A Framework for Logics. Rigidity, Finitism and an Encyclopedia of Logics*

A logic is understood here as an algebraic structure L = (FL,SL, 2L) that consists of a set FL of formulas (sentences) plus a class SL of structures and a relation of satisfaction 2L between them. We have the usual metalogical notions like logical consequence and logical equivalence, defined in the usual way. A logic is rigid, if there exists a set Fat ⊆ FL so that the logic can be reduced to the propositional logic that is defined over the set Fat of propositional constants, in an obvious way. If this reduction is recursive and the basic set Fat is finite, we call the rigid logic finitistic. – Finitistic logics are an important subclass of the class of all rigid logics, because of its metalogical merits: every rigid logic is decidable, regarding both satisfaction and logical consequence. I however discuss mainly the general rigid case here. The main advantage of rigid languages is that we can describe them in terms of set theory. In other words: rigid languages are not a construction of pure logics or meta-mathematics, respectively, but they are a proper construction of mathematics: we can take every rigid logic as a mathematical structure and we can describe every property of this logic in terms of this mathematical structure. – A construction, similar to this, is the well-known Henkin-trick that reduces richer languages to first-order logic. But, whereas in the case of the Henkin-reduction we have a language that is not a part of the mathematical universe in itself and thus has the well-known properties of expressive power on the one hand and incompleteness (in the sense of: not being able to express everything in the realm of set theory) on the other, in the case of reduction to propositional logic we have a less powerful language which however has the serious advantage that it is part of the mathematical universe and thus it is complete in a pretty obvious sense. Of course, rigid logics are useless for the purpose of mathematical foundation (because we have to assume mathematics to be able to define them). But they are extremely useful for practically every application that is not intended for the definition of mathematical languages. Thus rigid languages should be a good choice for philosophical logics of any kind, because we can define them in a pretty straightforward and unifying way. My first example is the rigid first-order logic RIGp(D,P, α) which is built over a (possibly infinite) set D of individuals, a finite or countable set P of predicates and a function α : P 7→ N that assigns to each predicate its ‘arity’. A structure S