A Galois connection between classical and intuitionistic logics. I: Syntax

In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by G\"odel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a simplified version of Isabelle's meta-logic

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". 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 the box 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 and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

A Galois connection between classical and intuitionistic logics. II: Semantics

Three classes of models of QHC, the joint logic of problems and propositions, are constructed, including a class of subset/sheaf-valued models that is related to solutions of some actual problems (such as solutions of algebraic equations) and combines the familiar Leibniz-Euler-Venn semantics of classical logic with a BHK-type semantics of intuitionistic logic. To test the models, we consider a number of principles and rules, which empirically appear to cover all "sufficiently simple" natural conjectures about the behaviour of the operators ! and ?, and include two hypotheses put forward by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these turns out to be either derivable in QHC or equivalent to one of only 13 principles and 1 rule, of which 10 principles and 1 rule are conservative over classical and intuitionistic logics. The three classes of models together suffice to confirm the independence of these 10 principles and 1 rule, and to determine the full lattice of implications between them, apart from one potential implication.Comment: 35 pages. v4: Section 4.6 "Summary" is added at the end of the paper. v3: Major revision of a half of v2. The results are improved and rewritten in terms of the meta-logic. The other half of v2 (Euclid's Elements as a theory over QHC) is expected to make part III after a revisio

Towards a Proof Theory of G\"odel Modal Logics

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments

Ecumenical modal logic

The discussion about how to put together Gentzen's systems for classical and intuitionistic logic in a single unified system is back in fashion. Indeed, recently Prawitz and others have been discussing the so called Ecumenical Systems, where connectives from these logics can co-exist in peace. In Prawitz' system, the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation, and the constant for the absurd, but they would each have their own existential quantifier, disjunction, and implication, with different meanings. Prawitz' main idea is that these different meanings are given by a semantical framework that can be accepted by both parties. In a recent work, Ecumenical sequent calculi and a nested system were presented, and some very interesting proof theoretical properties of the systems were established. In this work we extend Prawitz' Ecumenical idea to alethic K-modalities