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POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS’ TALE

Published online by Cambridge University Press:  26 April 2018

PAOLO PISTONE
PAOLO PISTONE*
Affiliation:
DIPARTIMENTO DI MATEMATICA E FISICA UNIVERSITÀ ROMA TRE L.GO S. LEONARDO MURIALDO 1 00146 ROME, ITALYE-mail:[email protected]
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Abstract

The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.

We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze (shown inconsistent by Russell’s paradox) and the one given by Martin-Löf for the intuitionistic type theory with a type of all types (shown inconsistent by Girard’s paradox).

The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.

Keywords

03A0503F0303F05second order logicimpredicativityintuitionistic type theorynormalization proofs
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 24 , Issue 1 , March 2018 , pp. 1 - 52
Copyright
Copyright © The Association for Symbolic Logic 2018 

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