- Functional predicate
In

formal logic and related branches ofmathematics , a**functional predicate**, or**function symbol**, is a logical symbol that may be applied to an object term to produce another object term.Functional predicates are also sometimes called "mappings", but that term has other meanings as well.In a model, a function symbol will be modelled by a function.Specifically, the symbol "F" in a

formal language is a functional symbol if,given any symbol "X" representing an object in the language, "F"("X") is again a symbol representing an object in that language.Intyped logic , "F" is a functional symbol with "domain" type**T**and "codomain" type**U**if, given any symbol "X" representing an object of type**T**, "F"("X") is a symbol representing an object of type**U**.One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply aconstant symbol.Now consider a model of the formal language, with the types

**T**and**U**modelled by sets [**T**] and [**U**] and each symbol "X" of type**T**modelled by an element ["X"] in [**T**] .Then "F" can be modelled by the set:$[F]\; :=ig\{(\; [X]\; ,\; [F(X)]\; ):\; [X]\; in\; [mathbf\{T\}]\; ig\},$which is simply a function with domain [**T**] and codomain [**U**] .It is a requirement of a consistent model that ["F"("X")] = ["F"("Y")] whenever ["X"] = ["Y"] .**Introducing new function symbols**In a treatment of predicate logic that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols.Introducing new function symbols from old function symbols is easy; given function symbols "F" and "G", there is a new function symbol "F" o "G", the "composition" of "F" and "G", satisfying ("F" o "G")("X") = "F"("G"("X")),

for all "X".Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of "F" matches the codomain type of "G", so this is required for the composition to be defined.One also gets certain function symbols automatically.In untyped logic, there is an "identity predicate" id that satisfies id("X") = "X" for all "X".In typed logic, given any type

**T**, there is an identity predicate id_{T}with domain and codomain type**T**; it satisfies id_{T}("X") = "X" for all "X" of type**T**.Similarly, if**T**is asubtype of**U**, then there is an inclusion predicate of domain type**T**and codomain type**U**that satisfies the same equation; there are additional function symbols associated with other ways of constructing new types out of old ones.Additionally, one can define functional predicates after proving an appropriate

theorem .(If you're working in aformal system that doesn't allow you to introduce new symbols after proving theorems, then you will have to use relation symbols to get around this, as in the next section.)Specifically, if you can prove that for every "X" (or every "X" of a certain type),there exists aunique "Y" satisfying some condition "P", then you can introduce a function symbol "F" to indicate this.Note that "P" will itself be a relational predicate involving both "X" and "Y".So if there is such a predicate "P" and a theorem:: For all "X" of type**T**, for some unique "Y" of type**U**, "P"("X","Y"),then you can introduce a function symbol "F" of domain type**T**and codomain type**U**that satisfies:: For all "X" of type**T**, for all "Y" of type**U**, "P"("X","Y")if and only if "Y" = "F"("X").**Doing without functional predicates**Many treatments of predicate logic don't allow functional predicates, only relational predicates.This is useful, for example, in the context of proving

metalogic al theorems (such asGödel's incompleteness theorem s), where one doesn't want to allow the introduction of new functional symbols (nor any other new symbols, for that matter).But there is a method of replacing functional symbols with relational symbols wherever the former may occur; furthermore, this is algorithmic and thus suitable for applying most metalogical theorems to the result.Specifically, if "F" has domain type

**T**and codomain type**U**, then it can be replaced with a predicate "P" of type (**T**,**U**).Intuitively, "P"("X","Y") means "F"("X") = "Y".Then whenever "F"("X") would appear in a statement, you can replace it with a new symbol "Y" of type**U**and include another statement "P"("X","Y").To be able to make the same deductions, you need an additional proposition::For all "X" of type**T**, for someunique "Y" of type**U**, "P"("X","Y").(Of course, this is the same proposition that had to be proved as a theorem before introducing a new function symbol in the previous section.)Because the elimination of functional predicates is both convenient for some purposes and possible, many treatments of formal logic do not deal explicitly with function symbols but instead use only relation symbols; another way to think of this is that a functional predicate is a "special kind of" predicate, specifically one that satisfies the proposition above.This may seem to be a problem if you wish to specify a proposition schema that applies only to functional predicates "F"; how do you know ahead of time whether it satisfies that condition?To get an equivalent formulation of the schema, first replace anything of the form "F"("X") with a new variable "Y".Then

