Higher-order Kripke Models

• Higher-order Kripke models are defined as a hierarchy where higher-level 'worlds' are themselves complete lower-order Kripke models rather than isolated points.
• The framework extends standard Kripke semantics to support intuitionistic and non-classical modal logics by interpreting modalities between full semantic structures.
• Equivalence results show that both partial and homogeneous models correspond to known birelational semantics (IK and MK), offering a modular approach to semantic representation.

Searching arXiv for recent and related work on higher-order Kripke models and generalized Kripke semantics. Higher-order Kripke models are a generalization of standard Kripke semantics in which the objects playing the role of “possible worlds” may themselves be lower-order Kripke models rather than ordinary points. In the framework introduced for intuitionistic and non-classical modal logics, standard Kripke models are reclassified as $0$-ary models, while an $n$-ary model for $n>0$ is a Kripke model whose domain consists only of $(n-1)$-ary Kripke models. The central semantic shift is that, just as classical modalities are interpreted by accessibility between classical propositional valuations, non-classical modalities are interpreted by accessibility between non-classical propositional models. For intuitionistic modal logic, this yields $1$-ary models equivalent to familiar birelational semantics for $IK$ or to a new logic $MK$, depending on the structural restrictions imposed on the $0$-ary layer (Barroso-Nascimento, 24 Jul 2025).

1. Hierarchical architecture

The general definition begins with a language $\mathbb{L}$ and a set of truth values $\mathcal{V}$. A $n$0-ary Kripke model is a sequence

$n$1

where $n$2 is a non-empty set of objects, $n$3 is a non-empty set of relations on $n$4, and $n$5 is a valuation. Standard Kripke models arise as special cases in which $n$6 consists of ordinary worlds and $n$7 contains the familiar accessibility or order relations. For $n$8, an $n$9-ary Kripke model is a sequence

$n>0$0

where $n>0$1 is a non-empty set of $n>0$2-ary Kripke models, $n>0$3 is a non-empty set of relations on $n>0$4, and $n>0$5 is a valuation. The defining feature is therefore that, for $n>0$6, every object in $n>0$7 is itself a lower-order Kripke model (Barroso-Nascimento, 24 Jul 2025).

This hierarchy extends to infinitary structures. An infinitary higher-order Kripke model is an infinite sequence

$n>0$8

such that $n>0$9 is a set of $(n-1)$0-ary Kripke models and, for each $(n-1)$1, every $(n-1)$2 has $(n-1)$3. The framework also distinguishes unirelational from multirelational variants, and finitely relational models from those with arbitrary sets of relations. Mathematically, the construction generalizes the type of objects admitted in a Kripke frame; conceptually, it preserves a possible-worlds reading while lifting “worlds” from valuations or points to complete lower-order semantic structures (Barroso-Nascimento, 24 Jul 2025).

A recurrent misconception is that higher-order Kripke models are merely Kripke frames with more relations. The formal hierarchy rejects that description: the novelty is not an additional relation on the same domain, but a change in the ontological type of the domain itself. For $(n-1)$4, worlds are models.

2. Intuitionistic base layer and birelational background

The point of departure in the intuitionistic case is the ordinary propositional Kripke semantics. An intuitionistic frame is $(n-1)$5 with $(n-1)$6 reflexive and transitive. An intuitionistic propositional model is $(n-1)$7, where $(n-1)$8 and

$(n-1)$9

Satisfaction is defined in the standard way, with implication governed by upward persistence: $1$0 Monotonicity holds: if $1$1 and $1$2, then $1$3 (Barroso-Nascimento, 24 Jul 2025).

Usual intuitionistic modal semantics then adds a second relation on the same set of worlds. For $1$4, a birelational model has the form $1$5, with $1$6 monotone along $1$7, together with Simpson’s frame conditions $1$8 and $1$9: $IK$0

$IK$1

The modal clauses are

$IK$2

$IK$3

For $IK$4, the semantics adds $IK$5 and optionally $IK$6, and the modal clauses become

$IK$7

In this presentation, $IK$8 is strictly stronger than $IK$9; it validates, among other formulas, $MK$0 and a disjunctive principle that $MK$1 does not validate (Barroso-Nascimento, 24 Jul 2025).

The higher-order construction is designed to relocate modality from this birelational layering on a single set of points to a genuinely new semantic level. Intuitionisticity remains in the $MK$2-ary models; modality is introduced by accessibility between those models.

