in the usual model theory (see §7.1.4). However, to formalize Kolmogorov’s interpretation of inference rules (which is a substantial part of his interpretation of intuitionistic logic, and seems to be uniquely possible in his approach), we need an alternative generalized model theory (see §7.3). It is nothing new for formulas and principles, but an entirely different interpretation of MATHEMATICAL SEMANTICS OF INTUITIONISTIC LOGIC 4 rules (and more complex meta-logical constructs) already in the case where the meta- logical connectives and quantifiers have a classical, two-valued interpretation (see §7.2 concerning this case). We also formulate an alternative interpretation of intuitionistic logic, called “modified BHK-interpretation”, which is compatible with the usual, Tarski-style model theory. A specific class of usual models of intuitionistic logic that fully meets the expectations of the modified BHK-interpretation is described. These are sheaf-valued, or “proof-relevant topological” models (§6) of intuitionistic logic not to be confused with the usual open set-valued “sheaf models” (see [13] and references there). As one could expect, sheaf- valued models turned out to be a special case of something well-known: the “categorical semantics” (cf. [121]), and more specifically Palmgren’s models of intuitionistic logic in locally cartesian closed categories with finite sums [156]. But unexpectedly, this very simple and natural special case with obvious relevance for the informal semantics of intu- itionistic logic does not seem to have been considered per se (apart from what amounts to the , fragment [11]). We prove completeness of intuitionistic logic with respect ∧ → to sheaf-valued models by showing that in a certain rather special case (sheaves over zero-dimensional separable metrizable spaces) they interpret the usual topological mod- els over the same space. Normally (e.g. for sheaves over Euclidean spaces or Alexandroff spaces) this is far from being so. On the other hand, our sheaf-valued models also interpret Medvedev–Skvortsov models (§3.13). 1.2. Informal semantics We adopt and develop Kolmogorov’s understanding of intuitionistic logic as the logic of schemes of solutions of mathematical problems (understood as tasks). In this approach, intuitionistic logic is viewed as an extension package that upgrades classical logic without removing it in contrast to the standard conception of Brouwer and Heyting, which regards intuitionistic logic as an alternative to classical logic that criminalizes some of its principles. The main purpose of the upgrade comes, for us, from Hilbert’s idea of equivalence between proofs of a given theorem, and from the intuition of this equivalence relation as capable of being nontrivial. This idea of “proof relevance” amounts to a categorification; thus, in particular, our sheaf-valued semantics of intuitionistic logic can be viewed as a categorification of the familiar Leibniz–Euler–Venn subset-valued semantics of classical logic. Traditional ways to understand intuitionistic logic (semantically) have been rooted either in philosophy with emphasis on the development of knowledge (Brouwer, Heyt- ing, Kripke) or in computer science with emphasis on effective operations (Kleene, Markov, Curry–Howard).
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