A Model of Connexive Set Theory

Abstract In this paper, we explore the possibility of constructing algebra-valued models for connexive set theories. In particular, we build an algebra-valued model on top of the four-element lattice $$\mathbb{M}\mathbb{C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> which semantically captures Wansing’s logic of material connexvity (MC). We show that the resulting model validates an axiom system that is classically equivalent to $$\textsf{ZF}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ZF</mml:mi> </mml:math> and that its underlying logic is a connexive logic. For this purpose, we tweak our semantic interpretation of set-membership and identity resulting in a model with a classical notion of identity and a non-classical notion of set-membership. Finally, in the conclusion, we discuss the role of our model within the ongoing debate between logical pluralism and monism.