Connecting Base-Extension Semantics, Beth Semantics, and Ecumenical Logic
Abstract
We investigate the relationship between base-extension semantics, Beth semantics, and Prawitz's ecumenical logic. We first present a base-extension semantics for the propositional fragment of Prawitz's system and prove that it is sound and complete. The semantics interprets intuitionistic and classical operators directly within a common proof-theoretic framework, without reducing the classical operators to their intuitionistic counterparts by means of a translation. We then analyse the effect of incorporating Beth-style clauses into base-extension semantics. When arbitrary bases are admitted and ⊥ is governed by atomic explosion, these clauses trivialise. By contrast, when ⊥ is treated as a basic sentence and attention is restricted to consistent bases, the Beth-style clauses for atoms and disjunction characterise, respectively, classical atoms and classical disjunction in Prawitz's system. We also obtain equivalent, path-independent formulations of the classical clauses for atoms, disjunction, and implication. These results reveal that the consistency constraint is not merely a technical restriction: within base-extension semantics, it transforms clauses originally associated with intuitionistic semantics into semantic characterisations of classical operators.
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BibTeX
@misc{barrosonascimento2026connecting,
title = {{Connecting Base-Extension Semantics, Beth Semantics, and Ecumenical Logic}},
author = {Victor Barroso-Nascimento and Joaquim T. Waddington and Elaine Pimentel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38871},
eprint = {2610.38871},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38871},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Connecting Base-Extension Semantics, Beth Semantics, and Ecumenical Logic
AU - Victor Barroso-Nascimento
AU - Joaquim T. Waddington
AU - Elaine Pimentel
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38871
UR - https://arxiv.org/abs/2610.38871
AB - We investigate the relationship between base-extension semantics, Beth semantics, and Prawitz's ecumenical logic. We first present a base-extension semantics for the propositional fragment of Prawitz's system and prove that it is sound and complete. The semantics interprets intuitionistic and classical operators directly within a common proof-theoretic framework, without reducing the classical operators to their intuitionistic counterparts by means of a translation. We then analyse the effect of incorporating Beth-style clauses into base-extension semantics. When arbitrary bases are admitted and ⊥ is governed by atomic explosion, these clauses trivialise. By contrast, when ⊥ is treated as a basic sentence and attention is restricted to consistent bases, the Beth-style clauses for atoms and disjunction characterise, respectively, classical atoms and classical disjunction in Prawitz's system. We also obtain equivalent, path-independent formulations of the classical clauses for atoms, disjunction, and implication. These results reveal that the consistency constraint is not merely a technical restriction: within base-extension semantics, it transforms clauses originally associated with intuitionistic semantics into semantic characterisations of classical operators.
ER -