Ontological Differentiation (OD) Overview

• Ontological Differentiation (OD) is a framework for explicitly separating distinct ontological roles by partitioning conceptual structures using tools like adjunctions, closure operators, and factorization.
• It employs category theory, Lawvere-style semantics, and typed logical systems to delineate levels such as meaning, object, name, and existence, thereby enabling coherent semantic integration.
• OD extends to lexical and dynamical domains with cancellation-based methods that quantify semantic differences and provide new diagnostics for phase-space and definitional analyses.

Searching arXiv for papers on ontological differentiation and related category-theoretic/semantic formulations. Ontological Differentiation (OD) designates a family of formal and methodological moves that make ontological distinctions explicit rather than presupposing a single undifferentiated representational domain. In the cited literature, OD is used to separate kinds of conceptual structure, assign different semantic notions to distinct formal levels, distinguish ontological types from logical predicates, and compare entities through cancellation-based expansions rather than through simple geometric or distributional proximity (Kent, 2024, Inoué, 20 Feb 2026, Saba, 2019, Garcia-Cuadrillero et al., 8 Jul 2025, García-Cuadrillero et al., 25 May 2026). This suggests that OD is best understood not as one doctrine with one canonical definition, but as a recurrent strategy: identify distinct ontological roles, formalize their separation, and specify principled mappings among them.

1. Recurring structure of the concept

Across the relevant literatures, OD has a stable core. It is used to make different ontologies and conceptual structures explicit, to distinguish them by their own internal operators or levels, and to relate them without collapsing them into a single privileged vocabulary. In the category-theoretic account, this is done through adjunctions, closure and interior operators, axes, kernels, and canonical factorizations (Kent, 2024). In Lawvere-style semantics, it appears as the separation of meaning, object, name, and existence into distinct structural levels (Inoué, 20 Feb 2026). In strongly typed natural-language semantics, it is realized as the distinction between ontological concepts as types and logical concepts as predicates (Saba, 2019). In lexical and dynamical applications, it becomes a cancellation-based comparison of recursive expansions or trajectories (Garcia-Cuadrillero et al., 8 Jul 2025, García-Cuadrillero et al., 25 May 2026).

Context	What is differentiated	Main formal device
Category theory and ontology integration	extent, intent, axis, closure, interior, kernels	adjunctions and factorization
Lawvere-style semantics	meaning, object, name, existence	object/morphism/element/internal-logic stratification
Typed semantics	ontological concepts vs logical concepts	strong typing and type unification
Lexical and dynamical OD	definitional or trajectorial components	recursive expansion and cross-cancellation

A common misconception is that OD requires fixing one correct ontology. The category-theoretic and pluralist formulations explicitly reject that conclusion. One paper states that, because representation is relative to purpose, it is more important to standardize methods and frameworks for relating ontologies than to standardize any particular choice of ontology; that standardization is identified with semantic integration (Kent, 2024). A related pluralist formulation argues that different representation systems may be incompatible at the level of basic notions and therefore should not be forced into a single ontological scheme (Novikov-Borodin, 2017).

2. Categorical formulations: abstract truth, factorization, and semantic integration

A technically explicit formulation of OD is developed in "The Characterization of Abstract Truth and its Factorization" (Kent, 2024). The framework uses two kinds of order-enriched categories. A conceptual structures category $\mathsf{C}$ is an order-enriched category with finite limits. A lattice of theories category is a conceptual structures category that is the complete category for an order-enriched fibration. In this setting, ontologies are represented as structured objects, often via adjunctions, and different communities' ontologies become different objects or different adjunctions in a common CS/LOT framework.

The central semantic device is the order-enriched adjunction

$$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$

with the adjunction condition

$$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$

This is treated as the formal embodiment of abstract truth. From such an adjunction the paper defines a closure on $A_0$,

$(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$

and an interior on $A_1$,

$(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$

so that what counts as closed or open is relative to the adjunction and hence relative to the ontology or conceptual structure at issue. OD here consists in making that relativity explicit rather than masking it behind a single global semantics.

