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Отмеченное субординатное натуральное исчисление для базовой интуиционистской кондициональной логики
Логические исследования. 2025. Т. 31. № 2. С. 143–168.
This article presents a labeled Fitch-style natural deduction system, 𝓕IntCK, for the basic propositional intuitionistic conditional logic IntCK introduced by G.K. Olkhovikov. The logic IntCK serves as a correct intuitionistic counterpart to Chellas' minimal conditional logic CK, designed to accommodate Lewis' strong and weak counterfactual conditionals within a single framework. In order to do this, IntCK features two independent logical connectives, namely □→ and ◇→, and is interpreted over birelational Kripke semantics with specific confluence conditions linking the intuitionistic and conditional accessibility relations. The 𝓕IntCK calculus employs labels, relational atoms, and the notion of labeled quasi-formulas to encode semantic notions and conditions, allowing the construction of derivations that reflect the truth and non-truth of formulas in possible worlds. The system is built upon a subordinate derivation framework, enabling an inductive definition of derivability (in the style of V.A. Smirnov). The inference rules of 𝓕IntCK are aligned with semantic principles: structural rules govern relational atoms, while logical rules determine the correct assertibility conditions for connectives, including the two counterfactual operators □→ and ◇→. Finally, the paper outlines the proof strategies for two metatheorems: weak completeness of 𝓕IntCK with respect to the class of all birelational intuitionistic conditional frames and deductive equivalence of 𝓕IntCK and IntCK.
Research target:
Philosophy, Ethics, and Religious Studies
Language:
Russian
Keywords: completenessтеория доказательствproof theoryCounterfactualsIntuitionistic Logicинтуиционистская логикаlabelsnatural deductionСonditional logiclabeled formulaerelational atoms birelational Kripke semanticsdeductive equivalenceисчисление естественного выводаконтрфактуалыусловная логикаметкиотмеченные формулыреляционные атомыбиреляционная семантика Крипкесемантическая полнотадедуктивная эквивалентность
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