Intuitionistic Neighbourhood Models

• Intuitionistic neighbourhood models are semantic structures that combine intuitionistic preorders with neighbourhood functions to evaluate modal formulas without relying solely on binary relations.
• They distinguish between minimal and maximal neighbourhoods, enabling separate interpretations of implication and modal operators in non-normal modal systems.
• They support modularity and robust metatheoretical results, bridging Kripke, bi-relational, and topological semantics while ensuring completeness and decidability.

Intuitionistic neighbourhood models are semantic structures for intuitionistic modal and non-normal modal logics in which modal content is evaluated by neighbourhood functions rather than solely by binary accessibility relations. Across the literature, they combine an intuitionistic ordering of information—typically a preorder $(W,\le)$—with one or more neighbourhood assignments and a hereditary or upward-closed valuation. This architecture supports systems in which $\Box$ and $\Diamond$ need not be interdefinable by classical duality, and it provides a natural semantics for intuitionistic monotone modal logics such as $IM$, for general families of intuitionistic non-normal modal logics, and for single-modality systems with minimal and maximal neighbourhoods (Dalmonte et al., 30 Jun 2026, Dalmonte et al., 2019, Witczak, 2017).

1. Basic semantic architecture

The most general formulation in the supplied literature is the intuitionistic neighbourhood frame

$\mathcal F=(W,\le,N_\Box,N_\Diamond),$

where $(W,\le)$ is a nonempty preorder and $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$ are neighbourhood functions. An intuitionistic neighbourhood model is then

$\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$

with $V\colon W\to\mathcal P(\mathit{Atm})$ hereditary in the sense that

$$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$

This framework was developed as a modular setting for families of intuitionistic non-normal modal logics and is explicitly intended to capture systems with one or both of $\Box$0 and $\Box$1 (Dalmonte et al., 2019).

A second line of development uses a single neighbourhood assignment $\Box$2 together with two derived sets: $\Box$3 In this presentation, $\Box$4 is the minimal or “intuitionistic” neighbourhood, while $\Box$5 is the maximal or “modal” neighbourhood. The minimal neighbourhood governs implication and persistence; the maximal neighbourhood governs the modal operator. This semantics appears in work on intuitionistic modal logic without the global superset axiom and in its multi-topological reformulation (Witczak, 2017, Witczak, 2019).

For intuitionistic monotone modal logic $\Box$6, the 2026 semantics is given by a constructive neighbourhood frame

$\Box$7

where $\Box$8 is a pre-ordered set of worlds and $\Box$9 assigns to each world a collection of neighbourhoods. A constructive neighbourhood model is a pair $\Diamond$0, where $\Diamond$1 assigns each atom an upset, i.e. a set closed upward under $\Diamond$2 (Dalmonte et al., 30 Jun 2026). This single-neighbourhood presentation is tailored to $\Diamond$3 and its extensions, and the paper presents it as a more direct analogue of classical neighbourhood semantics than the earlier intuitionistic first-order translation (Dalmonte et al., 30 Jun 2026, Groot, 18 Jul 2025).

2. Forcing and interpretation of the modal operators

In the Dalmonte–Grellois–Olivetti framework, the forcing clauses for the modalities are set-theoretic: $\Diamond$4

$\Diamond$5

where $\Diamond$6. The intuitionistic connectives $\Diamond$7 receive the usual Kripke clauses, so implication is evaluated over all $\Diamond$8 (Dalmonte et al., 2019). In this setting, $\Diamond$9 and $IM$0 are primitive, and their interaction is not fixed by classical duality.

In the single-neighbourhood semantics of Witczak, the forcing relation is defined so that implication depends on the minimal neighbourhood and modality on the maximal neighbourhood: $IM$1

$IM$2

The same framework also introduces further operators: $IM$3

$IM$4

$IM$5

Here the absence of a global superset axiom is a defining design choice, replaced by a relativized superset condition (Witczak, 2017).

For intuitionistic monotone modal logic, the translation-based semantics in (Groot, 18 Jul 2025) uses

$IM$6

and derives the corresponding clause for the dual $IM$7: $IM$8 The later constructive semantics for $IM$9 modifies this by quantifying over all future worlds $\mathcal F=(W,\le,N_\Box,N_\Diamond),$0: $\mathcal F=(W,\le,N_\Box,N_\Diamond),$1

$\mathcal F=(W,\le,N_\Box,N_\Diamond),$2

This constructive clause internalizes persistence at the modal level by requiring modal verification throughout the preorder above $\mathcal F=(W,\le,N_\Box,N_\Diamond),$3 (Dalmonte et al., 30 Jun 2026).

