Base-Extension Semantics (B‑eS) in Modal Logic

• Base-extension semantics (B‑eS) is a proof-theoretic method that defines validity relative to a base of atomic inference rules rather than truth in models.
• It extends classical modal logic (K, KT, K4, S4) by introducing a modal relation on bases, preserving inferential structure and ensuring soundness and completeness.
• B‑eS bridges traditional proof-theoretic validity with Kripke semantics, offering a versatile framework adaptable to intuitionistic, linear, and higher-order logics.

Searching arXiv for the cited paper and closely related Base-Extension Semantics work to ground the article in current literature. Base-extension semantics (B‑eS) is a proof-theoretic semantics in which the meaning and validity of formulas are determined relative to a “base” of atomic inference rules rather than by truth in models of possible worlds or set-theoretic structures. In the modal setting developed in “Base-extension Semantics for Modal Logic” (Eckhardt et al., 2024), B‑eS assigns semantic values to formulas by an inductive definition grounded in provability from a base and, for modal operators, in a relational structure on bases. The framework is developed for the classical propositional modal systems $K$, $KT$, $K4$, and $S4$, with $\square$ as the primary modal operator, and it establishes soundness and completeness theorems together with a duality result for $\square$ and a natural presentation of $\lozenge$ (Eckhardt et al., 2024). Related work situates B‑eS within a broader proof-theoretic program for intuitionistic propositional logic (Gheorghiu et al., 2022), intuitionistic sentential logic (Gheorghiu, 7 Mar 2025), categorical proof-theoretic semantics (Pym et al., 2023), classical linear logic (Barroso-Nascimento et al., 11 Apr 2025), intuitionistic modal logics (Buzoku et al., 9 Jul 2025), second-order logic (Gheorghiu et al., 11 Aug 2025), and linear-logic phase semantics (Piotrovskaya, 11 Jun 2026).

1. Inferentialist setting and basic conception

B‑eS belongs to proof-theoretic semantics, where meaning is based on inference rather than on truth in models. In the formulation of (Eckhardt et al., 2024), it is presented as a mathematical expression of the inferentialist interpretation of logic: bases encode non-logical inferential commitments about atoms, and logical constants are then defined by structural clauses over those bases. This contrasts with model-theoretic semantics, where formulas are evaluated relative to models, worlds, valuations, and accessibility relations.

In the classical non-modal starting point recalled in (Eckhardt et al., 2024), the language is

$\phi ::= p \mid \bot \mid \phi \to \phi$

with $p$ ranging over atomic sentences. A base rule is a pair $(L_j,p)$, written

$KT$0

where $KT$1 is a finite, possibly empty, set of atomic sentences and $KT$2 is atomic. A base $KT$3 is any countable collection of such rules, and $KT$4 is the closure of the empty set under the rules in $KT$5, that is, the set of atoms obtainable by repeated rule application (Eckhardt et al., 2024). This restriction to simple production rules is explicitly noted as crucial in obtaining a classical propositional logic in the base-extension framework.

The central semantic judgment is validity at a base. For the classical fragment, (Eckhardt et al., 2024) gives the clauses: $KT$6 A formula is valid if it is valid at every base (Eckhardt et al., 2024). In this architecture, atomic validity is proof-theoretic from the outset, while complex validity is obtained by induction over formula structure and universal quantification over base extensions.

This general picture extends beyond classical modal logic. In intuitionistic propositional logic, support $KT$7 is likewise defined relative to bases, with atomic support tied to derivability in the base and validity obtained by quantifying over all bases in a chosen basis (Gheorghiu et al., 2022). Gheorghiu later emphasized that, for intuitionistic sentential logic, soundness and completeness of B‑eS can be obtained directly from Mints’ clausal theorem, underscoring the role of proof-search in the framework (Gheorghiu, 7 Mar 2025).

2. Formal architecture: bases, extensions, and modal relations

For modal logic, (Eckhardt et al., 2024) extends the language to

$KT$8

with $KT$9 defined as $K4$0 and $K4$1 introduced either as $K4$2 or via an independent clause discussed below. The framework retains atomic bases, but adds a relation on bases that plays the role analogous to accessibility in Kripke semantics.

Instead of a frame $K4$3, modal B‑eS uses the set $K4$4 of all bases together with a modal relation $K4$5. Intuitively, $K4$6 means that $K4$7 is accessible from, or possible relative to, $K4$8 (Eckhardt et al., 2024). Modal behavior is not encoded inside the atomic rules themselves; it is carried entirely by this additional relational layer.

