The theory of truth tabular connectives, both truth functional and modal

Abstract

Briefly, the following things are done in this paper. An informal exposition of the significance of partial truth tables in modal logic is presented. The intuitive concept of a truth tabular sentence connective is examined and made precise. From among a vast assortment of truth tabular connectives, the semantically well-behaved ones (called regular connectives) are singled out for close investigation. The familiar concept of functional completeness is generalized to all sets of regular connectives, and the functional completeness (or incompleteness) of selected sets of connectives is established. Some modal analogues of Sheffer's stroke are presented, i.e., single connectives are introduced which serve to define not only the truth functional connectives but all the regular modal connectives as well. The notion of duality is extended to all regular connectives and a general duality theorem is proved. Lastly, simplified proofs are given of several metatheorems about the system S5.