Introducing New Function Symbols
In a treatment of predicate logic that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols. Introducing new function symbols from old function symbols is easy; given function symbols F and G, there is a new function symbol F o G, the composition of F and G, satisfying (F o G)(X) = F(G(X)), for all X. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of F matches the codomain type of G, so this is required for the composition to be defined.
One also gets certain function symbols automatically. In untyped logic, there is an identity predicate id that satisfies id(X) = X for all X. In typed logic, given any type T, there is an identity predicate idT with domain and codomain type T; it satisfies idT(X) = X for all X of type T. Similarly, if T is a subtype of U, then there is an inclusion predicate of domain type T and codomain type U that satisfies the same equation; there are additional function symbols associated with other ways of constructing new types out of old ones.
Additionally, one can define functional predicates after proving an appropriate theorem. (If you're working in a formal system that doesn't allow you to introduce new symbols after proving theorems, then you will have to use relation symbols to get around this, as in the next section.) Specifically, if you can prove that for every X (or every X of a certain type), there exists a unique Y satisfying some condition P, then you can introduce a function symbol F to indicate this. Note that P will itself be a relational predicate involving both X and Y. So if there is such a predicate P and a theorem:
- For all X of type T, for some unique Y of type U, P(X,Y),
then you can introduce a function symbol F of domain type T and codomain type U that satisfies:
- For all X of type T, for all Y of type U, P(X,Y) if and only if Y = F(X).
Read more about this topic: Functional Predicate
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