Algebra-valued models for LP-Set Theory

In this paper, we explore the possibility of constructing algebra-valued models of set theory based on Priest's Logic of Paradox. We show that we can build a non-classical model of ZFC which has as internal logic Priest's Logic of Paradox and validates Leibniz's law of indiscernibility of identicals. This is achieved by modifying the interpretation map for is an element of and = in our algebra-valued model. We end by comparing our model constructions to Priest's model-theoretic strategy and point out that we have a trade-off between a classical notion of identity and the validity of ZF and its theorems.