A Simplified and Improved Free-Variable Framework for Hilbert's epsilon as an Operator of Indefinite Committed Choice
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arXiv
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| Format: | Preprint |
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2011
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| _version_ | 1866914814991794176 |
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| author | Wirth, Claus-Peter |
| author_facet | Wirth, Claus-Peter |
| contents | Free variables occur frequently in mathematics and computer science with ad hoc and altering semantics. We present the most recent version of our free-variable framework for two-valued logics with properly improved functionality, but only two kinds of free variables left (instead of three): implicitly universally and implicitly existentially quantified ones, now simply called "free atoms" and "free variables", respectively. The quantificational expressiveness and the problem-solving facilities of our framework exceed standard first-order and even higher-order modal logics, and directly support Fermat's descente infinie. With the improved version of our framework, we can now model also Henkin quantification, neither using quantifiers (binders) nor raising (Skolemization). We propose a new semantics for Hilbert's epsilon as a choice operator with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports reductive proof search optimally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1104_2444 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | A Simplified and Improved Free-Variable Framework for Hilbert's epsilon as an Operator of Indefinite Committed Choice Wirth, Claus-Peter Artificial Intelligence Logic Free variables occur frequently in mathematics and computer science with ad hoc and altering semantics. We present the most recent version of our free-variable framework for two-valued logics with properly improved functionality, but only two kinds of free variables left (instead of three): implicitly universally and implicitly existentially quantified ones, now simply called "free atoms" and "free variables", respectively. The quantificational expressiveness and the problem-solving facilities of our framework exceed standard first-order and even higher-order modal logics, and directly support Fermat's descente infinie. With the improved version of our framework, we can now model also Henkin quantification, neither using quantifiers (binders) nor raising (Skolemization). We propose a new semantics for Hilbert's epsilon as a choice operator with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports reductive proof search optimally. |
| title | A Simplified and Improved Free-Variable Framework for Hilbert's epsilon as an Operator of Indefinite Committed Choice |
| topic | Artificial Intelligence Logic |
| url | https://arxiv.org/abs/1104.2444 |