Base-Extension Semantics (B-eS)

• Base-extension semantics is a proof-theoretic framework where formula meanings are defined inductively from a base of atomic inference rules rather than by model-theoretic truth conditions.
• It employs base-indexed support and inductive clauses to capture complex logical connectives and ensure soundness and completeness across systems like intuitionistic, modal, and substructural logics.
• Extensions of B-eS demonstrate its adaptability by grounding bilateral logics, dynamic epistemic updates, and higher-order constructs through atomic simulation and categorical reconstruction.

Base-extension semantics (B-eS) is a form of proof-theoretic semantics in which meaning is based on inference and the validity of formulae is defined inductively from a base of atomic rules rather than by truth in a model. Across the literature, B-eS is presented as a semantics of logical constants grounded in atomic provability, while proof-theoretic validity is treated as a semantics of arguments; the two approaches are closely related but not identical (Gheorghiu et al., 2022). The framework has been developed for intuitionistic and classical propositional logic, modal systems, bilateral logics, substructural logics, second-order logic, and dynamic epistemic settings, with recurring emphasis on soundness and completeness relative to proof systems and, in several cases, to model-theoretic semantics (Eckhardt et al., 2024).

1. Inferential setting and the role of bases

In B-eS, the basic semantic parameter is a base: a set of atomic inference rules, or more generally an atomic system. Several presentations also distinguish a basis, namely a set of bases or atomic systems (Gheorghiu et al., 2022). Atomic propositions are not assigned primitive truth values; instead, their meaning is fixed by what is derivable from the relevant atomic rules. This is the point at which B-eS differs from model-theoretic semantics and aligns with inferentialism and proof-theoretic semantics (Gheorghiu et al., 2022).

The framework is uniformly reductive. Meaning for complex formulae is reduced to support or validity conditions for their immediate constituents, and ultimately to atomic derivability in the base. In one standard propositional and modal formulation, a base is a countable collection of base rules, and its closure $\overline{\mathscr{B}}$ is the set of atomic sentences derivable from $\mathscr{B}$ by its rules (Eckhardt et al., 2024). In other formulations, especially for intuitionistic logic, bases are interpreted as collections of definite formulae or logic programs, so that atomic support becomes operationally identifiable with proof search in the associated program (Gheorghiu et al., 2022).

This inferential architecture is also why B-eS is often described as base-parametric. A formula is not evaluated absolutely at the first stage; it is evaluated relative to a base, and global validity is then obtained by quantifying over all bases in the chosen basis (Gheorghiu et al., 2022). A plausible implication is that B-eS is less a single semantics than a family of closely related proof-theoretic schemes, unified by the use of atomic bases and inductive clauses.

2. Support, inductive clauses, and non-truthfunctional connectives

The central semantic notion in much of the literature is support, usually written as a base-indexed judgment such as $\Gamma \Vdash_B \phi$ or $\Vdash_{\mathscr{B}} \phi$ (Gheorghiu et al., 2022). In a standard propositional clause-set, atomic support is given by atomic derivability: $$\Vdash_{\mathscr{B}} p \iff p \in \overline{\mathscr{B}}.$$ Implication is defined by quantifying over base extensions: $$\Vdash_{\mathscr{B}} \phi \rightarrow \psi \iff \forall \mathscr{C} \supseteq \mathscr{B},\; (\Vdash_{\mathscr{C}} \phi \Rightarrow \Vdash_{\mathscr{C}} \psi),$$ and falsum is supported when every atomic proposition is supported (Eckhardt et al., 2024).

For intuitionistic logic, disjunction and, in some variants, conjunction are not treated by ordinary truth-functional clauses. Sandqvist-style B-eS uses an elimination-rule or second-order presentation of disjunction, quantifying over all base extensions and atomic conclusions; categorical reconstructions explicitly identify this as a second-order or elimination-rule presentation rather than a coproduct interpretation (Pym et al., 2023). A related development for intuitionistic multiplicative linear logic adopts an elimination-rule-centered treatment more systematically and proposes a generalized elimination-rule presentation for conjunction in IPL as well (Gheorghiu et al., 2023).

This treatment is metatheoretically significant. The second-order clause for disjunction is repeatedly singled out as essential for completeness in intuitionistic settings (Pym et al., 2023). It also shows that B-eS is not simply a matter of adding formulae to a context. One paper gives the explicit counterexample

$$\phi := (a \to b \vee c) \to \big((a \to b) \vee (a \to c)\big),$$

which is not a theorem of IPL, yet there is a base $B$ such that $B \Vdash a \to b \vee c$ while $B \not\Vdash (a \to b) \vee (a \to c)$ (Gheorghiu et al., 2022). This blocks a naive identification of “base” with “context” and underscores that support is a proof-theoretic relation with its own structural conditions.

