Combination of Logics and Combination of Theories

Vladimir Vasyukov

Institute of Philosophy, Russian Academy of Sciences,

Russia

COMBINATION OF LOGICS AND COMBINATION OF THEORIES

Many researches in the field of combining logics has been directed at fibring, a more general combination mechanism proposed by D.Gabbay. Fibring can be applied beyond the universe of modal systems while adopting a basic universe of logic systems. And here it turned out, that the interesting general question seems to be the nature and the structure of the general universe of possible combinations of logical systems. The conception of Universal Logic allows to put forward the hypothesis concerning the structure of the universe of logical systems. A Universal Logic is a general theory of logics considered as the kind of mathematical structures by analogy with the treatment of algebras from the point of view of Universal Algebra.

Adopting the point of view of Universal Logic one can easy arrive at conclusion that a category-theoretical perspective, where logical systems will constitute main category Log of our interest, apparently provides us the background for the investigation of the global universe of Universal Logic. As is well known for any logical system there is the complete lattice of theories under the inclusion ordering and this lattice, in turn, allows us to define the very helpful notion of a theory space. Theory spaces and their morphisms constitute the category Tsp, with the usual identity and composition of functions, which can be connected with Log by means of a special functor Th. This functor gives us an opportunity to define a notion of equipollence of logical systems being very important for resolving the problem of identity of the different formulations of one and the same logical system.

The perspective above prompt us the following kinds of combinations of logical systems:

labeling or cofibring;

unconstraint possible translating;

constraint possible translating.

Labelings gives us products in Log, unconstraint possible translatings gives coexponentials (dual to the usual exponentials in Cartesian closed categories) in Log and constraint possible translatings gives exponentials in Log. But we need to take into account that in Log all this constructions works just up to the equivalence which is based exactly on the equipollence.

The last fact means that if we transfer categorical constructions from Log to Tsp then all equipollences will transform into usual categorical isomorphisms. As a consequence, we obtain in Tsp coproducts, products, coexponentials and exponentials with the respective diagrams. All of them will be the complete lattices according to the properties of Th.

It was shown that Log is a topos and a complement topos, i.e. a Cartesian coclosed category with a complement classifier. Since we can introduce with a help of Th a terminal object of Tsp, an initial object, a subobject classifier and a complement classifier then as a result we obtain that Tsp will be also a topos and a complement topos. Thus the general universe of possible combinations of logical systems turns out to be more than enough structured formation.