Constructive Modal Logic CK: Foundations

• Constructive Modal Logic CK is a minimal intuitionistic modal system that integrates primitive necessity (□) and possibility (◇) operators without classical dualities, forming the basis of constructive modal reasoning.
• It supports multiple sequent calculi, including Gentzen-type and nested systems, which ensure strong cut-elimination and rule invertibility for robust proof-theoretic analysis.
• CK features comprehensive semantic frameworks, such as birelational Kripke and neighbourhood semantics, that guarantee persistence, completeness, and practical applicability in game semantics and constructive mathematics.

Constructive Modal Logic CK (Constructive K) is the canonical constructive counterpart to the classical normal modal logic K. CK is foundational within the landscape of constructive and intuitionistic modal logics, providing a minimal yet robust setting for modal reasoning in a constructivist context. It is characterized by an intuitionistic propositional base, addition of necessity (□) and possibility (◇) as primitive modalities, and an absence of classical dualities between these modalities. The system supports comprehensive proof-theoretic, algebraic, and relational semantic analysis, with strong cut-elimination, admissibility, and meta-theoretic properties positioning it as the baseline for most constructive modal systems.

1. Formal Language and Axiomatization

CK is formulated over a propositional modal language generated by:

• Atoms $p, q, r, \ldots$,
• Connectives $\wedge, \vee, \to, \bot, \top$,
• Modalities $\Box$ (necessity) and $\Diamond$ (possibility).

Axiom Schemes and Rules

CK consists of the full set of intuitionistic propositional axioms (IPC), plus:

• Necessitation: From $\vdash A$, infer $\vdash \Box A$.
• K□: $\Box(A \to B) \to (\Box A \to \Box B)$.
• K◇: $\Box(A \to B) \to (\Diamond A \to \Diamond B)$ (Tabatabai et al., 2022, Groot et al., 7 Jan 2026).

No duality axiom (e.g., $\Diamond A \leftrightarrow \neg\Box\neg A$) is postulated, and no assumptions are made regarding the interaction between the two modalities beyond monotonicity via the K axioms. The underlying logic is intuitionistic rather than classical, so excluded middle and standard de Morgan laws do not hold in general (Dalmonte et al., 2019, Groot et al., 2024).

2. Proof Theory and Sequent Calculi

CK admits multiple sequent-style proof systems, notably both standard Gentzen-type calculi and nested/tree-structured sequent presentations.

Single- and Multi-succedent Calculi

• Single-succedent systems limit sequents to a unique formula in the succedent, in line with proof-theoretic constructivism and the hereditary property of Kripke semantics (Dalmonte, 2022, Dalmonte, 2023).
• Modal Rules involve internalization of K as structural rules:
  • (K□): From $\Gamma \Rightarrow p$, infer $\Box\Gamma \Rightarrow \Box p$, where $\Box\Gamma$ denotes the multiset with each formula boxed.
  • (K◇): From $\Gamma, A \Rightarrow B$, infer $\Box\Gamma, \Diamond A \Rightarrow \Diamond B$ (Tabatabai et al., 2022, Strassburger et al., 2015).
• Necessitation is formulated as a zero-premise modal right rule.
• Standard intuitionistic structural rules (weakening, contraction, cut) are admissible; cut is eliminable (Tabatabai et al., 2022, Strassburger et al., 2015, Dalmonte, 2022).

Nested Sequent Calculi

CK can also be presented via nested sequents, which enables modular extension to D, T, B, 4, 5 and yields syntactic cut-elimination proofs (Strassburger et al., 2015). Modal rules correspond to tree-expansion/reduction, with cut admissibility established via detailed multiset rank induction.

Focused and Curry-Howard Proof Theory

A focused proof system for the minimal CK fragment yields unique typing derivations for each normal proof term and establishes a bijection between proof normal forms and "winning innocent strategies" in associated games over proof arenas (Acclavio et al., 2023).

3. Algebraic, Kripke, and Neighbourhood Semantics

Birelational Kripke Semantics

CK frames are triples $(W, \leq, R)$, where:

• $(W, \leq)$: preorder of worlds (persistence),
• $R$: arbitrary accessibility relation.
• Forcing:
  • $$w \Vdash \Box A \iff \forall u \geq w,\, \forall v.\, (uRv \implies v \Vdash A)$$
  • $$w \Vdash \Diamond A \iff \forall u \geq w,\, \exists v.\, (uRv \wedge v \Vdash A)$$
• The class of CK models is closed under generated subframes, disjoint unions, and bounded morphic images (Groot et al., 7 Jan 2026, Groot et al., 2024, Dalmonte, 2023).

Canonical Model and Completeness

Strong completeness is demonstrated using a canonical model construction, where "segments" (pairs of prime theories and successor sets) ensure the validity of all CK axioms. The truth lemma holds via induction on formulas, and the model validates the axiomatization exactly (Groot et al., 2024, Groot et al., 7 Jan 2026).

Neighbourhood Semantics

CK admits a neighbourhood semantics corresponding to intuitionistic neighbourhood frames equipped with two closure operations:

• $N_{\Box}(w)$ closed under intersection and contains the whole upper set;
• $N_{\Diamond}(w)$ is upward closed.
• Forcing: $w \models \Box A \iff [A] \in N_{\Box}(w)$, $$w \models \Diamond A \iff W \setminus [A] \in N_{\Diamond}(w)$$.
• Monotonicity of neighbourhoods in the order ensures persistence (Dalmonte et al., 2019).

