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Completeness and decidability results for a logic of contrary-to-duty conditionals. (English) Zbl 1272.03096
Ten years after the publication of their chapter [“Deontic logic and contrary-to duties”, in: Handbook of philosophical logic. Vol. 8. 2nd ed. Dordrecht: Kluwer Academic Publishers (2002; Zbl 1055.03002), pp. 265–343], the authors revisit the scene with two main goals. The first is to elaborate further on the intuitions that lay behind the formal constructions of that chapter, with particular attention to the distinction between ‘actual’ and ‘primary’ (formerly ‘ideal’) obligations. These are associated with distinct kinds of possibility, ‘actual’ and ‘potential’, applicable to the actions of agents, represented by different diamond operators in the formal language. The second goal is to deepen formal work begun in [loc. cit.]. Using a maximality construction, the authors establish soundness and completeness for an axiom system in terms of a possible worlds semantics, both differing a little from their 2002 counterparts. Decidability is also established via the finite model property. In a footnote, it is noted that related theorems were established independently by D. Gabbay and K. Schlechta [“Critical analysis of the Carmo-Jones system of contrary-to-duty obligations”, Preprint, arXiv:1002.3021].

03B45 Modal logic (including the logic of norms)
03B25 Decidability of theories and sets of sentences
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