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Conditional Reasoning and the Shadows It Casts Onto the First-order Logic: the Nelsonian Case

Year: 2025
Type: article
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 CK $\mathsf {CK}$ sans serif upper C upper K introduced by Brian Chellas in [3]; however, the main result of the article 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 N 4 CK $\mathsf {N4CK}$ sans serif upper N Baseline sans-serif 4 sans serif upper C upper K , introduced in [11] into QN 4 $\mathsf {QN4}$ sans serif upper Q upper N sans serif 4 , the paraconsistent variant of Nelson’s first-order logic of strong negation. Thus N 4 CK $\mathsf {N4CK}$ sans serif upper N Baseline sans-serif 4 sans serif upper C upper K is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of CK $\mathsf {CK}$ sans serif upper C upper K on the non-classical basis provided by the propositional fragment of QN 4 $\mathsf {QN4}$ sans serif upper Q upper N sans serif 4 . Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which makes it possible to improve an older result [10, Proposition 7] by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic FSK d $\mathsf {FSK}^d$ sans serif upper F upper S upper K Superscript d
DOI: 10.1017/bsl.2025.10136
Language: English
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W4416933979
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