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In logic, a clause is a propositional formula formed from a finite collection of literals (atoms or their negations) and logical connectives.A clause is true either whenever at least one of the literals that form it is true (a disjunctive clause, the most common use of the term), or when all of the literals that form it are true (a conjunctive clause, a less common use of the term).
In modern logic only the contradictories in the square of opposition apply, because domains may be empty. (Black areas are empty, red areas are nonempty.) In first-order logic, the empty domain is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid.
Otherwise, when the formula contains an empty clause, the clause is vacuously false because a disjunction requires at least one member that is true for the overall set to be true. In this case, the existence of such a clause implies that the formula (evaluated as a conjunction of all clauses) cannot evaluate to true and must be unsatisfiable.
The resulting inference rule is refutation-complete, [6] in that a set of clauses is unsatisfiable if and only if there exists a derivation of the empty clause using only resolution, enhanced by factoring. An example for an unsatisfiable clause set for which factoring is needed to derive the empty clause is:
The incidence graph (clause-variable graph) is planar. At least one literal must be true. NP-c LSAT Linear SAT Each clause contains 3 literals, intersects at most one other clause, and the intersection is exactly one literal. At least one literal must be true. NP-c HORN-SAT Horn satisfiability Horn clauses (at most one positive literal).
A Horn clause with exactly one positive literal is a definite clause or a strict Horn clause; [2] a definite clause with no negative literals is a unit clause, [3] and a unit clause without variables is a fact; [4] A Horn clause without a positive literal is a goal clause. The empty clause, consisting of no literals (which is equivalent to ...
bottom, falsity, contradiction, falsum, empty clause propositional logic, Boolean algebra, first-order logic: denotes a proposition that is always false. The symbol ⊥ may also refer to perpendicular lines.
Because the logical or means a disjunction formula is true when either one or both of its parts are true, it is referred to as an inclusive disjunction. This is in contrast with an exclusive disjunction, which is true when one or the other of the arguments are true, but not both (referred to as exclusive or, or XOR).