The consistency of the axioms of abstraction and extensionality in a three-valued logic.

Abstract

The Abstraction Axiom I want to consider is the following one, which is based on the Lukasiewicz three-valued logic.(*) (Sy)(Ax)frεy+*φ(x, z u ..., z n ))

Extensionality and Restriction in Naive Set Theory

“…6 The principle of comprehension is a generalization of axiom 2.1: 4 Brady calls this the definition of identity, and reserves the name 'extensionality' for a substitution principle.…”

“…See [33](361) for discussion of expressing the extensionality axiom with a material biconditional. 6 Cantor goes on to relate his definition to the "Platonic ιδoς or ιδ α. "…”

Models for a paraconsistent set theory

“…By no means should this be interpreted as a defect of the paracomplete interpretation in comparison with the paraconsistent one, because the comprehension scheme as formulated here above does not express (in both cases actually) what it is intended to; in particular it does not fit in with our definition of comprehension given in Section 2.3, seeing that t∈ |ϕ ⊂ ⊃ ψ| M does not imply that |ϕ| M = |ψ| M . 9 Note that we are using the same notation for the connectives and their truth functions. 10 If need were, we would remind the reader that the truth degrees are subsets of {t, f}.…”

Paths to Triviality

“…1 John Slaney showed around the same time that there is a definite limit on how strong such a paraconsistent logic can be ( [51]); although the contraction axiom (A → (A → B)) → (A → B) is not derivable in 1 There were earlier attempts at showing that the naïve theories can be non-trivial, notably [8], [24], [31], [47], [48], [49] and [50]. However, these results either restrict abstraction or lack a decent conditional, one satisfying at least identity and modus ponens-A → A and A, A → B B-and so at best show that A & T A and A(a) & a ∈ {x|A} are intersubstitutable without delivering the biconditionals A ↔ T A and a ∈ {x|A} ↔ A( x /a).…”