3. One-ary models, partiality, homogeneity, and equivalence results

A $MK$3-ary model for intuitionistic modal logic has as its worlds intuitionistic propositional Kripke models. The broad notion is a general model $MK$4, where $MK$5 is a non-empty set of intuitionistic propositional models and $MK$6 is an accessibility relation between them. Two restricted forms are central. A partial model $MK$7 requires $MK$8 to be at least partially homogeneous: there is a reference model $MK$9 such that every $0$0 is at least a partial copy of $0$1. A homogeneous model $0$2 requires that all $0$3 share the same frame: $0$4 Partial models therefore allow subframes of a common reference frame, while homogeneous models vary only the valuations (Barroso-Nascimento, 24 Jul 2025).

For partial models, the modal clauses are $0$5-style. If $0$6, $0$7, and $0$8, then

$0$9

$\mathbb{L}$0

For homogeneous models, the modal clauses are the simpler $\mathbb{L}$1-style clauses

$\mathbb{L}$2

$\mathbb{L}$3

In the homogeneous case, no $\mathbb{L}$4-quantification appears in the modal clauses themselves; monotonicity is supplied by the underlying intuitionistic propositional semantics together with frame homogeneity (Barroso-Nascimento, 24 Jul 2025).

The paper proves bidirectional equivalence with the standard birelational semantics. From a higher-order model $\mathbb{L}$5, one defines

$\mathbb{L}$6

with valuation $\mathbb{L}$7, order

$\mathbb{L}$8

and modal relation

$\mathbb{L}$9

This construction turns partial models into birelational models satisfying the relevant $\mathcal{V}$0-conditions, and homogeneous models into strong models for $\mathcal{V}$1. Conversely, every birelational model is transformed into a suitable higher-order model by means of a full canonical structure rich enough to embed any countable frame as at least a partial copy. The resulting main theorem states that, for every sequent $\mathcal{V}$2, validity in all partial $\mathcal{V}$3-ary models is equivalent to validity in all birelational models for $\mathcal{V}$4, and validity in all homogeneous $\mathcal{V}$5-ary models is equivalent to validity in all strong models for $\mathcal{V}$6 (Barroso-Nascimento, 24 Jul 2025).

These equivalence results show that the higher-order semantics is not merely heuristic. For the intuitionistic case it is extensionally matched to established semantics, while reorganizing the semantic architecture around models-as-worlds.

4. Alternative timelines and modular semantics

The framework is accompanied by a specific intuitive reading. Under the standard intuitionistic interpretation, a frame $\mathcal{V}$7 can be read as a timeline of growing information: $\mathcal{V}$8 means that $\mathcal{V}$9 is a possible extension of the informational state $n$00. A $n$01-ary intuitionistic model is therefore a timeline. A $n$02-ary higher-order model $n$03 is then a collection of such timelines together with an accessibility relation between them. The intended reading is that $n$04 says that $n$05 is an alternative possible way the same mathematical or informational process could have unfolded. The paper’s illustrative example uses times $n$06, $n$07, and $n$08, with different propositional models over the same chain $n$09: one unproductive day, one in which $n$10 is proved in the morning and $n$11 in the evening, and one in which $n$12 is proved in the afternoon and $n$13 in the evening. On this reading, $n$14 at morning in one timeline means that there is an accessible alternative timeline in which $n$15 is already proved at morning (Barroso-Nascimento, 24 Jul 2025).

This “alternative timelines” reading is then generalized. The paper explicitly states that, more generally, the $n$16-ary models can be read as defining a concept of “alternative” for any substantive interpretation of the $n$17-ary models. If the $n$18-ary layer is interpreted as computational states, algebraic structures, or another Kripke-style semantic base, then the higher-order accessibility relation $n$19 becomes a relation between whole such structures.

The homogeneous $n$20-style clauses are called modular because they do not mention any specifics of the base logic: no $n$21, no special truth values, and no logic-specific frame parameters. They require only a notion of $n$22-ary model, a satisfaction relation for base formulas, and a relation between $n$23-ary models. This is used to propose a general schema for adding modalities to any non-classical logic with Kripke semantics. The paper states that Tarski models can be seen as degenerate Kripke models with a single world, so the same mechanism also recovers ordinary classical modal logic $n$24 when applied to singleton $n$25-ary bases. Standard frame conditions on $n$26 then recover stronger classical modal logics in the familiar way: reflexive $n$27 yields $n$28, and transitive $n$29 yields $n$30 (Barroso-Nascimento, 24 Jul 2025).

A common misunderstanding is that this framework abandons the possible-worlds intuition in favor of a purely technical encoding. The opposite is claimed: higher-order Kripke models remain relational and possible-worlds-like, but their worlds are whole semantic configurations rather than isolated points.

