Published in Philosophia Mathematica, 2022
This is a book review of a collection of Hellman’s papers, comissioned by Philosophia Mathematica.
Recommended citation: Scambler, Chris. (2022). "Geoffrey Hellman: Mathematics and its Logics." Philosophia Mathematica. 30 (2), 245-255. https://academic.oup.com/philmat/article/30/2/245/6514760
Published in Journal of Philosophical Logic, 2021
This paper presents and defends a modal version of set theory according to which there is only one size of infinity.
Recommended citation: Scambler, Chris. (2021). "Can All Things Be Counted." Journal of Philosophical Logic. 50 (5), 1079-1106. https://link.springer.com/content/pdf/10.1007/s10992-021-09593-w.pdf
Published in Journal of Philosophical Logic, 2021
In their article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc present novel and striking results about meta-inferential validity in various three valued logics. In the process, they have thrown open the door to a hitherto unrecognized domain of non-classical logics with surprising intrinsic properties, as well as subtle and interesting relations to various familiar logics, including classical logic. One such result is that, for each natural number n, there is a logic which agrees with classical logic on tautologies, inferences, meta-inferences, meta-meta-inferences, meta-meta-…(n - 3 times)-meta-inferences, but that disagrees with classical logic on n + 1-meta-inferences. They suggest that this shows that classical logic can only be characterized by defining its valid inferences at all orders. In this article, I invoke some simple symmetric generalizations of BPS’s results to show that the problem is worse than they suggest, since in fact there are logics that agree with classical logic on inferential validity to all orders but still intuitively differ from it. I then discuss the relevance of these results for truth theory and the classification problem.
Recommended citation: Scambler, Chris. (2021). "Classical Logic and the Strict-Tolerant Hierarchy." Journal of Philosophical Logic. 49, pages 1079–1089. https://link.springer.com/article/10.1007/s10992-019-09520-0
Published in Erkenntnis, 2021
This paper defends the Cantorian view that the existence of a bijection between two collections is necessary and sufficient for them to have the same size. It is a response to a case made to the contrary by Bruno Whittle.
Recommended citation: Scambler, Chris. (2021). "A Justification for the Quantificational Hume Principle." Erkenntnis. 86 (5), 1293–1308. https://link.springer.com/article/10.1007/s10670-019-00154-x
Published in Journal of Philosophical Logic, 2021
This paper generalizes some results of Barrio, Pailos and Szmuc from the finite to the transfinite. I show that there are subclassical systems of logic that agree with classical logic arbitrarily far through a hierarchy of metainferential validity relations but that fail to be classical eventually.
Recommended citation: Scambler, Chris. (2021). "Transfinite Meta-inferences." Journal of Philosophical Logic. 49, pages 1079–1089. https://link.springer.com/article/10.1007/s10992-020-09548-7
Published in Review of Symbolic Logic, 2020
In this paper I build on work of Philip Welch exploring the logical and philosophical properties of a novel kind of revenge paradox arising for Field’s theory of truth, the ‘ineffable liar’.
Recommended citation: Scambler, Chris. (2020). "Ineffability and Revenge." Review of Symbolic Logic. 13 (4), 797-809. https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/abs/ineffability-and-revenge/21F84612DD48D61C16E2410D2FB6FF89
Published in Synthese, 2020
In this paper, I develop a view on set-theoretic ontology I call Universe-Indeterminism, according to which there is a unique but indeterminate universe of sets. I argue that Solomon Feferman’s work on semi-constructive set theories can be adapted to this project, and develop a philosophical motivation for a semi-constructive set theory closely based on Feferman’s but tailored to the Universe-Indeterminist’s viewpoint. I also compare the emergent Universe-Indeterminist view to some more familiar views on set-theoretic ontology.
Recommended citation: Scambler, Chris. (2020). "An Indeterminate Universe of Sets." Synthese. 197, 545–573. https://link.springer.com/article/10.1007/s10992-019-09520-0