On Characterizing Elementary Logic

Abstract

This is an expository paper. My aim is to explain in a way accessible to the non-specialist a number of results that amount to characterizations of elementary logic (EL), i.e. first order predicate logic with identity. The characterizations that I have in mind can be described as follows: First I define the very general concept abstract logic (AL). Then I define an inclusion relation ⊆ between ALs. Intuitively L⊆L′ means that everything that can be expressed in L can also be expressed in L′ although perhaps in a different way. For example, in EL it can be said of a binary relation that it is transitive. Thus if EL⊆L, this can also be expressed in L. (I assume, of course, that EL is an AL) L and L′ are equivalent, L≡L’ if L⊆L′ and L′⊆L. The next step consists in choosing a number of properties P ₁,…,P _n of ALs such that EL has P ₁,…, P _n . An example of an interesting property of this type is the following which may be called the Lowenheim property, since it means that the original Löwenheim theorem holds for L: If a sentence of L has a model, then it has a countable (finite or denumerable) model. Finally, a characterization of EL is a (true) statement of the form: If L is an AL, EL⊆L, and L has P ₁,…, P _n , then L≡EL. Since all reasonable ALs contain EL, this may also be expressed by saying that EL is the strongest AL having P ₁,…, P _n . Of course, one can also speak of characterizations of logics other than EL and there are, in fact, results of this type but they will not be discussed here.