3. Structural conditions and corresponding axioms

A major feature of intuitionistic neighbourhood semantics is that modal principles are encoded by structural conditions on neighbourhood functions. In the two-neighbourhood setting, the basic conditions are monotonicity along $\mathcal F=(W,\le,N_\Box,N_\Diamond),$4,

$\mathcal F=(W,\le,N_\Box,N_\Diamond),$5

supplementation,

$\mathcal F=(W,\le,N_\Box,N_\Diamond),$6

closure under finite intersection,

$\mathcal F=(W,\le,N_\Box,N_\Diamond),$7

and containing the unit,

$\mathcal F=(W,\le,N_\Box,N_\Diamond),$8

for $\mathcal F=(W,\le,N_\Box,N_\Diamond),$9. When both modalities are present, one may further impose weak interaction,

$(W,\le)$0

negation-closure in either direction,

$(W,\le)$1

or strong interaction,

$(W,\le)$2

This modularity is used to recover all 24 logics of Dalmonte–Grellois–Olivetti, and the corresponding cut-free sequent calculi and filtration arguments yield uniform proofs of decidability and the finite-model property (Dalmonte et al., 2019).

In the $(W,\le)$3 setting, the frame conditions are stated directly on $(W,\le)$4. Besides persistence of atomic valuations,

$(W,\le)$5

the paper isolates the following conditions:

$(W,\le)$6

for the $(W,\le)$7-condition, corresponding to the axiom $(W,\le)$8;

$(W,\le)$9

for the $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$0-condition, corresponding to $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$1;

$$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$2

for the $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$3-condition, corresponding to $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$4;

$$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$5

for the $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$6-condition, corresponding to $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$7;

$$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$8

for the $$N_\Box,N_\Diamond\colon W\to\mathcal P(\mathcal P(W))$$9-condition in the monotone setting;

and the $\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$0-condition, or continuality,

$\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$1

A frame satisfying the last condition is called continual (Dalmonte et al., 30 Jun 2026).

The literature also identifies specific systems via neighbourhood properties. Constructive $\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$2 ($\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$3) is characterized by supplementation, finite intersection, and unit on $\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$4, supplementation on $\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$5, and strong interaction. $\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$6 adds the non-normal axiom

$\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$7

(“no-0”), semantically expressible by the condition that whenever $\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$8 does not contain the empty set, $\mathcal M=(W,\le,N_\Box,N_\Diamond,V),$9 contains the empty set; equivalently, one may impose weak interaction together with supplementation on both sides (Dalmonte et al., 2019).

4. Canonical models, proof theory, and metatheorems

The general metatheory follows the familiar soundness–completeness pattern, but with canonical constructions adapted to neighbourhood structure. For the Dalmonte–Grellois–Olivetti family, if $V\colon W\to\mathcal P(\mathit{Atm})$0 is one of the Hilbert systems and $V\colon W\to\mathcal P(\mathit{Atm})$1 the class of models satisfying exactly the corresponding frame conditions, then soundness states: $V\colon W\to\mathcal P(\mathit{Atm})$2 for every $V\colon W\to\mathcal P(\mathit{Atm})$3 and every world $V\colon W\to\mathcal P(\mathit{Atm})$4. Completeness states the converse: if $V\colon W\to\mathcal P(\mathit{Atm})$5 is valid in all models in $V\colon W\to\mathcal P(\mathit{Atm})$6, then $V\colon W\to\mathcal P(\mathit{Atm})$7. The canonical model uses prime $V\colon W\to\mathcal P(\mathit{Atm})$8-theories as worlds and defines

$V\colon W\to\mathcal P(\mathit{Atm})$9

after which a Truth Lemma is proved by induction on formulas (Dalmonte et al., 2019).

The canonical construction for $$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$0 is more specialized. Maximal consistent sets are replaced by segments $$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$1, where $$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$2 is an $$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$3-prime theory and $$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$4 is a family of sets of prime theories satisfying the usual neighbourhood existence conditions for $$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$5 and $$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$6. The canonical universe is

$$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$7

ordered by

$$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$8

with neighbourhoods

$$w\le v\text{ and }p\in V(w)\Longrightarrow p\in V(v).$$9

and valuation

$\Box$00

The truth lemma

$\Box$01

is then proved by induction, and one checks that $\Box$02 is continual and satisfies exactly the extra frame conditions that were assumed (Dalmonte et al., 30 Jun 2026).

For the single-neighbourhood logic $\Box$03, soundness is proved for the system consisting of IPC together with $\Box$04 and $\Box$05 for $\Box$06, modus ponens, and the rule $\Box$07. Completeness uses a canonical model $\Box$08 whose worlds are prime $\Box$09-theories and whose minimal and maximal neighbourhoods are defined by

$\Box$10

$\Box$11

with

$\Box$12

A Truth Lemma yields completeness, and by passing to the bi-relational setting and applying filtration, the paper establishes the finite model property and decidability (Witczak, 2017).

