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Predicate logic
 
 
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In mathematical logic
Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
, predicate logic is the generic term for symbolic formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
s like first-order logic
First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
, second-order logic
Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
, many-sorted logic
Many-sorted logic

Many-sorted logic can reflect formally our intention, not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming....
 or infinitary logic
Infinitary logic

An infinitary logic is a logic that allows infinitely long Proposition and/or infinitely long Mathematical proof. Some infinitary logics may have different properties from those of standard first-order logic....
. This formal system is distinguished from other systems in that its formulas contain variable
Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
s which can be quantified
Quantification

Quantification has two distinct meanings. In mathematics and empirical science, it refers to human acts, known as counting and measuring that map human sense observations and experiences into element s of some Set of numbers....
. Two common quantifiers are the existential
Existential quantification

In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. In laymen's terms, it simply refers to something....
 ∃ and universal
Universal quantification

In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing.The resulting statement is a universally quantified statement, and we have universally quantified over the predicate....
 ∀ quantifiers. The variables could be elements in the universe
Domain of discourse

The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in deductive logic, especially predicate logic....
, or perhaps relations or functions over the universe. For instance, an existential quantifier over a function symbol would be interpreted as modifier "there is a function".

In informal usage, the term "predicate logic" occasionally refers to first-order logic
First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
.






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In mathematical logic
Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
, predicate logic is the generic term for symbolic formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
s like first-order logic
First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
, second-order logic
Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
, many-sorted logic
Many-sorted logic

Many-sorted logic can reflect formally our intention, not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming....
 or infinitary logic
Infinitary logic

An infinitary logic is a logic that allows infinitely long Proposition and/or infinitely long Mathematical proof. Some infinitary logics may have different properties from those of standard first-order logic....
. This formal system is distinguished from other systems in that its formulas contain variable
Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
s which can be quantified
Quantification

Quantification has two distinct meanings. In mathematics and empirical science, it refers to human acts, known as counting and measuring that map human sense observations and experiences into element s of some Set of numbers....
. Two common quantifiers are the existential
Existential quantification

In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. In laymen's terms, it simply refers to something....
 ∃ and universal
Universal quantification

In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing.The resulting statement is a universally quantified statement, and we have universally quantified over the predicate....
 ∀ quantifiers. The variables could be elements in the universe
Domain of discourse

The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in deductive logic, especially predicate logic....
, or perhaps relations or functions over the universe. For instance, an existential quantifier over a function symbol would be interpreted as modifier "there is a function".

In informal usage, the term "predicate logic" occasionally refers to first-order logic
First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
. Some authors consider the predicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from an informal, more intuitive development.

Footnotes