On Blass translation for Leśniewski's propositional ontology and modal logics (Q2115291)

The paper continues a project -- see, e.g. [\textit{T. Inoué}, Bull. Sect. Log., Univ. Łódź, Dep. Log. 24, No. 4, 223--233 (1995; Zbl 0842.03012)] -- aiming at sharpening our understanding of Leśniewski's ontology using modal logic. The target is the study ``A faithful modal interpretation of propositional ontology'', by \textit{A. Blass} [Math. Japon. 40, No. 2, 217--223 (1994; Zbl 0809.03014)]. In this paper a faithful interpretation of Leśniewski's epsilon in modal logic K is given. According to that interpretation the construct ``\(\epsilon ab\)'' (reading: ``\(a\) is \(b\)'') is analysed as ``\(p_a \wedge \Box (p_a\supset p_b) \wedge ( p_b \supset \Box (p_b\supset p_a)\)'', where \(p_a\) and \(p_b\) are propositional variables corresponding to the name variables \(a\) and \(b\), respectively. The paper under review gives an alternative proof of the faithfulness of the proposed interpretation. Given a non-theorem \(\phi\) in Leśniewski's system, one needs to define a suitable Kripke model falsifying the modal translation of \(\phi\). This is achieved by means of the notion of a Hintikka formula [\textit{T. Inoué}, Bull. Sect. Log., Univ. Łódź, Dep. Log. 24, No. 2, 105--114 (1995; Zbl 0829.03005)]. After having proved the faithfulness of the interpretation, the paper extends the faithfulness result to a number of well-known modal logics, including von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logics, and the modal logic of provability GL. The reader may wish to compare with alternative embeddings, like those into first-order logic with equality [\textit{A. Ishimoto}, Stud. Log. 36, 285--299 (1978; Zbl 0391.03004)].