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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2210.05491 |
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| _version_ | 1866909570260008960 |
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| author | Köpp, Nils Petrakis, Iosif |
| author_facet | Köpp, Nils Petrakis, Iosif |
| contents | We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and Gödel's system $\mathbf{T}$, by defining simultaneously strong negation $A^{\mathbf{N}}$ of a formula $A$ and strong negation $P^{\mathbf{N}}$ of a predicate $P$ in TCF. As a special case of the latter, we get strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of double negation elimination for strong negation in TCF. We introduce the so-called tight formulas of TCF i.e., formulas implied by the weak negation of their strong negation, and the relative tight formulas. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05491 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Strong negation in the theory of computable functionals TCF Köpp, Nils Petrakis, Iosif Logic Logic in Computer Science We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and Gödel's system $\mathbf{T}$, by defining simultaneously strong negation $A^{\mathbf{N}}$ of a formula $A$ and strong negation $P^{\mathbf{N}}$ of a predicate $P$ in TCF. As a special case of the latter, we get strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of double negation elimination for strong negation in TCF. We introduce the so-called tight formulas of TCF i.e., formulas implied by the weak negation of their strong negation, and the relative tight formulas. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics. |
| title | Strong negation in the theory of computable functionals TCF |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2210.05491 |