Lattice of Theories (LOT)

• Lattice-of-Theories (LOT) is a structured family of order-theoretic constructions that organizes theories by entailment, closure, and model semantics.
• LOT frameworks employ concept lattices, Galois connections, and fixed-point axioms to unify syntactic and semantic dimensions across logical systems.
• LOT enables theory combination and empirical approximations through geometric methods, bridging classical, many-valued, and modal logical paradigms.

Searching arXiv for the cited LOT papers to ground the article in the listed literature. arxiv_search.query({"search_query":"id:(Grilletti et al., 2020) OR id:(Kent, 2018) OR id:(Kent, 2018) OR id:(Przybocki et al., 21 Nov 2025) OR id:(Berlioz et al., 28 Jan 2026) OR id:(Conradie et al., 2019)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}) arXiv search returned the following relevant records:

• (Berlioz et al., 28 Jan 2026) — "The Latent Space of Equational Theories"
• (Przybocki et al., 21 Nov 2025) — "Characterizing Sets of Theories That Can Be Disjointly Combined"
• (Grilletti et al., 2020) — "Lattices of Intermediate Theories via Ruitenburg's Theorem"
• (Conradie et al., 2019) — "Modelling competing theories"
• (Kent, 2018) — "Conceptua: Institutions in a Topos"
• (Kent, 2018) — "The IFF Approach to the Lattice of Theories" Lattice-of-Theories (LOT) denotes a family of order-theoretic constructions in which theories are organized by entailment, model classes, or closure under a semantic relation. In current arXiv literature, LOT is not a single invariant object but a recurrent pattern: it appears as the concept lattice of theories obtained from Tarski satisfaction, as a lattice of $\chi$-variants of intermediate logics generated by fixed-point axioms, as a complete lattice of theory-combination properties for decidable first-order theories, as a lattice of axiomatic and empirical theories in many-valued modal semantics, and as an empirical geometric approximation to the classical equational LOT (Kent, 2018, Kent, 2018, Grilletti et al., 2020, Przybocki et al., 21 Nov 2025, Conradie et al., 2019, Berlioz et al., 28 Jan 2026).

1. Order-theoretic core

At its most general, a LOT is a partially ordered collection of theories equipped with meet and join operations. What varies across frameworks is the choice of elements and the direction of order. In the institution/FCA tradition, the elements are closed intents or formal concepts, and the order on theories is reverse inclusion of intents, equivalently inclusion of extents. In the intermediate-logic setting, the elements are intermediate logics or their $\chi$-variants ordered by inclusion of theories. In the disjoint-combination setting, the elements are closed sets of decidable theories ordered by inclusion. In the equational and lattice-based modal settings, theories are again ordered by entailment or inclusion of axioms, with lattice operations defined by closure under consequence (Kent, 2018, Kent, 2018, Grilletti et al., 2020, Przybocki et al., 21 Nov 2025, Conradie et al., 2019, Berlioz et al., 28 Jan 2026).

The resulting meet and join operations are correspondingly presentation-dependent. In the truth-concept-lattice formulation, join on intents is intersection of theories and meet is semantic closure of their union. In the classical equational LOT, by contrast, $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$ and $T_1 \vee T_2 = Cn(T_1 \cup T_2)$. In the lattice of closed theory-combination properties, $\bigwedge_i X_i = \bigcap_i X_i$ and $\bigvee_i X_i = Cl(\bigcup_i X_i)$. This comparison shows that LOT is best understood as a structural schema linking consequence, models, and closure, rather than as a single fixed lattice presentation.

2. Satisfaction, concept lattices, and institutional LOT

A canonical LOT construction arises from Tarski satisfaction. In an institution $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$, each signature $\Sigma$ has a sentence set $\mathrm{Sen}(\Sigma)$, a model category $\mathrm{Mod}(\Sigma)$, and a satisfaction relation $\chi$0. For a signature morphism $\chi$1, the satisfaction condition is

$\chi$2

Fixing $\chi$3, one obtains the truth classification with tokens $\chi$4, types $\chi$5, and incidence $\chi$6. FCA then supplies the derivation Galois connection

$\chi$7

$\chi$8

with

$\chi$9

Formal concepts are pairs $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$0 with $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$1 and $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$2, and the intents $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$3 are precisely theories closed under logical consequence relative to $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$4 (Kent, 2018).

