15 Binary Logics, Orthologics and their Relations to Normal Modal Logics Yutaka Miyazaki

abstract. We study the relation between the class Ext(O) of orthologics and the class Next(KTB) of normal modal logics over KTB by means of embeddings of logics. First we introduce binary logics, which are generalizations of orthologics, as logics that are embeddable into some normal modal logics. We investigate the embeddability relation between the class of binary logics and the class of all normal modal logics, and establish some preservation results. Then we analyze the relation between Ext(O) and Next(KTB) and show that there exists a continuum of normal modal logics over KTB, and that any tabular orthologic is embeddable into infinitely many normal modal logics over KTB. 1 Binary logics as generalizations of orthologics In 1974, R.I.Goldblatt showed that the orthologic O can be embedded into the normal modal logic KTB by a translation of formulas [2]. Our aim is to investigate the structure of the lattice of orthologics and that of normal modal logics by means of embeddability relations between these classes of logics. The "basic " orthologic O can be thought of as the logic of ortholattices. The term "ortholattice " is an abbreviation of ortho-complemented lattice and a prime example of an ortholattice is the lattice of all subspaces of a finite dimensional vector space with orthogonal complement. The most characteristic feature of ortholattices is that they do not in general have the distributive law. Because of this feature, we face some serious difficulties in studying ortholattices and orthologics. One of such difficulties in syntactical analysis of orthologics is the lack of a good implication connective [6]. Although the success of Kripke semantics brought about remarkable developments in the research of modal logics, we do not know as much about normal modal logics containing the axiom B (q oe 23q) as about intermediate logics and modal logics containing S4. This is mainly because symmetric Kripke frames are less tractable than transitive ones. A completely new ide