Extensionalist explanation and solution of Russell's Paradox

(Logic & Philosophy of Mathematics, English)

n this paper, I propose an answer to the open question about the, so-called, explanation of Russell's paradox. In the debate about this paradox, there are traditionally two main and incompatible positions: the Cantorian explanation and the Predicativist one. I briefly rehearse the reasons why both these positions can be neglected and propose a third, Extensionalist, one, with a related solution.

The Extensionalist explanation identifies the key of Russell's Paradox in a proposition about the extensions: ∀F∃x(x = ext(F)) that allows to derive, from the existence of Russell's concept, the existence of Russell's extension. This proposition is a theorem of classical logic whose derivation presupposes the classical treatment of identity and quantification. So, we can explain Russell's paradox by the (inappropriate) classical correlation between concepts and extensions and, in particular, in the assumption (provided by classical logic) that the correlation is defined on the whole second order domain.

The solution related to the Extensionalist explanation consists in a re- formulation of Frege's theory, in which classical first order logic is replaced with negative free logic to allow the derivation of parts of Peano Arithmetic as a logical theory of extensions. We can analyse three different versions of this free fregean system that share the logical part of the axiomatization (FL) and differ each other only by the non-logical axioms ( E-BLV: ∀F ∀G(ext(F ) = ext(G) ↔ E !(ext(F )) ∧ E !(ext(G)) ∧ Πx(F x ↔ Gx)); P- BLV: ∀F∀G(ext(F) = ext(G) ↔ (φ(F) ∧ φ(G)Πx(Fx ↔ Gx)) T-BLV): ∀F ∀G(ext(F ) = ext(G) ↔ (φ(F ) ∧ φ(G) ∧ Πx(F x ↔ Gx))). All these systems prevent to derive Russell's Paradox and allow to derive different parts of Peano Arithmetic.

References

_1_ Boolos, G. (1986). Saving Frege from contradiction, Aristotelian Society, Supplementary Volume 87, 137-151.

_2_ Cocchiarella, N. B. (1992). Cantor's power-set Theorem versus Frege's double correlation Thesis, History and Philosophy of logic, 13, 179-201.

_3_ Payne, J. (2013). Abstraction relations need not be reflexive, Thought, 2, 137-147.

_4_ Uzquiano, G. (forthcoming). Impredicativity and Paradox.

Chair: Fabian Heimann

Time: 17:40-18:10, 18 September 2019 (Wednesday)

Location: SR 1.007

Ludovica Conti

(University of Pavia, Italia)

My name is Ludovica Conti. At present, I am a PhD candidate of Northwestern Philosophy Consortium - FINO (University of Pavia, Italy).

My current PhD project concerns the explanations and solutions of Russell's Paradox in Frege's philosophy of mathematics and in the abstractionist programs.

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Last update: 2014-04-01.