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Implicit commitment in a general setting.

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Title: Implicit commitment in a general setting.
Authors: ŁeŁyk, Mateusz1 (AUTHOR), Nicolai, Carlo2 (AUTHOR)
Source: Journal of Logic & Computation. Sep2024, Vol. 34 Issue 6, p1136-1158. 23p.
Subjects: Incompleteness theorems, Philosophy of mathematics, Logic programming, Open-ended questions, Motivation (Psychology)
Abstract: Gödel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those implicit assumptions. This notion of implicit commitment motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn't been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, we carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system |$S$|⁠ , it can be derived that a uniform reflection principle for |$S$| —stating that all numerical instances of theorems of |$S$| are true—must be contained in |$S$| 's implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Implicit commitment in a general setting.
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  Data: <searchLink fieldCode="AR" term="%22ŁeŁyk%2C+Mateusz%22">ŁeŁyk, Mateusz</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Nicolai%2C+Carlo%22">Nicolai, Carlo</searchLink><relatesTo>2</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Sep2024, Vol. 34 Issue 6, p1136-1158. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Incompleteness+theorems%22">Incompleteness theorems</searchLink><br /><searchLink fieldCode="DE" term="%22Philosophy+of+mathematics%22">Philosophy of mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Logic+programming%22">Logic programming</searchLink><br /><searchLink fieldCode="DE" term="%22Open-ended+questions%22">Open-ended questions</searchLink><br /><searchLink fieldCode="DE" term="%22Motivation+%28Psychology%29%22">Motivation (Psychology)</searchLink>
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  Data: Gödel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those implicit assumptions. This notion of implicit commitment motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn't been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, we carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system |$S$|⁠ , it can be derived that a uniform reflection principle for |$S$| —stating that all numerical instances of theorems of |$S$| are true—must be contained in |$S$| 's implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1093/logcom/exad025
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        Text: English
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      – SubjectFull: Philosophy of mathematics
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      – SubjectFull: Logic programming
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      – SubjectFull: Open-ended questions
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              Text: Sep2024
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