Abstract
Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional part of ‘strict finitistic logic’ from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model.
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Acknowledgements
I would like to thank Rosalie Iemhoff for her everyday, most insightful and helpful pieces of advice. I am also grateful to Amirhossein Akbar Tabatabai for his penetrating comments: indeed, it was he who first suggested the relation with IPC. I sincerely thank an anonymous referee for extremely helpful comments and suggestions. Some of the rudimentary versions of the present article were presented at the OZSW Conference 2020, the 3rd Workshop on Proof Theory and its Applications in 2021 and the Dutch Logic PhD Day 2022. This study was conducted under the doctoral supervision by Professor Rosalie Iemhoff at Utrecht University.
Funding
This study was funded under the name of ‘Graduate Scholarship for Degree Seeking Students’ by Japan Student Services Organisation.
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Yamada, T. Wright’s Strict Finitistic Logic in the Classical Metatheory: The Propositional Case. J Philos Logic (2023). https://doi.org/10.1007/s10992-022-09698-w
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DOI: https://doi.org/10.1007/s10992-022-09698-w
Keywords
- Strict finitism
- Crispin Wright
- Constructivism
- Finitism
- Classical reconstruction