The LOT is the concept lattice of these formal concepts, ordered by

$T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$5

Hence, on theories, join is intersection of theories and meet is the closure of the union. In the IFF formulation, the same construction appears as the truth concept lattice of the truth classification $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$6. Closed theories are intents $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$7 paired with extents $T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$8, and entailment is represented by closure:

$T_1 \wedge T_2 = Cn(T_1) \cap Cn(T_2)$9

Top is $T_1 \vee T_2 = Cn(T_1 \cup T_2)$0, the set of logical truths, and bottom is $T_1 \vee T_2 = Cn(T_1 \cup T_2)$1, the inconsistent closed theory (Kent, 2018).

The categorical refinement in the institutional setting is that derivation factors through the concept lattice of theories. The derivation adjunction between powersets factors as a reflection and a coreflection through an axis $T_1 \vee T_2 = Cn(T_1 \cup T_2)$2, and this axis is exactly $T_1 \vee T_2 = Cn(T_1 \cup T_2)$3. The paper states this as the factorization of abstract truth through LOT by an extent reflection and an intent coreflection, and gives the restricted equivalence

$T_1 \vee T_2 = Cn(T_1 \cup T_2)$4

LOT is also functorial under signature morphisms, yielding induced adjunctions between signature-indexed theory lattices, and the Grothendieck construction assembles these fibers into a category $T_1 \vee T_2 = Cn(T_1 \cup T_2)$5; under mild cocompleteness assumptions on $T_1 \vee T_2 = Cn(T_1 \cup T_2)$6, this category is cocomplete, so semantic integration of ontologies is represented by colimits in $T_1 \vee T_2 = Cn(T_1 \cup T_2)$7 (Kent, 2018).

3. Intermediate logics and fixed-point lattices

A different LOT construction is developed for intermediate logics. An intermediate logic $T_1 \vee T_2 = Cn(T_1 \cup T_2)$8 is any set of formulas between intuitionistic propositional calculus $T_1 \vee T_2 = Cn(T_1 \cup T_2)$9 and classical propositional calculus $\bigwedge_i X_i = \bigcap_i X_i$0, closed under modus ponens and uniform substitution. The class $\bigwedge_i X_i = \bigcap_i X_i$1 of intermediate logics forms a bounded lattice under inclusion, with meet given by intersection and join by least intermediate logic containing both (Grilletti et al., 2020).

Fix a univariate formula $\bigwedge_i X_i = \bigcap_i X_i$2 and write $\bigwedge_i X_i = \bigcap_i X_i$3. The central construction is the $\bigwedge_i X_i = \bigcap_i X_i$4-logic generated by a set $\bigwedge_i X_i = \bigcap_i X_i$5: the smallest set $\bigwedge_i X_i = \bigcap_i X_i$6 containing $\bigwedge_i X_i = \bigcap_i X_i$7, all substitution instances of formulas in $\bigwedge_i X_i = \bigcap_i X_i$8, the fixed-point axiom scheme $\bigwedge_i X_i = \bigcap_i X_i$9 for every atom $\bigvee_i X_i = Cl(\bigcup_i X_i)$0, and closed under modus ponens. If $\bigvee_i X_i = Cl(\bigcup_i X_i)$1 is the intermediate logic generated by $\bigvee_i X_i = Cl(\bigcup_i X_i)$2, then $\bigvee_i X_i = Cl(\bigcup_i X_i)$3. For each fixed $\bigvee_i X_i = Cl(\bigcup_i X_i)$4, the collection $\bigvee_i X_i = Cl(\bigcup_i X_i)$5 of $\bigvee_i X_i = Cl(\bigcup_i X_i)$6-variants forms a bounded lattice, and the map $\bigvee_i X_i = Cl(\bigcup_i X_i)$7 is a lattice homomorphism satisfying

$\bigvee_i X_i = Cl(\bigcup_i X_i)$8

The key input is Ruitenburg’s theorem: for a univariate $\bigvee_i X_i = Cl(\bigcup_i X_i)$9, there exists $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$0 such that

$$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$1

This makes fixed points of $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$2 syntactically tractable and yields the characterization

$$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$3

Up to IPC-equivalence, univariate formulas have only six possible fixed-point shapes:

$$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$4

and since $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$5 there are exactly five distinct lattices $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$6.

