Concrete functions in logic

Question

In logic in CS by huth and ryan, I have seen the word **concrete functions /elements ** while a model is being explained. I cannot get what the difference between a normal function and concrete one is. Can some one elaborate on it?

Answer 1

It is a way (not very useful, IMO) to make a clear distinction between the symbols of the language, i.e. the set $\mathcal F$ of function symbols and the set $\mathcal P$ of predicate symbols and the "objects" of the domain of the model $\mathcal M$ :

"concrete objects" can be mathematical objects, like numbers and "concrete functions" can be "ordinary" mathematical functions, like sum).

See page 125 :

The distinction between $f$ and $f^{\mathcal M}$ and between $P$ and $P^{\mathcal M}$ is most important.

The symbols $f$ and $P$ are just that: symbols, whereas $f^{\mathcal M}$ and $P^{\mathcal M}$ denote a concrete function (or element) and relation in a model $\mathcal M$, respectively.

This means that $f^{\mathcal M}$ and $P^{\mathcal M}$ are not part of the formal language of predicate logic but are part of the meta-language we are using to define the formal system itself and its semantical interpretation.