Conditional reasoning and the shadows it casts onto the first-order logic: the Nelsonian case

TL;DR

This paper develops a translation of conditional logic formulas into first-order logic, demonstrating that Nelsonian conditional logic can be faithfully embedded into a paraconsistent first-order logic, extending classical modal logic results.

Contribution

It introduces a standard translation for conditional logic into first-order logic and shows that Nelsonian conditional logic can be embedded into a paraconsistent logic, bridging classical and non-classical logics.

Findings

Faithful embedding of $ extsf{N4CK}$ into $ extsf{QN4}$.

Extension of translation methods to modal logic.

Improvement of existing modal logic embedding results.

Abstract

We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf{CK}$ introduced by Brian Chellas; however, the main result of the paper is that a classically equivalent reformulation of these notions (i.e. of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf{N4CK}$, introduced in arXiv:2311.02361 into $\mathsf{QN4}$, the paraconsistent variant of Nelson's first-order logic of strong negation. Thus $\mathsf{N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf{CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf{QN4}$. Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which allows to improve an older result by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf{FSK}^d$ into $\mathsf{QN4}$.