These five cases have sharply different behavior. $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$7 because the fixed-point axiom $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$8 is trivial. $$I = (\mathrm{Sig}, \mathrm{Sen}, \mathrm{Mod}, \models)$$9 and $\Sigma$0 are singleton lattices: the axioms $\Sigma$1 and $\Sigma$2 force all atoms to denote $\Sigma$3 and $\Sigma$4, respectively, and $\Sigma$5 as theories. The negative-variant case $\Sigma$6 imposes $\Sigma$7 and yields the negative variants, or DNA-logics; an explicit example is

$\Sigma$8

Here the $\Sigma$9-core consists of regular elements, forming a Boolean algebra in the signature $\mathrm{Sen}(\Sigma)$0, so for any $\mathrm{Sen}(\Sigma)$1, the logic $\mathrm{Sen}(\Sigma)$2 makes the $\mathrm{Sen}(\Sigma)$3-fragment classical, while disjunction may remain non-classical. Moreover,

$\mathrm{Sen}(\Sigma)$4

where $\mathrm{Sen}(\Sigma)$5. The dense-element variant $\mathrm{Sen}(\Sigma)$6 imposes $\mathrm{Sen}(\Sigma)$7; algebraically, the $\mathrm{Sen}(\Sigma)$8-core is the set of dense elements, and

$\mathrm{Sen}(\Sigma)$9

The construction has an algebraic dual. If $\mathrm{Mod}(\Sigma)$0 is the unary polynomial induced by $\mathrm{Mod}(\Sigma)$1 on a Heyting algebra $\mathrm{Mod}(\Sigma)$2, the $\mathrm{Mod}(\Sigma)$3-core is

$\mathrm{Mod}(\Sigma)$4

for some sufficiently large $\mathrm{Mod}(\Sigma)$5, equivalently the set of fixed points of $\mathrm{Mod}(\Sigma)$6. Restricting valuations to $\mathrm{Mod}(\Sigma)$7 yields $\mathrm{Mod}(\Sigma)$8-semantics, and

$\mathrm{Mod}(\Sigma)$9

On the algebraic side, $\chi$00-varieties are exactly the classes of Heyting algebras closed under subalgebras, homomorphic images, products, and core superalgebras, and the paper proves

$\chi$01

This produces a full duality between the logical and algebraic LOTs (Grilletti et al., 2020).

4. LOT for disjoint theory combination

In the theory-combination setting, LOT is not a lattice of individual first-order theories but a lattice of theory-combination properties. Let $\chi$02 be the class of decidable first-order theories over countable signatures, and for $\chi$03 define

$\chi$04

Here $\chi$05 and $\chi$06 are combinable when their disjoint union $\chi$07 is decidable. The operator $\chi$08 is antitone and satisfies the antitone Galois connection

$\chi$09

Therefore $\chi$10 is a closure operator, and the class $\chi$11 of closed subsets of $\chi$12 forms a complete lattice under inclusion with

$\chi$13

On closed sets, $\chi$14 is an antitone involution, so the lattice is isomorphic to its dual (Przybocki et al., 21 Nov 2025).

The semantic basis is Fontaine’s lemma: for decidable theories $\chi$15 and $\chi$16, $\chi$17 is decidable iff there is an algorithm that, given cubes $\chi$18 and $\chi$19, decides whether the spectra intersect:

$\chi$20

The LOT then organizes classical combination properties by exact Galois duality. Stable infiniteness is a closed symmetric node:

$\chi$21

Shiny theories, which are smooth, have the finite model property, and have a computable minimal model function, are exactly the largest class combinable with all decidable theories:

$\chi$22

The paper also proves

$\chi$23

showing that politeness does not extend indiscriminate combination beyond stable infiniteness.

Several additional closed nodes are paired by $\chi$24. Gentleness and computable finite spectra satisfy

$\chi$25

Smoothness plus computable spectra and infinite decidability satisfy

$\chi$26

Computable spectra is a symmetric fixed point:

$\chi$27

and the paper presents this as a new sharp symmetric combination theorem generated by lattice operations, with

$\chi$28

There are parallel dualities for $\chi$29-shiny and $\chi$30-decidable classes, and the free-filter families $\chi$31-QG and co-$\chi$32-QG yield chains and antichains of size $\chi$33. The resulting LOT answers several long-standing open questions negatively: no weaker symmetric property below stable infiniteness suffices for a Nelson-Oppen-style method applying to all pairs in that class, politeness is unsalvageable in the indiscriminate sense, and there is no single $\chi$34 whose combinability would characterize gentleness for all decidable theories (Przybocki et al., 21 Nov 2025).

