Consistency and Decidability in Some Paraconsistent Arithmetics

Abstract


 
 
 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. 
 
 

True, Untrue, Valid, Invalid, Provable, Unprovable

“…There are paraconsistent theories of arithmetic that would seem to do most of what one wants here (see. e.g., Weber, 2021, ch.6), but there are also many other better studied paraconsistent arithmetics with inconsistent models that leave the required properties underdetermined (Ferguson, 2019;Tedder, 2021), cf. (Weber, 2022, §3).…”

Paraconsistency in Mathematics

“…Then a sound and complete axiomatization of paraconsistent arithmetic can be given in this logic; cf. Tedder (2021). Another, more radical, direction is to move away from the idea of "generalizing" classical logic and considering instead contra-classical logics -in particular, connexive logics.…”