Predicate (mathematical logic) explained
, the symbol
is a predicate that applies to the
individual constant
. Similarly, in the formula
, the symbol
is a predicate that applies to the individual constants
and
.
According to Gottlob Frege, the meaning of a predicate is exactly a function from the domain of objects to the truth-values "true" and "false".
In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula
would be true on an
interpretation if the entities denoted by
and
stand in the relation denoted by
. Since predicates are
non-logical symbols, they can denote different relations depending on the interpretation given to them. While
first-order logic only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections of objects defined by other predicates.
Predicates in different systems
A predicate is a statement or mathematical assertion that contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables’ value or values.
- In propositional logic, atomic formulas are sometimes regarded as zero-place predicates.[1] In a sense, these are nullary (i.e. 0-arity) predicates.
- In first-order logic, a predicate forms an atomic formula when applied to an appropriate number of terms.
- In set theory with the law of excluded middle, predicates are understood to be characteristic functions or set indicator functions (i.e., functions from a set element to a truth value). Set-builder notation makes use of predicates to define sets.
- In autoepistemic logic, which rejects the law of excluded middle, predicates may be true, false, or simply unknown. In particular, a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.
- In fuzzy logic, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.
See also
External links
Notes and References
- Book: Lavrov. Igor Andreevich. Larisa. Maksimova. Larisa Maksimova . Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. 2003. Springer. New York. 0306477122. 52.