5. Variants, incompleteness, and conjectural scope

Beyond the intuitionistic case, the framework is developed as a family of variants. The paper distinguishes unirelational and multirelational higher-order models; finitely relational models, in which the relational signature is finite at every level; partial and homogeneous $n$31-ary models; and infinitary towers. It then formulates a sequence of conjectures about expressive power and completeness (Barroso-Nascimento, 24 Jul 2025).

The conjectures are progressively stronger. One states that there is at least one Kripke-incomplete normal modal logic that becomes complete with respect to some class of $n$32-ary Kripke models for some $n$33. Another states that all normal modal logics are complete with respect to some class of $n$34-ary models for some $n$35. The paper then proposes “pushing up relations”: if a logic is complete with respect to finitely relational $n$36-ary models, then it is conjectured to be complete with respect to some class of unirelational $n$37-ary models for some $n$38. Corresponding conjectures are stated for at least one Kripke-incomplete logic and for all normal modal logics. An infinitary analogue is also proposed: if a logic is complete with respect to some class of infinitary higher-order models using only finitely relational models at each level, then it is complete with respect to a class of infinitary models using only unirelational ones. The boldest conjecture is that every propositional modal logic, normal or not, is complete with respect to some class of infinitary higher-order Kripke models (Barroso-Nascimento, 24 Jul 2025).

These claims are presented as conjectures rather than theorems. The rationale given is that higher-order valuation functions can depend on the combined behavior of many lower-order models, so the resulting semantics may be strictly more expressive than ordinary $n$39-ary Kripke frames. This suggests, though does not establish, that higher-order Kripke models might offer a relational response to Kripke-incompleteness without abandoning a possible-worlds reading.

The same section clarifies another misconception. Higher-order Kripke semantics is not identical to neighborhood semantics, algebraic semantics, or general frames, even when it is motivated by incompleteness phenomena. Its distinctiveness lies in keeping accessibility explicitly relational while allowing worlds of worlds.

6. Related enriched and higher-order Kripke semantics

The phrase “higher-order Kripke model” is used in more than one way in the recent literature, and adjacent work helps locate the models-as-worlds framework within a broader semantic landscape. In awareness logic, a “lattice of Kripke models, induced by atom subset inclusion” separates uncertainty from unawareness by combining ordinary accessibility relations with awareness maps on a lattice of restrictions; that framework is shown equivalent to both HMS and FH models and inherits completeness results for explicit knowledge and for the Logic of General Awareness (Belardinelli et al., 2021). A closely related earlier formulation presents “a model based on a lattice of Kripke models, induced by atom subset inclusion, in which uncertainty and unawareness are separate,” again via transformations preserving formula satisfaction (Belardinelli et al., 2020).

In categorical semantics, a model of higher-order modal logic in an elementary topos interprets the proposition type not by $n$40 but by “a suitable complete Heyting algebra $n$41,” and the canonical map $n$42 both interprets equality and induces a modal operator as a comonad. That framework explicitly claims that “the usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed” by this notion (Awodey et al., 2014). In homotopical semantics, Quillen model structures on presheaf toposes arising from tree unravellings of Kripke models yield a “homotopy theory for modal logic,” revisiting modal preservation theorems and the Hennessy–Milner property from a homotopical perspective (Reggio, 2023).

Other enrichments shift the higher-order content into worlds themselves. In the semantics of the Perfectly Transparent Equilibrium, adapted multimodal Kripke structures include epistemic and logical accessibility relations, Lewisian closest-state functions, non-normal worlds, and levels of logical omniscience that decrease under counterfactual deviation; the resulting hierarchy is explicitly described as higher-order because level $n$43 conditions are defined through level $n$44 worlds (Fourny, 2018). In separation logic for higher-order store, a “Kripke model where worlds live in a recursively defined ultrametric space” is used to validate nested Hoare triples and selected higher-order frame rules, with worlds solving a recursive equation of the form $n$45 (Schwinghammer et al., 2011). In provability semantics, “provability models” are introduced as a new Kripke-style semantics that “combine features of Kripke models with the assignment of logics to individual worlds,” and are used for $n$46, $n$47, $n$48, $n$49, $n$50, and $n$51 (Mojtahedi et al., 8 Oct 2025).

Within this broader family, the distinctive feature of higher-order Kripke models in the strict sense of models-as-worlds remains sharp. Standard Kripke models are reclassified as $n$52-ary, modality over non-classical bases is reconstructed as accessibility between $n$53-ary models, and the resulting $n$54-ary semantics recovers known intuitionistic systems while yielding a modular schema for non-classical modal semantics more generally (Barroso-Nascimento, 24 Jul 2025).