3. Completeness methods, proof search, and categorical reconstruction

A major theme in the B-eS literature is adequacy: support in all bases should coincide with derivability in the intended proof system. For IPL, this is stated as

$\mathscr{B}$0

with proofs using specially constructed simulation bases that assign fresh atoms to subformulae and encode introduction and elimination behavior at the atomic level (Gheorghiu et al., 2022). The same flattening or simulation idea recurs in modal, bilateral, sequent-calculus, and linear-logic developments.

Several papers reinterpret this completeness machinery operationally. Bases can be encoded as definite clauses, and atomic proof in a base can then be identified with provability in the corresponding logic program. On this reading, completeness for IPL is linked to uniform proof search, while negation is illuminated via negation-as-failure: denial of a proposition is understood not as assertion of its negation but as failure to find a proof of it (Gheorghiu et al., 2022). Another paper shows that Sandqvist’s unusual completeness proof closely parallels Mints’ resolution-based method for intuitionistic logic, and concludes that soundness and completeness of B-eS follow directly from Mints’ theorem (Gheorghiu, 7 Mar 2025).

The framework has also been reconstructed categorically. In presheaf semantics, formulas are interpreted as functors, and validity is expressed by natural transformations. Within this reconstruction, Sandqvist’s disjunction is recovered as a second-order or parametric construction rather than as categorical coproduct; from the perspective of validity, it can also be viewed as the natural disjunction in a category of sheaves (Pym et al., 2023). This is presented not as a replacement of B-eS but as a demonstration of the naturality of its constructions.

The relation between B-eS and proof-theoretic validity remains a live issue. One paper argues that B-eS for IPL encapsulates the declarative content of an elimination-based version of proof-theoretic validity (Gheorghiu et al., 2022). Another, working with monotonic introduction-based variants, shows that the atomic Kreisel–Putnam rule

$\mathscr{B}$1

is uniformly valid in monotonic proof-theoretic validity, thereby implying incompleteness of intuitionistic logic over that framework as well (d'Aragona, 22 Mar 2025). This suggests that not every proof-theoretic semantics based on atomic systems reproduces exactly the theorems of intuitionistic logic.

4. Modal and dynamic extensions

Modal B-eS introduces relational structure on bases. For the classical modal systems $\mathscr{B}$2, $\mathscr{B}$3, $\mathscr{B}$4, and $\mathscr{B}$5, validity is indexed both by a base and by a relation $\mathscr{B}$6 on bases, designed to parallel Kripke accessibility (Eckhardt et al., 2024). In that setting, the clause for $\mathscr{B}$7 is

$\mathscr{B}$8

The corresponding clause for $\mathscr{B}$9 is existential over accessible bases at every extension, and the paper proves the duality

$\Gamma \Vdash_B \phi$0

(Eckhardt et al., 2024).

For $\Gamma \Vdash_B \phi$1, B-eS is shown equivalent to Kripke validity and to Hilbert derivability (Eckhardt et al., 2024). The same paper also states that the semantics, in its current form, is not complete with respect to euclidean modal logics. More specifically, it does not validate the S5 axiom $\Gamma \Vdash_B \phi$2 (Eckhardt et al., 2024). That limitation was not the last word. A later paper develops a B-eS for multi-agent S5 using agent-indexed S5-modal relations on bases and establishes soundness and completeness, explicitly framing $\Gamma \Vdash_B \phi$3 as the knowledge operator $\Gamma \Vdash_B \phi$4 (Eckhardt et al., 2024).

Dynamic epistemic extensions continue this line. In inferentialist public announcement logic, the key difficulty is that announcements of the form $\Gamma \Vdash_B \phi$5 do not merely remove worlds, as in Kripke semantics, but update modal relations on bases. The resulting notion of effective update can be non-deterministic, so support for announcement formulae is universally quantified over all effective updates satisfying the required conditions (Eckhardt et al., 2024). The three-player card game and the muddy children puzzle are used to show that, in this framework, all relevant information must be explicit in the bases; nothing can remain tacitly encoded in a world structure (Eckhardt et al., 2024).