The paper defines $K4$9 to be a modal relation iff, for all bases $S4$0, the following hold (Eckhardt et al., 2024):

1. If $S4$1, then there exists some $S4$2 with $S4$3 and $S4$4, and for all $S4$5 with $S4$6, $S4$7.
2. If $S4$8, then for all $S4$9 with $\square$0, $\square$1.
3. For all $\square$2: if $\square$3 is consistent and $\square$4, then either $\square$5 is maximally-consistent, or there exists $\square$6 with $\square$7.
4. For all $\square$8: if $\square$9, then for all $\square$0, $\square$1.

A $\square$2-modal relation is one that additionally satisfies the frame conditions corresponding to $\square$3, such as reflexivity for $\square$4 and transitivity for $\square$5 (Eckhardt et al., 2024). This relational apparatus is the decisive formal addition that makes modal B‑eS possible.

The choice of these clauses is significant. Clause (3) allows modal information to be transferred to maximally-consistent bases, which function as proof-theoretic analogues of canonical worlds. Clause (4) enforces downward closure on the left coordinate: if $\square$6 is possible at $\square$7, then it remains possible at every subset of $\square$8 (Eckhardt et al., 2024). This suggests that strengthening a base may restrict possibilities but cannot create new ones.

3. Semantic clauses for $\square$9, $\lozenge$0, $\lozenge$1, and $\lozenge$2

Given a modal logic $\lozenge$3, a base $\lozenge$4, and a $\lozenge$5-modal relation $\lozenge$6, (Eckhardt et al., 2024) defines modal validity as follows: $\lozenge$7 A formula is $\lozenge$8-valid iff it is valid at all bases and all $\lozenge$9-modal relations (Eckhardt et al., 2024).

The defining feature of the $\phi ::= p \mid \bot \mid \phi \to \phi$0-clause is the joint quantification over supersets $\phi ::= p \mid \bot \mid \phi \to \phi$1 and $\phi ::= p \mid \bot \mid \phi \to \phi$2-successors $\phi ::= p \mid \bot \mid \phi \to \phi$3. This is the modal analogue of base-extension reasoning: modality is evaluated not merely at the current base but across all extensions of that base and all accessible bases from those extensions (Eckhardt et al., 2024). This differs structurally from ordinary Kripke semantics, where one quantifies only over accessible worlds from the current world. The extra quantification is exactly what preserves the inferentialist, extension-sensitive architecture of B‑eS.

The paper establishes a monotonicity theorem: $\phi ::= p \mid \bot \mid \phi \to \phi$4 and also an explosion principle: $\phi ::= p \mid \bot \mid \phi \to \phi$5 (Eckhardt et al., 2024). These preserve the underlying classical behavior of the base-extension setting.

For $\phi ::= p \mid \bot \mid \phi \to \phi$6, (Eckhardt et al., 2024) first introduces it definitionally as $\phi ::= p \mid \bot \mid \phi \to \phi$7, but then provides an independent semantic clause: $\phi ::= p \mid \bot \mid \phi \to \phi$8 The paper proves the duality lemma: $\phi ::= p \mid \bot \mid \phi \to \phi$9 and conversely,

$p$0

(Eckhardt et al., 2024). The “natural presentation of $p$1” mentioned in the abstract refers precisely to this independently formulated existential-over-accessible-bases clause.

A related development for intuitionistic modal logics likewise gives $p$2 an elimination-style, second-order flavor rather than a naive existential Kripke-style clause, emphasizing that B‑eS systematically interprets modal operators through inferential use rather than by direct truth-conditions (Buzoku et al., 9 Jul 2025). This suggests a common methodological principle across classical and intuitionistic modal B‑eS.

4. Canonical behavior, maximally-consistent bases, and correspondence with Kripke semantics

A major structural result in (Eckhardt et al., 2024) is that maximally-consistent bases serve as proof-theoretic analogues of classical valuations and, in the modal setting, of worlds in canonical Kripke models. A base $p$3 is maximally-consistent iff it is consistent and for every base rule $p$4, either $p$5 or $p$6 is inconsistent (Eckhardt et al., 2024).

At such bases, modal validity simplifies substantially. The paper proves: $p$7 for maximally-consistent $p$8 (Eckhardt et al., 2024). The reason is that every proper superset of a maximally-consistent base is inconsistent, and inconsistent bases validate everything by explosion, so the extra quantification over supersets becomes vacuous. At this point the modal clause is pointwise identical in shape to the Kripke clause for $p$9.

This correspondence is central to the completeness proof. The paper proves that for $(L_j,p)$0, the following are equivalent:

1. $(L_j,p)$1 is valid in B‑eS for $(L_j,p)$2;
2. $(L_j,p)$3 is valid in Kripke semantics for $(L_j,p)$4;
3. $(L_j,p)$5 is a theorem of the Hilbert system for $(L_j,p)$6 (Eckhardt et al., 2024).

