Universal Logic

Structuring the Universe of Universal Logics

Vladimir Vasyukov

Academy of Sciences - Moscow - Russia

The usual definition of universal logic as a general theory of logical structures means under a structure an object of the following type S = <$, i>, where $ is the domain of the structure (it is just the set) and i is the type of the structure which is the sequence of relations defined on the domain (it includes functions and not only relations between elements of the domain, but relations between parts of the domain, etc.). This gives us an opportunity to clarify what does it mean for a logic to be equivalent or translatable into another one. Moreover, the last notion enable us to collect the structures into the category whose arrows are the translations of structures. It turns out, that such category has its own structure allowing us to consider the products and the co-products of the logical structures as well as the co-exponentials of those.

The usual definition of universal logic as a general theory of logical structures means under a structure an object of the following type S = <$, i>, where $ is the domain of the structure (it is just the set) and i is the type of the structure which is the sequence of relations defined on the domain (it includes functions and not only relations between elements of the domain, but relations between parts of the domain, etc.). This gives us an opportunity to clarify what does it mean for a logic to be equivalent or translatable into another one. Moreover, the last notion enable us to collect the structures into the category whose arrows are the translations of structures. It turns out, that such category has its own structure allowing us to consider the products and the co-products of the logical structures as well as the co-exponentials of those.