For monotone modal logic via translation, completeness is also tied to a two-sorted intuitionistic first-order language with sorts $\Box$13 for worlds and $\Box$14 for neighbourhoods, relations $\Box$15 and $\Box$16, and a standard translation

$\Box$17

The result is a correspondence

$\Box$18

which supplies a completeness proof for the translation-based semantics (Groot, 18 Jul 2025).

5. Relations to Kripke, bi-relational, and topological semantics

Intuitionistic neighbourhood semantics is closely related to, but not identical with, Kripke and bi-relational semantics. In the classical case, taking $\Box$19 to be equality recovers the usual Scott–Montague neighbourhood semantics for classical non-normal logics, and classical monotone modal logic $\Box$20 is usually given by neighbourhood frames $\Box$21 (Dalmonte et al., 2019, Dalmonte et al., 30 Jun 2026). The 2026 $\Box$22 paper states explicitly that every classical monotone neighbourhood frame $\Box$23 becomes a constructive one by taking $\Box$24 to be the identity relation (Dalmonte et al., 30 Jun 2026).

Relational semantics appears as a special case of neighbourhood semantics. In the $\Box$25 comparison, relational models arise when each $\Box$26 is a principal upset $\Box$27. More generally, if one imposes closure under supersets and binary intersections on each $\Box$28 and defines a relation $\Box$29 by

$\Box$30

then one regains a Kripke-style relation (Dalmonte et al., 30 Jun 2026). This supports the view that constructive neighbourhood frames generalize ordinary bi-relational semantics for intuitionistic normal modal logics such as $\Box$31 and $\Box$32.

The single-neighbourhood approach admits an explicit translation into bi-relational models. Every $\Box$33-frame $\Box$34 determines a relation

$\Box$35

so that

$\Box$36

Conversely, from a bi-relational frame $\Box$37 satisfying

$\Box$38

one recovers neighbourhoods by

$\Box$39

$\Box$40

Under these translations, minimal neighbourhood corresponds to $\Box$41 and maximal neighbourhood to $\Box$42, and the forcing clauses agree pointwise (Witczak, 2017).

A further reformulation is topological. Witczak’s multi-topological semantics replaces neighbourhood structure by a family of spaces $\Box$43, where $\Box$44 is a distinguished nonempty open set. The interpretation of implication and $\Box$45 is given by

$\Box$46

$\Box$47

The paper proves equivalence in both directions: from an $\Box$48-model to a multi-topological model and from a minimal-open multi-topological frame back to a neighbourhood model, with world-by-world agreement

$\Box$49

in both translations (Witczak, 2019).

6. Variants, invariance, and interpretive significance

The literature does not present a single uniform notion of intuitionistic neighbourhood model; rather, it develops several closely related semantics, each designed for a particular fragment or family of logics. The two-neighbourhood semantics of (Dalmonte et al., 2019) is bimodal and explicitly modular. The minimal/maximal-neighbourhood semantics of (Witczak, 2017) and (Witczak, 2019) isolates the interaction between intuitionistic persistence and modal reachability by splitting the neighbourhood profile into $\Box$50 and $\Box$51. The constructive neighbourhood semantics of $\Box$52 in (Dalmonte et al., 30 Jun 2026) is tuned to monotone modality in an intuitionistic setting and is presented as a faithful intuitionistic variant of classical monotone modal logic $\Box$53.

Several recurring misconceptions are addressed by these developments. One is that intuitionistic modal semantics must be relational. The neighbourhood literature shows instead that relational semantics is only a special case, while neighbourhoods capture monotone but not necessarily normal modal behaviour more naturally (Dalmonte et al., 30 Jun 2026, Dalmonte et al., 2019). Another is that $\Box$54 and $\Box$55 should be related by classical duality. The general framework of intuitionistic non-normal modal logics explicitly studies interaction principles weaker than duality, and systems such as $\Box$56 validate intuitionistically meaningful axioms that fail classical duality (Dalmonte et al., 2019).

The invariance theory of the single-neighbourhood approach reinforces the semantic robustness of the framework. Bounded morphisms $\Box$57 are defined by preservation of atoms together with

$\Box$58

and they preserve truth of all formulas. Behavioral equivalence is defined via common bounded morphic images, and bisimulation is formulated by matching both minimal and maximal neighbourhood structure. The paper also defines $\Box$59-bisimulation and proves that two worlds are $\Box$60-bisimilar iff they agree on all formulas of degree $\Box$61 (Witczak, 2017).

From a broader perspective, intuitionistic neighbourhood models function as a semantic bridge. They connect classical neighbourhood semantics to intuitionistic information orderings, relate bi-relational and topological presentations, support canonical-model completeness proofs, and underpin proof-theoretic results such as cut-free calculi, decidability, and finite-model constructions (Dalmonte et al., 2019, Dalmonte et al., 30 Jun 2026, Witczak, 2017). A plausible implication is that their main technical value lies not in replacing relational semantics universally, but in providing the correct ambient category for intuitionistic modal systems whose modal behaviour is monotone or non-normal rather than fully normal.