5. Bilateral semantics, refutation, and incompatibility

A bilateral extension of B-eS takes proof and refutation as primitive. In the version developed for the bilateral dual intuitionistic logic $\Gamma \Vdash_B \phi$6, a bilateral atomic system contains both atomic proof rules and atomic refutation rules, and semantic support is correspondingly split into positive and negative judgments, $\Gamma \Vdash_B \phi$7 and $\Gamma \Vdash_B \phi$8 (Barroso-Nascimento et al., 2 May 2025). The semantics is proved sound and complete with respect to a bilateral natural deduction system, and structural dualities between proof rules and refutation rules are lifted to dualities for deductions and bases. On that basis, bilateral semantic harmony is formulated as a restatement of the syntactic horizontal inversion principle (Barroso-Nascimento et al., 2 May 2025).

A later development sharpens the constructive reading. It introduces epistemically adequate bases satisfying logical consistency, unit completeness, and epistemic consistency, together with explicit constructions of proofs and refutations (Barroso-Nascimento et al., 19 Oct 2025). In that setting, no formula can be both provable and refutable in the same epistemically adequate base: $\Gamma \Vdash_B \phi$9 The paper presents this as a central innovation over earlier bilateral frameworks such as Wansing’s $\Vdash_{\mathscr{B}} \phi$0 and Nelson’s $\Vdash_{\mathscr{B}} \phi$1 (Barroso-Nascimento et al., 19 Oct 2025).

This bilateral strand also changes the interpretation of denial. In the IPL-oriented logic-programming treatment, denial is associated with failure to find a proof (Gheorghiu et al., 2022). In the later bilateral system with incompatible proofs and refutations, refutation is instead an explicit construction, and it is shown to correspond to David Nelson’s constructive falsity (Barroso-Nascimento et al., 19 Oct 2025). The two lines are not identical, but together they show that B-eS can support both unilateral and genuinely bilateral proof-theoretic semantics.

6. Substructural, classical, sequent, and higher-order generalizations

B-eS has been extended well beyond ordinary propositional logic. For intuitionistic multiplicative linear logic, it is presented as the first exploration of proof-theoretic semantics for a substructural logic, with support indexed by multisets of atoms so that resource information is transmitted through the semantics (Gheorghiu et al., 2023). The logic of bunched implications requires a further step: base rules and support must carry bunched structure, because contexts are bunches rather than sets or multisets. The resulting semantics systematically combines the additive behavior of IPL with the multiplicative behavior of IMLL and is proved sound and complete for BI (Gu et al., 2023).

Classical linear logic prompted a different modification. For the multiplicative-additive fragment, one paper gives a proof-theoretic semantics in which support is oriented toward derivability of $\Vdash_{\mathscr{B}} \phi$2, thereby capturing a specifically classical notion of refutation (Barroso-Nascimento et al., 11 Apr 2025). A later paper establishes an equivalence between this B-eS perspective and phase semantics by defining bidirectional maps between bases and phase spaces and by providing new B-eS clauses for the exponentials of linear logic (Piotrovskaya, 11 Jun 2026). This suggests that proof-theoretic and algebraic presentations of linear-logic validity can be inter-translated without loss at the level of semantic content.

The sequent-calculus perspective yields yet another variant. Here B-eS is defined using atomic systems based on sequent calculus rather than natural deduction, and multiple-conclusion sequents are said to make the framework more suited to classical semantics. In that formulation, the semantic clauses are derived solely from right introduction rules, and the presence or absence of atomic cut rules changes the semantics even though completeness is obtained in both cases (Barroso-Nascimento et al., 24 May 2025). A plausible implication is that the exact proof calculus chosen for the base is not an inessential presentation detail but part of the semantic architecture.

Higher-order extensions continue the same pattern. For second-order logic, B-eS is based on atomic systems encoding inferential commitments, with classical and intuitionistic versions obtained by varying the class of atomic systems. The resulting semantic consequence relations are sound and complete for corresponding Hilbert-style calculi and are shown equivalent to Henkin semantics (Gheorghiu et al., 11 Aug 2025). In that setting, second-order quantification is interpreted as systematic substitution rather than set-theoretic commitment (Gheorghiu et al., 11 Aug 2025).

Taken together, these developments present B-eS not as a single finished semantics but as a proof-theoretic program. Its common core is stable: atomic bases, inductive support, and adequacy to proof systems. Its boundaries remain active: elimination-style clauses versus simpler clauses, monotonic versus non-monotonic variants, unilateral versus bilateral treatments, and the extent to which model-theoretic notions such as Kripke accessibility, phase spaces, or Henkin semantics can be recovered inside an inferentialist framework.