Four intuitionistic modal connectives

Abstract: We introduce the syntax and the semantics of intuitionistic modal logics based on a diamond connective à la Prenosil, its dual box connective, a diamond connective à la Wijesekera and its dual box connective. We analyze the modal definability of some elementary classes of frames. We study the complete axiomatizability of the sets of valid formulas determined by these classes of frames. We prove the decidability of the minimal intuitionistic modal logic determined by the class of all frames.

• The paper presents a Hilbert-style proof system that incorporates four distinct modal operators within an intuitionistic propositional logic framework.
• It develops a unified relational semantics that preserves the heredity property and analyzes modal interdefinability across various frame classes.
• The study demonstrates decidability via an embedding into a decidable fragment of first-order logic and outlines avenues for future complexity and axiomatization research.

Four Intuitionistic Modal Connectives: Syntax, Semantics, and Axiomatization

Introduction and Motivation

The paper "Four intuitionistic modal connectives" (2606.07348) presents a systematic analysis of the modal landscape for intuitionistic propositional logic (IPL) by formally introducing, axiomatising, and characterising the interplay between four distinct modal operators: $\lozenge$, $\square$, $\blacklozenge$, and $\blacksquare$. This work addresses core issues in intuitionistic modal logic (IML), notably the proliferation of non-equivalent modal semantics linked to variations in the accessibility relations and their interaction with the intuitionistic preorder. The study is explicitly motivated by the lack of consensus regarding standard semantics for possibility and necessity in IML outside the special settings examined in earlier literature (e.g., Fischer Servi, Wijesekera). The authors pursue a generalized, minimalistic setting without restricting to forward/backward confluent frames, leading to a richer expressivity analysis and more general results regarding axiomatizability and decidability.

Syntax and Relational Semantics

The base language extends the traditional signature of IPL ($\to, \top, \bot, \vee, \wedge$) with four modal operators:

• $\lozenge$: Possibility in the style of P\v{r}enosil,
• $\square$: Necessity, dual to $\lozenge$,
• $\blacklozenge$: Possibility in the style of Wijesekera,
• $\blacksquare$: Necessity as the dual of $\square$0.

Additionally, the language allows for a dual implication $\square$1. The semantics are defined over quadruples $\square$2, where $\square$3 is a non-empty preorder, $\square$4 is a binary relation, and $\square$5 a valuation monotonic w.r.t. $\square$6. The truth conditions of the modal operators are formulated to maintain the Heredity Property for arbitrary frames, not only the confluent subclasses. Notably:

• $\square$7: $\square$8 iff $\square$9, $\blacklozenge$0 such that $\blacklozenge$1 and $\blacklozenge$2.
• $\blacklozenge$3: $\blacklozenge$4 iff $\blacklozenge$5 with $\blacklozenge$6, $\blacklozenge$7.
• $\blacklozenge$8: $\blacklozenge$9 iff $\blacksquare$0 with $\blacksquare$1 and $\blacksquare$2.
• $\blacksquare$3: $\blacksquare$4 iff $\blacksquare$5 with $\blacksquare$6, $\blacksquare$7.

The semantics is designed so that the Heredity Property (monotonicity of satisfaction in the preorder) holds for all four modalities globally.

Expressivity, Interdefinability, and Frame Definability

A key result is the analysis of the eliminability and interdefinability of these connectives. While for some frame classes (e.g., forward and downward confluent frames) some connectives can be eliminated, in the general class of all frames the modalities are not interdefinable; this is substantiated with precise frame-theoretic counterexamples.

The paper thoroughly investigates modal definability of elementary frame classes (serial, reflexive, symmetric, transitive, Euclidean, deterministic, confluence variants). The authors prove that positive modal definitions exist for certain frame classes (e.g., seriality and forward confluence), but exhibit modal undefinability results for symmetry, reflexivity, and others, using canonical model-theoretic constructions.

Axiomatization and Canonical Model Construction

A central contribution is the presentation of a Hilbert-style proof system covering all four modalities. The proof system is formulated over "signed formulas" (i.e., decorated by $\blacksquare$8 or $\blacksquare$9, in the Rauszer/Lindenbaum tradition), and the logic is closed under uniform substitution and a specified set of modal inference rules and axioms. The completeness argument involves an elaborate canonical model construction over "balanced clips"—triples comprised of a filter, ideal, and two formulas—executing delicate case analyses for each logical connective.

The completeness theorems are detailed for the minimal logic over all frames ($\to, \top, \bot, \vee, \wedge$0), as well as for logics determined by additional modal axioms corresponding to important frame properties (e.g., serial, reflexive, confluent). For several frame classes, the resulting logics are shown to be finitely axiomatizable.

Decidability and Complexity

A strong technical result is the proof that the minimal IML with all four modal connectives is decidable. The core of the argument is an embedding of the logic into the monadic two-variable guarded fragment of first-order logic, a fragment known to be decidable. The paper provides explicit translations from modal formulas to this fragment with tight bounds on formula size (linear overhead). As a corollary, the logics for serial and reflexive frames inherit this decidability. The authors, however, leave open the precise complexity class, indicating future work on this front.

Additionally, the paper establishes that the following meta-theoretical problems are undecidable: modal definability, positive definability, and negative definability of elementary frame classes with respect to the language.

Implications and Future Work

This study delivers a comprehensive foundation for the algebraic and proof-theoretic study of modal expansions of IPL, applicable to a much broader class of frames than usually considered. The explicit modal axiomatizations for logics of serial and reflexive frames provide tools for further algebraic and computational investigations. The analysis of modal (un)definability clarifies the limits of the expressive power of these languages in the intuitionistic context.

The technical machinery introduced (notably, the use of balanced clips in the canonical model and the systematization via signed formulas) paves the way for possible new approaches to cut elimination and to the finite model property in IML, as well as for the development of decision procedures and automated reasoning tools.

Outstanding open problems include:

• Finitary axiomatization for the logics of symmetric and transitive frames,
• Characterization of the complexity of the membership problem,
• Alternative axiomatisations replacing infinitary inference rules with finite axiom sets,
• Relationships between the minimal IML and the fusion of tense and modal logics.

The technical innovations and semantic clarity delivered here importantly inform future work in non-classical modal logics, constructive temporal logics, and applications to type theory.

Conclusion

The paper achieves a systematic, general analysis of intuitionistic modal logics with four modalities, substantially extending the expressivity landscape and foundational proof theory beyond commonly studied restriction classes. The results clarify both the scope and limitations of intuitionistic modal languages with four modal connectives: demonstrating decidability, modal undefinability results for various frame properties, and providing explicit, complete axiomatizations for target frame classes. The techniques and concepts developed offer a robust basis for further research in constructive modal logic, its algebraic semantics, and computation.