Metalogic and the Overgeneration Argument

Mind 128 (511):761-793 (2019)
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A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic.

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Author Profiles

Luca Incurvati
University of Amsterdam
Salvatore Florio
University of Birmingham

Citations of this work

Unrestricted Quantification and the Structure of Type Theory.Salvatore Florio & Nicholas K. Jones - 2021 - Philosophy and Phenomenological Research 102 (1):44-64.
Logical Truth.Mario Gomez-Torrente - 2014 - Stanford Encyclopedia of Philosophy.

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References found in this work

Modal Logic as Metaphysics.Timothy Williamson - 2013 - Oxford, England: Oxford University Press.
Tractatus logico-philosophicus.Ludwig Wittgenstein - 1922 - Filosoficky Casopis 52:336-341.
Tractatus Logico-Philosophicus.Ludwig Wittgenstein - 1956 - Revista Portuguesa de Filosofia 12 (1):109-110.
Quantifiers and propositional attitudes.Willard van Orman Quine - 1956 - Journal of Philosophy 53 (5):177-187.

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