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One-place Predicate
In mathematical logic, a first-order predicate is a predicate that takes only individual(s) constants or variables as argument(s).. Compare second-order predicate and higher-order predicate. This is not to be confused with a one-place predicate or monad, which is a predicate that takes only one argument. For example, the expression "is a planet" is a one-place predicate, while the expression "is father of" is a two-place predicate. See also *First-order predicate calculus *Monadic predicate calculus In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symb ... References Predicate logic Concepts in logic {{Logic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate (logic)
In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula P(a), the symbol P is a predicate that applies to the individual constant a. Similarly, in the formula R(a,b), the symbol R is a predicate that applies to the individual constants a and b. 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 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 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 defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-order Predicate
In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument. Compare higher-order predicate. The idea of second order predication was introduced by the German mathematician and philosopher Frege. It is based on his idea that a predicate such as "is a philosopher" designates a concept, rather than an object. Sometimes a concept can itself be the subject of a proposition, such as in "There are no Bosnian philosophers". In this case, we are not saying anything of any Bosnian philosophers, but of the concept "is a Bosnian philosopher" that it is not satisfied. Thus the predicate "is not satisfied" attributes something to the concept "is a Bosnian philosopher", and is thus a second-level predicate. This idea is the basis of Frege's theory of number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-order Predicate
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic. The term "higher-order logic" is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the ''theory of simple types'', also called the ''simple theory of types''. Leon Chwistek and Frank P. Ramsey proposed this as a simplification of ''ramified theory of types'' specified in the ''Principia Mathematica'' by Alfred North Whitehead and Bertrand Russell. ''Simple types'' is sometimes also meant to exclude polymorphic and dependent types. Quantification scope First-order logic quantifies only variables that range over individuals; ''second-order logic'', also quantifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Predicate Calculus
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monadic Predicate Calculus
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols. All atomic formulas are thus of the form P(x), where P is a relation symbol and x is a variable. Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments. Expressiveness The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. It is so weak that, unlike the full predicate calculus, it is decidable—there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains). Löwenheim, L. (1915) "Über Möglichkeiten im Relativkalkül," ''Mathematische Annalen'' 76: 447-470. Translated as "On possibilities in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
