logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula

P(a)

, the symbol

P

is a predicate which applies to the individual constant

a

. Similarly, in the formula

R(a,b)

R

is a predicate which applies to the individual constants

a

and

b

. In the

semantics of logic In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment. Overview The truth cond ...

, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula

R(a,b)

would be true on an interpretation if the entities denoted by

a

and

b

stand in the relation denoted by

R

. Since predicates are

non-logical symbol In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes al ...

s, they can denote different relations depending on the interpretation used to interpret them. While

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

only includes predicates which apply to individual constants, other logics may allow predicates which apply to other predicates.

Predicates in different systems

* In

propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...

s are sometimes regarded as zero-place predicates In a sense, these are nullary (i.e. 0- arity) predicates. * In

, a predicate forms an atomic formula when applied to an appropriate number of terms. * In

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

with

excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...

, predicates are understood to be characteristic functions or set indicator functions (i.e., functions from a set element to a

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pro ...

). Set-builder notation makes use of predicates to define sets. * In autoepistemic logic, which rejects the

law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...

, 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, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

References

External links

Introduction to predicates
{{Authority control Predicate logic Propositional calculus Basic concepts in set theory Fuzzy logic Mathematical logic

Predicates in different systems

See also

References

External links