Algebraic Semantics and Duality

CK-algebras are Heyting algebras enriched with two unary operations $\Box, \Diamond$ satisfying:

• $\Box\top = \top$, $\Box a \wedge \Box b = \Box(a \wedge b)$,
• $\Diamond a \leq \Diamond(a \vee b)$, $\Box a \wedge \Diamond b \leq \Diamond(a \wedge b)$. CK enjoys a categorical duality, with CK-algebras dual to descriptive CK-frames. Sahlqvist correspondence and a Goldblatt-Thomason theorem apply to this setting, supporting completeness, canonicity, and modal definability (Groot et al., 7 Jan 2026).

4. Meta-theoretic Properties and Decidability

Cut Elimination and Structural Admissibility

CK's standard sequent calculi all admit analytic (i.e., syntactic) cut elimination by double induction on formula complexity and proof height. All logical rules are height-preserving invertible, aiding both meta-theoretic analysis and automated proof search (Tabatabai et al., 2022, Strassburger et al., 2015, Dalmonte, 2022).

Feasible Admissibility and Complexity

All Visser's (disjunction) rules are feasibly admissible in CK: there exists a polynomial-time algorithm that, given a proof of a premise, yields a proof of the conclusion. CK is PSPACE-complete, with proof search reducible to S4⊕K (Tabatabai et al., 2022, Dalmonte, 2023).

Disjunction Property and Conservativity

CK possesses the disjunction property and, for most diamond-axiom extensions (excluding certain combined $\mathsf{N}_\Diamond, \mathsf{I}_\Box$ cases), extensions of CK are conservative over the pure box fragment (Groot et al., 2024).

5. Extensions, Fragments, and the Position of CK

Extensions and Fragments

• IK (Intuitionistic K): CK plus diamond-axioms such as $\bot \to \Diamond \bot$, $(A \vee B) \to (\Diamond A \vee \Diamond B)$, and $(A \to \Box B) \to \Box(A \to B)$.
• CKX / IKX: Further addition of classical modal axioms T, B, 4, 5 (both a- and b-versions).
• □-Only and ◇-Only Fragments: E.g., CK$_\Box$ for IPC$+$K□, BLL (basic lax logic) for the diamond-only fragment (Tabatabai et al., 2022, Groot et al., 2024, Dalmonte, 2023).

CK is strictly weaker than both classical K and intuitionistic IK; it is the minimal constructive normal modal logic supporting feasible admissibility and constructive (analytic) sequent calculi (Tabatabai et al., 2022, Strassburger et al., 2015).

Relation to Minimal and Paraconsistent Modal Logics

CK can be viewed as the minimal logic K over the intuitionistic or minimal propositional base (as opposed to classical), occupying a foundational role within the constructive modal logic hierarchy (Dalmonte, 2023). Paraconsistent extensions of CK are also feasible, modeling contradictory but non-trivial contexts with suitably adjusted bi-relational frames (Gao et al., 25 Aug 2025).

6. Applications, Game Semantics, and Curry-Howard Correspondence

Game and Denotational Semantics

CK admits a game semantics using "winning innocent strategies" on modal arenas, providing a fully complete correspondence between normal proof terms and such strategies. Combinatorial proofs are represented as arena nets plus skew fibrations—each CK proof corresponds to a unique denotational invariant (Acclavio et al., 2021, Acclavio et al., 2023).

Curry-Howard-Lambek Correspondence

There is a precise Curry-Howard correspondence for the minimal fragment of CK, with modal λ-calculi corresponding to cut-free sequent and focused proofs. Strong normalization, confluence, and canonicity have been established (Acclavio et al., 2023).

Connections to Conditionals and Fundamental Modal Logic

CK serves as the basis for constructive conditional systems such as CCK (adding conditional operators with suitable monotonicity and frame conditions) (Dalmonte et al., 3 Jul 2025), and is provably equivalent to the algebraic/fundamental modal logic of constructive mathematics, as analyzed via representation theorems (Holliday, 2024).

---

Key References:

• (Tabatabai et al., 2022) Universal Proof Theory: Feasible Admissibility in Intuitionistic Modal Logics
• (Strassburger et al., 2015) On Nested Sequents for Constructive Modal Logics
• (Groot et al., 7 Jan 2026) Duality for Constructive Modal Logics: from Sahlqvist to Goldblatt-Thomason
• (Acclavio et al., 2023) Canonicity of Proofs in Constructive Modal Logic
• (Groot et al., 2024) Semantical Analysis of Intuitionistic Modal Logics between CK and IK
• (Dalmonte, 2023) Minimal modal logics, constructive modal logics and their relations
• (Dalmonte, 2022) Wijesekera-style constructive modal logics
• (Acclavio et al., 2021) Towards a Denotational Semantics for Proofs in Constructive Modal Logic
• (Gao et al., 25 Aug 2025) Paraconsistent Constructive Modal Logic
• (Holliday, 2024) Modal logic, fundamentally
• (Dalmonte et al., 2019) Intuitionistic Non-Normal Modal Logics: A general framework