The main constructive move is the polar factorization of an adjunction. Given

$g:A_0\rightleftharpoons A_1,$

the paper constructs an axis $\diamondsuit(g)$ and shows that

$g=\mathsf{ref}_g\circ \mathsf{ref}_g^{\propto},$

where $$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$0 is an extent reflection and $$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$1 is an intent coreflection. The same ontology is thereby decomposed into left side, right side, and a canonical intermediate conceptual axis. The paper also defines closed and open polar factorizations through $$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$2 and $$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$3, linked by an isomorphism

$$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$4

This yields several formally distinct but equivalent “cuts” through the same conceptual universe.

OD also governs mappings between ontologies. For a morphism of adjunctions

$$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$5

the paper constructs an axis adjunction

$$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$6

which separates the mapping into a left transformation, a central axis transformation, and a right transformation. At the category level, reflections and coreflections form a factorization system on the category of adjunctions, and the paper gives an equivalence

$$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$7

The article describes this not as loss of information but as a structured reorganization of ontological information. It also presents LOT-level “diamond diagrams,” where adjunctions factor through kernels, liftings, and equivalences, further differentiating primitive objects, kernels, and canonical axes within a single ontology.

The significance of this categorical program is methodological. Representation is taken to be relative to purpose, so semantic integration should standardize ways of relating ontologies rather than dictating one ontology. In that sense, OD is the structural precondition for ontology interoperability (Kent, 2024).

3. Structural stratification: meaning, object, name, and existence

A different but closely related formulation appears in "On the Category-Theoretic Independence of Meaning, Object, Name and Existence" (Inoué, 20 Feb 2026). Here OD is the assignment of four notions to distinct categorical levels in a Lawvere-style semantics. The setting is a semantic functor

$$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$8

typically with $$g=\langle \check{g},\hat{g}\rangle : A_0 \rightleftharpoons A_1,$$9 a topos. Types are interpreted as objects, terms as arrows, formulas as subobjects or arrows into $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$0, and quantifiers via adjoints between subobject lattices.

The four strata are sharply separated. An object is an object of $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$1 interpreting a type $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$2, namely $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$3. A name is a global element

$$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$4

or more generally a generalized element $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$5. Meaning is morphism-level or subobject-level data: if $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$6 is a term in context $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$7, then its meaning is

$$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$8

if $$a\circ \check{g}\le b \;\;\text{iff}\;\; a\le b\circ \hat{g}.$$9 is a formula, its meaning is the corresponding subobject or characteristic map into $A_0$0. Existence is split into external existence, given by the existence of a global element $A_0$1, and internal existence, given by the topos-theoretic existential quantifier

$A_0$2

The crucial separation is between internal existence and global naming. The paper proves that for a topos object $A_0$3, internal inhabitedness is equivalent to $A_0$4 being epimorphic, while having a global name requires a map $A_0$5. In general, the former does not imply the latter. The concrete witness is in $A_0$6, using the sheaf of local sections of the nontrivial covering

$A_0$7

The resulting sheaf $A_0$8 is internally inhabited because $A_0$9 is epimorphic, but it has no global element because the covering has no global section. The paper summarizes the point as the key separation between internal existence and global naming.

On this basis it proves a category-theoretic independence theorem: meaning, object, name, and existence are pairwise and collectively independent in the sense of non-recoverability. No notion is uniformly recoverable from the others by an equivalence-invariant construction across semantic environments. This gives OD a strong anti-reductionist form. The differentiated levels are not merely stylistic distinctions in language; they encode different structural data in the categorical universe (Inoué, 20 Feb 2026).

4. Typed ontologies in natural-language semantics

In natural-language semantics, OD is developed as a distinction between ontological concepts and logical concepts. "No Adjective Ordering Mystery, and No Raven Paradox, Just an Ontological Mishap" argues that logical semantics went wrong by not distinguishing what Cocchiarella calls first-intension concepts from second-intension concepts (Saba, 2019). The corresponding framework, ONTOLOGIK, treats ontological concepts as types in a strongly typed ontology and logical concepts as predicates over those types. Examples include

$(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$0

for ontological typing, and

$(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$1

for typed predicates.

