Contraction, Infinitary Quantifiers, and Omega Paradoxes
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Our main goal is to investigate whether the infinitary rules for the quantifiers endorsed by Elia Zardini in a recent paper are plausible. First, we will argue that they are problematic in several ways, especially due to their infinitary features. Secondly, we will show that even if these worries are somehow dealt with, there is another serious issue with them. They produce a truth-theoretic paradox that does not involve the structural rules of contraction.
KeywordsSubstructural logic Infinitary quantifiers Paradoxes Truth
Earlier versions of the material in this paper have been presented in 2016 at the 3rd CLE-Buenos Aires Logic Group Workshop in Buenos Aires and in 2017 at the Munich Center for Mathematical Philosophy (MCMP) in Munich. We are very grateful to the members of those audiences. We also owe thanks to two reviewers of this journal (and one of another) for their insightful comments and suggestions. Special thanks go to Elia Zardini for patiently discussing some of the ideas in this paper with us. We are also grateful to Eduardo Barrio, Natalia Buacar, Marcelo Coniglio, Carlo Nicolai, Lavinia Picollo, Damián Szmuc, and Paula Teijeiro for their contributions. This paper could not have been written without the financial aid of the National Scientific and Technical Research Council (CONICET).
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