Combining Algebraizable Logics

The general methodology of “algebraizing” logics (cf. [2], [4]) is used here for combining different logics. The combination of logics is represented as taking the colimit of the constituent logics in the category of algebraizable logics. The cocompleteness of this category as well as its isomorphism to the corresponding category of certain first-order theories are proved. In this paper we translate the “combining logics” problem to the problem of “combining” certain theories of usual first order logic. We prove that the category of a special class of logics, called algebraizable logical systems (see Def. 1.1 below), is isomorphic to the category of the corresponding first order theories. We also show that these categories are cocomplete. Some directions in which the approach chosen can perhaps be generalized are pointed out in the last section.