5. Many-valued modal LOT and competing theories

In lattice-based modal logic, LOT arises in three intertwined forms. The paper on competing theories states that LOT is not named explicitly there, but its lattice-theoretic machinery induces a logical LOT of axiomatic extensions, a variable-set LOT of empirical theories, and a concept lattice associated with a reflexive many-valued graph (Conradie et al., 2019).

The logical LOT is standard. If $\chi$35 is the set of axioms or theorems of a theory $\chi$36 extending the basic normal lattice-based modal logic $\chi$37, then

$\chi$38

with

$\chi$39

The paper isolates two special extensions:

$\chi$40

with $\chi$41 extending $\chi$42. Semantics is given on reflexive many-valued graphs over a complete, frame-distributive and dually frame-distributive commutative residuated lattice

$\chi$43

For a graph-based $\chi$44-frame $\chi$45, the complex algebra

$\chi$46

is a complete normal lattice expansion in which $\chi$47 is completely meet-preserving and $\chi$48 is completely join-preserving. The main completeness theorem states that, for $\chi$49, $\chi$50 is sound and complete with respect to the corresponding class of graph-based $\chi$51-frames.

The empirical LOT is indexed by sets of relevant variables. Each theory $\chi$52 determines a relation $\chi$53 encoding similarity relative to $\chi$54, and the powerset of variables forms a complete lattice ordered by inclusion. The modalities $\chi$55 and $\chi$56 induced by $\chi$57 act on the concept lattice and preserve meets or joins, respectively. In the paper’s case study, the theories $\chi$58, $\chi$59, and $\chi$60 correspond to ancient, modern calories, and hormonal or macronutrient response. Worlds are databases, $\chi$61 records local similarity, and modal evaluation captures how a hypothesis changes when viewed through the lens of another theory. This yields a formal semantics of competing theories in which partial compatibility and theory-ladenness are represented by truth degrees in the underlying residuated lattice (Conradie et al., 2019).

6. Latent and empirical approximations of LOT

A recent line of work constructs a geometric approximation to the classical LOT for equational logic. In the underlying exact order, an equational theory $\chi$62 over a binary-operation signature is ordered by entailment:

$\chi$63

equivalently $\chi$64. The classical lattice operations are

$\chi$65

The paper then embeds theories into a latent space using Stone pairings over random finite magmas. For a finite magma $\chi$66 and a formula $\chi$67,

$\chi$68

Using the ET dataset of $\chi$69 magma equations and a sample of $\chi$70 random magmas, one forms the Stone matrix $\chi$71, quotients by coordinate permutations through Stone spectra, and performs PCA on the centered row covariance to obtain a three-dimensional latent embedding (Berlioz et al., 28 Jan 2026).

The resulting geometry correlates strongly with proof-theoretic structure. The ET implication graph contains $\chi$72 implications among $\chi$73 equations; quotienting by provable equivalence yields $\chi$74 vertices and $\chi$75 arrows. Reversible edges are much shorter in the latent space than strict implication edges: the reported mean lengths are approximately $\chi$76 for reversible edges, $\chi$77 for strict atomic edges, and $\chi$78 for general strict edges. The first principal component $\chi$79 is strongly correlated with expectation, $\chi$80 with variance, and the $\chi$81-axis separates conjugate equations by an emergent reflection symmetry around $\chi$82, with self-conjugate laws near $\chi$83. A Hasse-like diagram is obtained by contracting each reversible clique to its center of mass and drawing atomic arrows between clique centers.

The framework also proposes sample-based surrogates for logical operations. For composite theories represented by binary satisfaction vectors $\chi$84 over the sampled magmas,

$\chi$85

while

$\chi$86

in the sense that any sampled model of $\chi$87 or $\chi$88 satisfies the meet. A proposed statistical test for implication estimates conditional satisfaction by

$\chi$89

and uses a Wilson score interval to decide whether the lower confidence bound exceeds a threshold. The paper is explicit that this geometry is only an empirical approximation: finite-random-magma statistics can diverge from general validity, and the current experiments rely on proof-theoretic ground truth rather than statistically inferred edges (Berlioz et al., 28 Jan 2026).

Across these settings, LOT functions as a unifying device for relating theories to one another through order, closure, and duality. In some cases it is exact and algebraic, as with concept lattices, $\chi$90-logics, and Galois lattices of theory-combination properties; in others it is semantic and many-valued; in still others it is empirical and geometric. This suggests that the enduring content of LOT lies in the systematic organization of theory space, together with explicit operations that connect syntactic strength, semantic extent, and transformations between theories.