Second-order logic

In mathematical logic, second-order logic is an extension of either propositional logic or first-order logic which contains variables in predicate positions (rather than only in term positions, as in first-order logic), and quantifiers binding them. So:

Related Topics:
Mathematical logic - Propositional logic - First-order logic - Predicate - Term - Binding

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:orall F F(mathrm{jones}) lor eg F(mathrm{jones}),

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

which might express the principle of bivalence with respect to Jones: For every property, either Jones has it or he doesn't.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It permits of various interpretations; frequently it is thought of as containing quantification over subsets of a domain, or functions from the domain into itself, rather than only over individual members of the domain. Thus, for example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:orall x exists y x+y=0

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

but one needs second-order logic to assert the least-upper-bound property of the real numbers:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:orall Asubseteq R

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

and insert in place of the dots a statement that if A is nonempty and has an upper bound in R then A has a least upper bound in R.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

(table of mathematical symbols)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~