Cut-free Deductive System for Continuous Intuitionistic Logic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910012597600256 |
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| author | Geoffroy, Guillaume |
| author_facet | Geoffroy, Guillaume |
| contents | We introduce and develop propositional continuous intuitionistic logic and propositional continuous affine logic via complete algebraic semantics. Our approach centres on AC-algebras, which are algebras $USC(\mathcal{L})$ of sup-preserving functions from $[0,1]$ to an integral commutative residuated complete lattice $\mathcal{L}$ (in the intuitionistic case, $\mathcal{L}$ is a locale). We give an algebraic axiomatisation of AC-algebras in the language of continuous logic and prove, using the Macneille completion, that every Archimedean model embeds into some AC-algebra. We also show that (i) $USC(\mathcal{L})$ satisfies $v \dot + v = 2v$ exactly when $\mathcal{L}$ is a locale, (ii) involutiveness of negation in $USC(\mathcal{L})$ corresponds to that in $\mathcal{L} $, and that (iii) adding those conditions recovers classical continuous logic. For each variant -affine, intuitionistic, involutive, classical -we provide a sequent style deductive system and prove completeness and cut admissibility. This yields the first sequent style formulation of classical continuous logic enjoying cut admissibility. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26849 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cut-free Deductive System for Continuous Intuitionistic Logic Geoffroy, Guillaume Logic in Computer Science Logic We introduce and develop propositional continuous intuitionistic logic and propositional continuous affine logic via complete algebraic semantics. Our approach centres on AC-algebras, which are algebras $USC(\mathcal{L})$ of sup-preserving functions from $[0,1]$ to an integral commutative residuated complete lattice $\mathcal{L}$ (in the intuitionistic case, $\mathcal{L}$ is a locale). We give an algebraic axiomatisation of AC-algebras in the language of continuous logic and prove, using the Macneille completion, that every Archimedean model embeds into some AC-algebra. We also show that (i) $USC(\mathcal{L})$ satisfies $v \dot + v = 2v$ exactly when $\mathcal{L}$ is a locale, (ii) involutiveness of negation in $USC(\mathcal{L})$ corresponds to that in $\mathcal{L} $, and that (iii) adding those conditions recovers classical continuous logic. For each variant -affine, intuitionistic, involutive, classical -we provide a sequent style deductive system and prove completeness and cut admissibility. This yields the first sequent style formulation of classical continuous logic enjoying cut admissibility. |
| title | Cut-free Deductive System for Continuous Intuitionistic Logic |
| topic | Logic in Computer Science Logic |
| url | https://arxiv.org/abs/2510.26849 |