The Hilbert system includes classical propositional axioms, the modal axiom $(L_j,p)$7,

$(L_j,p)$8

modus ponens, and necessitation, with $(L_j,p)$9 and $KT$00 added when appropriate (Eckhardt et al., 2024). The proof that these axioms and rules are B‑eS-valid proceeds by showing, among other things, that:

• modus ponens is B‑eS-valid,
• necessitation preserves B‑eS-validity,
• $KT$01 is B‑eS-valid,
• reflexivity yields validity of $KT$02,
• transitivity yields validity of $KT$03 (Eckhardt et al., 2024).

For soundness with respect to Kripke semantics, the paper starts from a Kripke countermodel $KT$04 with a world $KT$05 falsifying $KT$06, and constructs:

• a family of bases $KT$07, one for each world $KT$08,
• a modal relation $KT$09 on these bases, such that for all formulas $KT$10,

$KT$11

(Eckhardt et al., 2024). This demonstrates that any Kripke failure can be reflected as a B‑eS failure. The construction uses fresh atoms $KT$12 to separate worlds, builds atomic bases $KT$13, extends them to maximally-consistent bases $KT$14, and defines $KT$15 so as to mirror $KT$16 while satisfying the modal-relation conditions (Eckhardt et al., 2024).

This suggests that B‑eS is not merely analogous to Kripke semantics at the level of validity, but structurally close to it once maximally-consistent bases are isolated.

5. Range of applicability across logics

Although (Eckhardt et al., 2024) focuses on classical modal logics $KT$17, $KT$18, $KT$19, and $KT$20, B‑eS is part of a broader research program spanning multiple proof-theoretic and modal settings.

For intuitionistic propositional logic, Sandqvist’s base-extension semantics is presented in (Gheorghiu et al., 2022) as a support relation $KT$21 defined over bases that are atomic systems, with validity quantified over a basis of such systems. That paper interprets bases as collections of definite formulae and links support to uniform proof-search in hereditary Harrop logic, thereby connecting B‑eS to logic programming and negation-as-failure (Gheorghiu et al., 2022). Gheorghiu later sharpened this picture by showing that, for intuitionistic sentential logic, soundness and completeness of B‑eS follow directly from Mints’ resolution-based clausal theorem, making proof-search central to the semantics (Gheorghiu, 7 Mar 2025).

For intuitionistic modal logics, (Buzoku et al., 9 Jul 2025) develops B‑eS systematically for Simpson’s labelled natural deduction systems. There, bases involve labelled atoms and relational assumptions, and support clauses for $KT$22 and $KT$23 are designed to match the labelled natural deduction rules. The overall pattern parallels (Eckhardt et al., 2024), but in an intuitionistic rather than classical environment (Buzoku et al., 9 Jul 2025).

For classical linear logic, (Barroso-Nascimento et al., 11 Apr 2025) extends B‑eS to the multiplicative-additive fragment of classical linear logic by introducing support judgments indexed by atomic multisets and defining connectives through elimination-style clauses centered on derivability of $KT$24. A subsequent paper proves an equivalence between this linear-logic B‑eS and phase semantics, and defines B‑eS clauses for exponentials $KT$25 and $KT$26 (Piotrovskaya, 11 Jun 2026). In that setting, bases are pairs $KT$27 with derivability over multisets of atoms, and the correspondence with phase semantics is established by explicit translations in both directions (Piotrovskaya, 11 Jun 2026).

For the logic of bunched implications, (Gu et al., 2023) develops a B‑eS in which bases carry bunched structure and support is indexed by bunches of atoms, reflecting BI’s combination of additive and multiplicative contexts. The paper stresses that BI requires a more complex notion of derivability in a base and a richer notion of support than either IPL or IMLL (Gu et al., 2023).

For second-order logic, (Gheorghiu et al., 11 Aug 2025) generalizes B‑eS to a semantics grounded in atomic systems and substitution over predicate constants, yielding soundness and completeness results equivalent to Henkin-style second-order logic. That paper presents B‑eS as a proof-theoretic alternative to both full and Henkin model-theoretic semantics and distinguishes classical and intuitionistic second-order logic by varying the class of atomic systems (Gheorghiu et al., 11 Aug 2025).

This distribution across logics shows that B‑eS is not a single semantics for a single calculus but a reusable proof-theoretic methodology. A plausible implication is that its core architectural motif—atomic inferential bases plus inductive support clauses plus extension conditions—functions as a general inferential template adaptable to modal, substructural, and higher-order settings.

6. Limitations, controversies, and later developments

A notable limitation of the modal framework in (Eckhardt et al., 2024) is that it is not complete for Euclidean modal logics. The paper proves that even if $KT$28 is Euclidean, it is not the case that

$KT$29

In particular, axiom $KT$30,

$KT$31

fails in the current B‑eS formulation (Eckhardt et al., 2024). The counterexample is constructed by taking a maximally-consistent base $KT$32, choosing suitable bases $KT$33, and defining a Euclidean relation $KT$34 such that $KT$35 but $KT$36 (Eckhardt et al., 2024). The paper interprets this as showing that clauses (c) and (d) in the definition of modal relation are not strong enough to capture Euclideanity in the same way as ordinary Kripke accessibility.

