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On non-self-referential fragments of modal logics. (English) Zbl 1422.03037
Summary: Justification logics serve as “explicit” modal logics in a way that, formula \(\phi\) is a modal theorem if and only if there is a justification theorem, called a realization of \(\phi\), gained by replacing modality occurrences in \(\phi\) by (justification) terms with structures explicitly explaining their evidential contents. In justification logics, terms stand for justifications of (propositions expressed by) formulas, and as a kind of atomic terms, constants stand for that of (justification) axioms. R. Kuznets [Lect. Notes Comput. Sci. 5010, 228–239 (2008; Zbl 1143.03010); Theory Comput. Syst. 46, No. 4, 636–661 (2010; Zbl 1216.03037)] has shown that in order to realize (i.e., offer a realization of) some modal theorems, it is necessary to employ a self-referential constant, that is, a constant that stands for a justification of an axiom containing an occurrence of the constant itself. Based on existing works, including some of the author’s, this paper treats the collection of modal theorems that are non-self-referentially realizable as a fragment (called non-self-referential fragment) of the modal logic, and verifies: (1) that fragment is not closed in general under modus ponens; and (2) that fragment is not “conservative” in general when going from a smaller modal logic to a larger one.
MSC:
03B45 Modal logic (including the logic of norms)
03B42 Logics of knowledge and belief (including belief change)
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