A comparison of three kinds of monotonic proof-theoretic semantics and the base-incompleteness of intuitionistic logic
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| Format: | Preprint |
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2025
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| author | d'Aragona, Antonio Piccolomini |
| author_facet | d'Aragona, Antonio Piccolomini |
| contents | I deal with two approaches to proof-theoretic semantics: one based on argument structures and justifications, which I call reducibility semantics, and one based on consequence among (sets of) formulas over atomic bases, called base semantics. The latter splits in turn into a standard reading, and a variant of it put forward by Sandqvist. I prove some results which, when suitable conditions are met, permit one to shift from one approach to the other, and I draw some of the consequences of these results relative to the issue of completeness of (recursive) logical systems with respect to proof-theoretic notions of validity. This will lead me to focus on a notion of base-completeness, which I will discuss with reference to known completeness results for intuitionistic logic. The general interest of the proposed approach stems from the fact that reducibility semantics can be understood as a labelling of base semantics with proof-objects typed on (sets of) formulas for which a base semantics consequence relation holds, and which witness this very fact. Vice versa, base semantics can be understood as a type-abstraction of a reducibility semantics consequence relation obtained by removing the witness of the fact that this relation holds, and by just focusing on the input and output type of the relevant proof-object. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A comparison of three kinds of monotonic proof-theoretic semantics and the base-incompleteness of intuitionistic logic d'Aragona, Antonio Piccolomini Logic Logic in Computer Science I deal with two approaches to proof-theoretic semantics: one based on argument structures and justifications, which I call reducibility semantics, and one based on consequence among (sets of) formulas over atomic bases, called base semantics. The latter splits in turn into a standard reading, and a variant of it put forward by Sandqvist. I prove some results which, when suitable conditions are met, permit one to shift from one approach to the other, and I draw some of the consequences of these results relative to the issue of completeness of (recursive) logical systems with respect to proof-theoretic notions of validity. This will lead me to focus on a notion of base-completeness, which I will discuss with reference to known completeness results for intuitionistic logic. The general interest of the proposed approach stems from the fact that reducibility semantics can be understood as a labelling of base semantics with proof-objects typed on (sets of) formulas for which a base semantics consequence relation holds, and which witness this very fact. Vice versa, base semantics can be understood as a type-abstraction of a reducibility semantics consequence relation obtained by removing the witness of the fact that this relation holds, and by just focusing on the input and output type of the relevant proof-object. |
| title | A comparison of three kinds of monotonic proof-theoretic semantics and the base-incompleteness of intuitionistic logic |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2501.03297 |