Paola Fontana
Università della Svizzera italiana
Some have proposed (Weber, 2010; Priest, 2006) to develop naïve set theory in a paraconsistent logic. The aim of this work is to assess whether the paraconsistent strategies are suitable for this purpose and to analyse the motivations behind them.
(1) First, I will reconstruct Priest’s (1994) argument for the Principle of Uniform Solution: all the paradoxes share the same structure; hence they require the same solution. Standard solutions are inadequate because they address paradoxes in different ways. This is a motivation for the adoption of paraconsistent logic. Another one is that of capturing our intuitive notions, such as the naïve conception of sets.
(2) I will then present three kinds of strategy for developing a paraconsistent set theory. The material strategy takes the conditional in Extensionality and Comprehension to be the material conditional, and rejects the disjunctive syllogism. The relevant strategy takes the conditional to be a relevant one. The model-theoretic strategy aims to show that there are models of the naïve set theory with Priest’s Logic of paradox, that are also models of ZF. The point is to have a model that countenances ZF axioms and to use classical logic to derive its theorems (Incurvati, 2020).
(3) We might now ask whether these strategies really have the advantages they promise to bring: (a) they are supposed to satisfy the Principle of Uniform Solution and (b) to capture our intuitions. However, if we work in set theory, we have different purposes with respect to semantics: we might need different solutions to the paradoxes depending on our theoretical goals. As for (b), let us notice that disjunctive syllogism, modus ponens or rules such as Contraposition, which are arguably intuitive, fails in the strategies mentioned above.
It seems that paraconsistentists need to revise their motivations and explore other options to support their proposal.
References
Incurvati, Luca (2020). Conceptions of Set and the Foundations of Mathematics. Cambridge University Press.
Priest, G. (1994). The Structure of the Paradoxes of Self-Reference. Mind, 103(409), 25–34.
Priest, Graham (2006). In contradiction: a study of the transconsistent. New York: Oxford University Press.
Weber, Zach (2010). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic 3 (1):71-92.
Chair: Gabriel Levc
Time: September 8th, 14:40-15:10
Location: SR 1.006
SOPhiA 2026 