Consistency - First-order Logic

First-order Logic

A set of formulas in first-order logic is consistent (written Con) if and only if there is no formula such that and . Otherwise is inconsistent and is written Inc.

is said to be simply consistent if and only if for no formula of, both and the negation of are theorems of .

is said to be absolutely consistent or Post consistent if and only if at least one formula of is not a theorem of .

is said to be maximally consistent if and only if for every formula, if Con then .

is said to contain witnesses if and only if for every formula of the form there exists a term such that . See First-order logic.