This point is methodologically important because it prevents a simplistic identification of “relation on bases” with “accessibility relation on worlds.” The relational layer in B‑eS must interact correctly with the extension structure of bases, and this interaction can block ordinary frame-correspondence phenomena. The later paper “Base-extension Semantics for S5 Modal Logic” (Eckhardt et al., 2024) addresses precisely this difficulty by developing a B‑eS for multi-agent S5 that augments the relation on bases with additional structural conditions and uses epistemic operators $KT$37 as primitives (Eckhardt et al., 2024). That later development indicates that the incompleteness result for Euclidean logics in (Eckhardt et al., 2024) was not a terminal obstacle but a problem requiring a refined relational design.

Another line of development concerns the local computational meaning of support. “Support is Search” (Gheorghiu, 13 Mar 2026) shows, for intuitionistic propositional logic, that support in a fixed base coincides with proof-search in a second-order hereditary Harrop logic program, via a continuation-passing-style encoding of formulas as goals. The paper’s main theorem is: $KT$38 (Gheorghiu, 13 Mar 2026). This suggests that at least in the intuitionistic propositional setting, B‑eS admits a direct computational interpretation: support is not only a semantic judgment but also a proof-search problem. While (Eckhardt et al., 2024) does not pursue this operational direction for modal logic, the result points toward potential implementations of modal B‑eS via logic programming or related proof-search technologies.

A separate controversy concerns the relation between B‑eS and more traditional proof-theoretic validity in the sense of Dummett and Prawitz. “From Proof-theoretic Validity to Base-extension Semantics for Intuitionistic Propositional Logic” (Gheorghiu et al., 2022) argues that Sandqvist’s B‑eS for IPL encapsulates the declarative content of an elimination-rule-based notion of proof-theoretic validity. The paper’s main claim is that B‑eS is not a rival to proof-theoretic validity but its declarative face (Gheorghiu et al., 2022). This suggests that B‑eS may best be understood not merely as an alternative semantics, but as a semantic recasting of proof-theoretic notions of inferential meaning.

Finally, categorical work has shown that B‑eS can be reconstructed in presheaf and sheaf-theoretic settings. “Categorical Proof-Theoretic Semantics” (Pym et al., 2023) relates Sandqvist’s B‑eS for IPL to presheaf categories and shows that Sandqvist’s treatment of disjunction corresponds to a natural disjunction in a category of sheaves (Pym et al., 2023). This suggests that B‑eS is compatible not only with proof-search and rule-based viewpoints but also with higher-level categorical reconstructions.

7. Significance

B‑eS provides a proof-theoretic semantics in which atomic validity is determined by derivability from a base, complex validity is built inductively, and modal structure is represented by relations on bases rather than by valuations on worlds. In “Base-extension Semantics for Modal Logic” (Eckhardt et al., 2024), this framework is shown to recover the classical propositional modal systems $KT$39, $KT$40, $KT$41, and $KT$42, establishing equivalence among B‑eS validity, Kripke validity, and Hilbert derivability. The same paper also proves a duality result for $KT$43 and an independently motivated $KT$44-clause, while identifying a specific limit of the present formulation in its failure for Euclidean logics (Eckhardt et al., 2024).

Across the surrounding literature, B‑eS has emerged as a general inferential method rather than a one-off semantics. It has been adapted to intuitionistic propositional logic (Gheorghiu et al., 2022), resolution-based analyses of sentential logic (Gheorghiu, 7 Mar 2025), intuitionistic modal logics (Buzoku et al., 9 Jul 2025), classical linear logic (Barroso-Nascimento et al., 11 Apr 2025), phase semantics equivalence (Piotrovskaya, 11 Jun 2026), BI (Gu et al., 2023), second-order logic (Gheorghiu et al., 11 Aug 2025), and computational proof-search interpretations (Gheorghiu, 13 Mar 2026). Later work on S5 shows that even the Euclidean failure identified in (Eckhardt et al., 2024) can motivate refined modal-relation designs rather than abandonment of the framework (Eckhardt et al., 2024).

The resulting picture is that B‑eS occupies a distinctive position between proof theory and semantics. It is proof-theoretic because its primitives are inferential bases and support judgments; it is semantic because it defines validity abstractly and proves soundness and completeness theorems; and it remains close enough to Kripke-style structure that canonical-model arguments and world/base correspondences can be recovered when the relational conditions are strong enough. This suggests that B‑eS is best viewed as a family of inferential semantics whose central technical achievement is to replace truth-at-a-world with validity-at-a-base while preserving rigorous correspondence with established logical systems.