Logical Theory and Semantic Analysis pp 129-146 | Cite as

# 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 *AL*s. 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 *AL*s 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 *AL*s 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.

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