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.
Read more about this topic: Consistency
Famous quotes containing the word logic:
“We want in every man a long logic; we cannot pardon the absence of it, but it must not be spoken. Logic is the procession or proportionate unfolding of the intuition; but its virtue is as silent method; the moment it would appear as propositions and have a separate value, it is worthless.”
—Ralph Waldo Emerson (1803–1882)