From Phase Semantics to Base-extension Semantics (and back)

Ekaterina Piotrovskaya

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arXiv · 2606.13855

From Phase Semantics to Base-extension Semantics (and back)

Ekaterina Piotrovskaya

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Abstract

Linear logic admits a wide range of semantic presentations reflecting its resource-sensitive notion of consequence. One well-known example is phase semantics: an algebraic semantics in which formulas are interpreted in phase models, consisting of a phase space formed by a commutative monoid and a fixed subset, with respect to which an orthogonality relation is defined, and a valuation. A rather different and much more recent approach is given by base-extension semantics, which defines validity by inductively extending a provability relation on a base -- a set of inference rules over atomic propositions. We establish an equivalence between the two semantics by first defining bidirectional maps between bases and phase models, and then constructing an isomorphism between a phase model (resp. base) and its image under the composition of these maps. As a further contribution, we define the base-extension semantics clauses for the exponentials of linear logic.

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Ekaterina Piotrovskaya. 2026-06-11. From Phase Semantics to Base-extension Semantics (and back). https://arxiv.org/abs/2606.13855

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