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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).
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.
Empty categories exist in contrast to overt categories which are pronounced. [1] When representing empty categories in tree structures, linguists use a null symbol (∅) to depict the idea that there is a mental category at the level being represented, even if the word(s) are being left out of overt speech.
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:
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 ...
There are settings, such as inclusive logic, where empty domains are permitted. Moreover, if a class of algebraic structures includes an empty structure (for example, there is an empty poset), that class can only be an elementary class in first-order logic if empty domains are permitted or the empty structure is removed from the class.
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 proof is a refutation if the last clause is the empty clause. In SLD, all of the clauses in the sequence are goal clauses, and the other parent is an input clause. In SL resolution, the other parent is either an input clause or an ancestor clause earlier in the sequence.