Literal (mathematical logic)


In mathematical logic, a literal is an atomic formula or its negation.
The definition mostly appears in proof theory, e.g. in conjunctive normal form and the method of resolution.
Literals can be divided into two types:
The polarity of a literal is positive or negative depending on whether it is a positive or negative literal.
For a literal, the complementary literal is a literal corresponding to the negation of,
we can write to denote the complementary literal of. More precisely, if then is and if then is.
In the context of a formula in the conjunctive normal form, a literal is pure if the literal's complement does not appear in the formula.
In Boolean functions, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal. For example, if, and are variables then the expression contains three literals and the expression contains four literals. However, the expression would also be said to contain four literals, because although two of the literals are identical these qualify as two separate occurrences.

Examples

In propositional calculus a literal is simply a propositional variable or its negation.
In predicate calculus a literal is an atomic formula or its negation, where an atomic formula is a predicate symbol applied to some terms, with the terms recursively defined starting from constant symbols, variable symbols, and function symbols. For example, is a negative literal with the constant symbol 2, the variable symbols x, y, the function symbols f, g, and the predicate symbol Q.