universally quantify over each "Y" immediately after the corresponding "X" is introduced (that is, after "X" is quantified over, or at the beginning of the statement if "X" is free), and guard the quantification with "P"("X","Y").Finally, make the entire statement a material consequence of the uniqueness condition for a functional predicate above.Let us take as an example the

axiom schema of replacement inZermelo-Fraenkel set theory .(This example usesmathematical symbols .)This schema states (in one form), for any functional predicate "F" in one variable::$forall\; A,\; exists\; B,\; forall\; C,\; C\; in\; A\; ightarrow\; F(C)in\; B.$ First, we must replace "F"("C") with some other variable "D": :$forall\; A,\; exists\; B,\; forall\; C,\; C\; in\; A\; ightarrow\; D\; in\; B.$ Of course, this statement isn't correct; "D" must be quantified over just after "C": :$forall\; A,\; exists\; B,\; forall\; C,\; forall\; D,\; C\; in\; A\; ightarrow\; Din\; B.$ We still must introduce "P" to guard this quantification: :$forall\; A,\; exists\; B,\; forall\; C,\; forall\; D,\; P(C,D)\; ightarrow\; (C\; in\; A\; ightarrow\; D\; in\; B).$ This is almost correct, but it applies to too many predicates; what we actually want is: :$(forall\; X,\; exists\; !\; Y,\; P(X,Y))\; ightarrow\; (forall\; A,\; exists\; B,\; forall\; C,\; forall\; D,\; P(C,D)\; ightarrow\; (C\; in\; A\; ightarrow\; D\; in\; B)).$ This version of the axiom schema of replacement is now suitable for use in a formal language that doesn't allow the introduction of new function symbols. Alternatively, one may interpret the original statement as a statement in such a formal language; it was merely an abbreviation for the statement produced at the end.**See also***

Logical connective

*Logical constant

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Functional Grammar**— (FG) ist eine linguistische Theorie, die Ende der 1970er Jahre von Simon Cornelis Dik in Amsterdam entwickelt wurde, ausdrücklich als Gegenmodell zum Standard Modell der Transformationsgrammatik von Noam Chomsky. Nach dem Tod Diks 1995 wurde die… … Deutsch Wikipedia**Functional**— Func tion*al (f[u^][ng]k sh[u^]n*al), a. 1. Pertaining to, or connected with, a function or duty; official. [1913 Webster] 2. (Pathology, Physiol.) Pertaining to the function of an organ or part, or to the functions in general; involving or… … The Collaborative International Dictionary of English**Functional disease**— Functional Func tion*al (f[u^][ng]k sh[u^]n*al), a. 1. Pertaining to, or connected with, a function or duty; official. [1913 Webster] 2. (Pathology, Physiol.) Pertaining to the function of an organ or part, or to the functions in general;… … The Collaborative International Dictionary of English**predicate calculus**— Logic. See functional calculus. Also called predicate logic. [1945 50] * * * Part of modern symbolic logic which systematically exhibits the logical relations between propositions involving quantifiers such as all and some. The predicate calculus … Universalium**functional calculus**— noun a system of symbolic logic that represents individuals and predicates and quantification over individuals (as well as the relations between propositions) • Syn: ↑predicate calculus • Hypernyms: ↑symbolic logic, ↑mathematical logic, ↑formal… … Useful english dictionary**functional calculus**— the branch of symbolic logic that includes the sentential calculus and that deals with sentential functions and quantifiers and with logical relations between sentences containing quantifiers. Also called predicate calculus, predicate logic.… … Universalium**functional calculus**— noun Date: 1933 predicate calculus … New Collegiate Dictionary**predicate calculus**— noun Date: 1950 the branch of symbolic logic that uses symbols for quantifiers and for arguments and predicates of propositions as well as for unanalyzed propositions and logical connectives called also functional calculus compare propositional… … New Collegiate Dictionary**functional calculus**— func′tional cal′culus n. math. pho the branch of symbolic logic that includes the sentential calculus and that deals with sentential functions and quantifiers and with logical relations between sentences containing quantifiers Also called… … From formal English to slang**predicate calculus**— pred′icate cal′culus n. pho functional calculus • Etymology: 1945–50 … From formal English to slang