Predicate (mathematical logic): Difference between revisions
→See also: Add a link to "Truth value" |
The word "constants" in formal logic, refers to 0-ary function symbols, whereas this explanation probably had in mind some "single element of the domain". So it was misleading, and it's more accurate to say that first-order logic predicates are characterized by having a fixed arity, each coordinate being substituted by elements of the domain, in the usual semantic scheme. |
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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". |
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". |
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In the [[semantics of logic]], predicates are interpreted as [[relation (mathematics)|relation]]s. For instance, in a standard semantics for first-order logic, the formula <math>R(a,b)</math> would be true on an [[interpretation (logic)|interpretation]] if the entities denoted by <math>a</math> and <math>b</math> stand in the relation denoted by <math>R</math>. Since predicates are [[non-logical symbol]]s, they can denote different relations depending on the interpretation given to them. While [[first-order logic]] only includes predicates that apply to |
In the [[semantics of logic]], predicates are interpreted as [[relation (mathematics)|relation]]s. For instance, in a standard semantics for first-order logic, the formula <math>R(a,b)</math> would be true on an [[interpretation (logic)|interpretation]] if the entities denoted by <math>a</math> and <math>b</math> stand in the relation denoted by <math>R</math>. Since predicates are [[non-logical symbol]]s, they can denote different relations depending on the interpretation given to them. While [[first-order logic]] only includes predicates that apply to a fixed number of objects, other logics may allow predicates that apply to other predicates. |
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== Predicates in different systems == |
== Predicates in different systems == |
Revision as of 17:42, 15 May 2024
In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula , 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 a fixed number of objects, other logics may allow predicates that apply to 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
- Classifying topos
- Free variables and bound variables
- Multigrade predicate
- Opaque predicate
- Predicate functor logic
- Predicate variable
- Truthbearer
- Truth value
- Well-formed formula
References
- ^ Lavrov, Igor Andreevich; Maksimova, Larisa (2003). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. New York: Springer. p. 52. ISBN 0306477122.