The paper’s diagnosis of an “ontological mishap” is that standard first-order semantics represents both kind terms and property terms as predicates of the same logical kind, as in $(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$2 and $(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$3. OD corrects this by moving kind terms into the type system. Thus “Julie is articulate” becomes

$(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$4

which aligns the implicit and explicit typing content. The same distinction is used to reinterpret the raven hypothesis as

$(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$5

thereby dissolving the paradox generated when raven is treated as a predicate on a par with black.

The key operational mechanism is type unification. When multiple type constraints meet in one scope, the system combines them by a unification operator, written with a dot. Compatible cases collapse to the more specific type, as in

$(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$6

When type constraints clash, the clash can trigger missing-text completion. The canonical example is: $(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$7 where $(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$8 is constrained as $(\mbox{-})^{\bullet_g}=\check{g}\circ \hat{g},$9, $A_1$0, and $A_1$1. The paper simplifies

$A_1$2

so the type clash is resolved by introducing an implicit person associated with the omelet through a salient relation. OD here is not merely taxonomic; it drives inference.

"A Note on Ontology and Ordinary Language" develops a related strongly typed logic with explicit distinctions among entity, physical, artifact, human, animal, event, activity, process, attribute, and content, and with actual versus conceptual existence marked as $A_1$3 versus $A_1$4 (0704.3886). The basic unification clause is

$A_1$5

with an enriched case allowing mediating relations when a salient relation exists. This supports analyses such as “book review” versus “book proposal,” where the former presupposes an actual book and the latter a conceptual book; “john attended the seminar” versus “john cancelled the seminar,” where attendance forces actual eventhood but cancellation does not; and “john read a book and then he burned it,” represented by distinguishing the book as physical artifact from its content via $A_1$6.

A recurrent implication in these papers is that OD in semantics is not an embellishment on predicate logic. It is a reallocation of semantic work from ad hoc lexical polysemy and untyped predicates to a typed ontology whose categories, existence modes, and admissible relations are made formally explicit (Saba, 2019, 0704.3886).

5. Lexical OD and its trajectorial extension

A separate line of work treats OD as a distance-like measure grounded in recursive definitional structure. "Ontological differentiation as a measure of semantic accuracy" defines a lexicon $A_1$7 as a set of inter-defined concepts and introduces a read function $A_1$8 for recursive definitional expansion (Garcia-Cuadrillero et al., 8 Jul 2025). For a concept $A_1$9,

$(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$0

An OD computation is parameterized by a cancellation rule $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$1, a termination rule $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$2, and a score based on cancellation counts $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$3: $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$4 The main variant is Strong Ontological Differentiation (SOD), in which cancellations are triggered only by overlaps between elements originating from different starting concepts. Weak OD and Great OD define looser cancellation and stopping rules.

This lexical OD is used to compare pairwise semantic divergence and path coherence in definitional graphs extracted from the Simple English Wiktionary. The paper reports weak correlations between direct pairwise OD scores and cosine similarities across $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$5~2 million word pairs sampled from a pool representing over 50\% of the entries in the Wiktionary lexicon. The reported Spearman correlations between raw SOD and raw cosine are approximately $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$6, $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$7, and $(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$8 for ground-filtered, random-removal, and targeted-removal datasets, respectively. It also reports that Semantic Navigation paths consistently exhibit significantly lower cumulative OD scores than shortest paths. In the symmetric networks, SN is better in 59.82\%, 62.42\%, and 58.22\% of cases across the three dataset conditions, with shortest paths better in 23.22\%, 22.78\%, and 26.30\%, and ties in the remainder (Garcia-Cuadrillero et al., 8 Jul 2025).

