Speaking about Transitive Frames in Propositional Languages

This paper is a comparative study of the propositional intuitionistic (non-modal) and classical modal languages interpreted in the standard way on transitive frames. It shows that, when talking about these frames rather than conventional quasi-orders, the intuitionistic language displays some unusual features: its expressive power becomes weaker than that of the modal language, the induced consequence relation does not have a deduction theorem and is not protoalgebraic. Nevertheless, the paper develops a manageable model theory for this consequence and its extensions which also reveals some unexpected phenomena. The balance between the intuitionistic and modal languages is restored by adding to the former one more implication. 1 Both modal and intuitionistic propositional languages may be regarded as talking about quasi-orders F = hW;Ri, R a re exive and transitive binary relation on a set W . The modal languageML with the primitive connectives, say, ^, _, !, ? and 2 is interpreted in F by means of valuations V of propositional variables in the power-set 2 of W and the truth-relation j= in the following way: x j= p i x 2 V(p); p a variable; (1) x 6j= ?; (2) x j= ' ^ i x j= ' and x j= ; (3) x j= ' _ i x j= ' or x j= ; (4) x j= '! i x j= ' implies x j= ; (5) x j= 2' i 8y 2 W (xRy ) y j= '): (6) School of Information Science, JAIST, Tatsunokuchi, Ishikawa 923{12, Japan; e-mail: [email protected]. Institut f ur Informatik, Universitat Leipzig, Augustus-Platz 10-11, 04109 Leipzig, Germany; e-mail: [email protected]. zFachbereich Mathematik, FU Berlin, Arnimallee 3, 14195 Berlin, Germany; e-mail: [email protected].