Twist-Valued Models for Three-valued Paraconsistent Set Theory

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF)

Consistency and Decidability in Some Paraconsistent Arithmetics

The standard style of argument used to prove that a theory is unde- cidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent set- ting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent models seem likely to differ in terms of complexity. In this paper, I begin to explore this terrain, working, particularly, in incon- sistent theories of arithmetic couched in three-valued paraconsistent logics which have strong (i.e. detaching) conditionals

Consistency and Decidability in Some Paraconsistent Arithmetics

The standard style of argument used to prove that a theory is unde- cidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent set- ting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent models seem likely to differ in terms of complexity. In this paper, I begin to explore this terrain, working, particularly, in incon- sistent theories of arithmetic couched in three-valued paraconsistent logics which have strong (i.e. detaching) conditionals

Towards a Paraconsistent Quantum Set Theory

In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as developed by Isham, Butterfield and Doring, amongst others. Towards this end, we will study algebraic valued set-theoretic structures whose truth values correspond to the clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space. In particular, we will attempt to recreate, in these new structures, Takeuti's original isomorphism between the set of all Dedekind real numbers in a suitably constructed model of set theory and the set of all self adjoint operators on a chosen Hilbert space.Comment: In Proceedings QPL 2015, arXiv:1511.0118