"Strong Trajectorial Ontological Differentiation" transfers the same logic from lexical networks to dynamical systems (García-Cuadrillero et al., 25 May 2026). Instead of recursive definition expansion, trajectorial OD uses chronological evolution. A dynamical system

$(\mbox{-})^{\circ_g}=\hat{g}\circ \check{g},$9

is discretized on an $g:A_0\rightleftharpoons A_1,$0-dimensional grid, giving coordinate sequences

$g:A_0\rightleftharpoons A_1,$1

Component-wise cancellation is then defined cross-trajectory: a component $g:A_0\rightleftharpoons A_1,$2 is canceled if the same value has appeared in the corresponding component of the other trajectory at some level $g:A_0\rightleftharpoons A_1,$3. The process terminates at the first level $g:A_0\rightleftharpoons A_1,$4 where some state in either trajectory becomes fully canceled. After post-processing cancellations up to $g:A_0\rightleftharpoons A_1,$5, the scalar score is

$g:A_0\rightleftharpoons A_1,$6

where $g:A_0\rightleftharpoons A_1,$7 is the number of uncanceled components at level $g:A_0\rightleftharpoons A_1,$8. The paper focuses on the reversed-time variant FinSTOD, obtained by applying STOD to time-reversed trajectories.

The authors emphasize that STOD does not rely on the study of the tangent flow and compare it with FTLE, FLI, and LD across five systems: a linear hyperbolic saddle, the simple pendulum, Lorenz’63, the forced Duffing oscillator, and the time-dependent double gyre. The reported conclusion is that FinSTOD shows excellent performance in the identification of phase-space structures and adds a new derivative-free diagnostic to the “chaotic toolbox” (García-Cuadrillero et al., 25 May 2026).

Taken together, these works extend OD from ontology representation and semantic typing to explicit distance constructions over discrete expansions. This suggests a unifying abstract pattern: entities are represented by ordered expansions, overlaps are treated as cancellations, and the residual uncanceled structure quantifies difference (Garcia-Cuadrillero et al., 8 Jul 2025, García-Cuadrillero et al., 25 May 2026).

6. Incompatible representations, physical theories, and ontological pluralism

A broader philosophical formulation appears in "Ontological Systems In Cognition" (Novikov-Borodin, 2017). The paper begins from the Existing $g:A_0\rightleftharpoons A_1,$9, understood as everything able to influence the cognizer, and develops ontology through partitions and spaces. A representation system is a system of partitions based on a space, and spaces determine the initial notions and definitions of the corresponding system. Compatible systems are reducible to a common basic partition; incompatible systems are not. The paper states that representation systems based on incompatible spaces are incompatible with each other on the basis of any space, so they have different ontology. Objects from one system can appear as “strange” or “off-site” from the standpoint of another. Quantum mechanical objects and cosmological dark matter and dark energy are treated as cases where entities from incompatible or coherent spaces appear paradoxical when forced into one ontology.

This pluralist picture culminates in a multi-polar view of the world. Different scientific, philosophical, religious, and social systems are said to be incompatible at the level of basic notions yet interconnected at a higher level. The methodological point is not eliminative. Contradictions are addressed by separating incompatible ontologies rather than by compelling them into one consistent representational frame (Novikov-Borodin, 2017).

A more domain-specific analogue is found in "Ontology of the Theory of Relativity" (Escobedo, 2023). Although it does not use the explicit label OD, it develops a formal apparatus for separating formal, empirical, and ontological levels of physical theory. A model is defined as

$\diamondsuit(g)$0

and a physical theory as

$\diamondsuit(g)$1

where $\diamondsuit(g)$2 is empirical interpretation into the set $\diamondsuit(g)$3 of sensible properties and $\diamondsuit(g)$4 is ontological interpretation into the set $\diamondsuit(g)$5 of ontologically real beings. The paper distinguishes logical consistency, empirical consistency, and ontological consistency, and formulates a principle of ontological independence: physical reality is invariant with respect to its description. This is then used to compare Newtonian mechanics, special relativity, and general relativity by differentiating their ontological commitments: absolute space and time versus Minkowski spacetime, force-based gravity versus curvature, frame-dependent quantities versus ontological invariants such as proper time, $\diamondsuit(g)$6, and geodesic motion.

These pluralist and physical-theory accounts reinforce a central theme already visible in the categorical and semantic literature: OD is not principally a search for more ontological items. It is a discipline of explicit separation. Its aim is to mark where categories, layers, or representational systems differ; to specify the operators, invariants, or relations that make those differences exact; and to preserve semantic or theoretical coherence without erasing ontological plurality (Novikov-Borodin, 2017, Escobedo, 2023).