Functional predicate: Difference between revisions

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{{Short description|Symbol representing a mathematical concept}}

In [[formal logic]] and related branches of [[mathematics]], a '''functional predicate''', or '''function symbol''', is a logical symbol that may be applied to an object term to produce another object term.

{{Unreferenced|date=December 2009}}

Functional predicates are also sometimes called ''mappings'', but that term has other meanings as well.

In [[formal logic]] and related branches of [[mathematics]], a '''functional predicate''',{{cn|reason=This seems to me a particularly unusual naming.|date=July 2025}} or '''function symbol''', is a logical symbol that may be applied to an object term to produce another object term.

In a [[model (logic)|model]], a function symbol will be modelled by a [[function]].

Functional predicates are also sometimes called '''mappings''', but that term has other[[Mapping (mathematics)|additional meanings asin wellmathematics]].

In a [[model (logic)|model]], a function symbol will be modelled by a [[function (mathematics)|function]].

Specifically, the symbol <i>''F</i>'' in a [[formal language]] is a functional symbol if, [[given any]] symbol <i>''X</i>'' representing an object in the language, <i>''F</i>''(<i>''X</i>'') is again a symbol representing an object in that language.

In [[typed logic]], <i>''F</i>'' is a functional symbol with ''___domain'' type <b>'''T</b>''' and ''codomain'' type <b>'''U</b>''' if, given any symbol <i>''X</i>'' representing an object of type <b>'''T</b>''', <i>''F</i>''(<i>''X</i>'') is a symbol representing an object of type <b>'''U</b>'''.

One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in [[0 (number)|zero]] variables is simply a [[Logical constant|constant]] symbol.

Now consider a model of the formal language, with the types <b>'''T</b>''' and <b>'''U</b>''' modelled by [[setSet (mathematics)|sets]]s [<b>'''T</b>'''] and [<b>'''U</b>'''] and each symbol <i>''X</i>'' of type <b>'''T</b>''' modelled by an element [<i>''X</i>''] in [<b>'''T</b>'''].

Then <i>''F</i>'' can be modelled by the set

: [<imath>[F</i>] := \big\{([<i>X</i>],[<i>F</i>(<i>X</i>)]) : [<i>X</i>] \in [<b>\mathbf{T}]\big\},</bmath>]},

which is simply a [[function (mathematics)|function]] with ___domain [<b>'''T</b>'''] and codomain [<b>'''U</b>'''].

It is a requirement of a consistent model that [<i>''F</i>''(<i>''X</i>'')] = [<i>''F</i>''(<i>''Y</i>'')] whenever [<i>''X</i>''] = [<i>''Y</i>''].

== 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. Given the function symbols ''F'' and ''G'', one can introduce a new function symbol ''F'' ∘ ''G'', the ''[[function composition|composition]]'' of ''F'' and ''G'', satisfying (''F'' ∘ ''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 <i>''F</i>'' matches the codomain type of <i>''G</i>'', so this is required for the composition to be defined.

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 <i>F</i> and <i>G</i>, there is a new function symbol <i>F</i> o <i>G</i>, the ''composition'' of <i>F</i> and <i>G</i>, satisfying (<i>F</i> o <i>G</i>)(<i>X</i>) = <i>F</i>(<i>G</i>(<i>X</i>)), [[for all]] <i>X</i>.

Of course, the right side of this equation doesn't make sense in typed logic unless the ___domain type of <i>F</i> matches the codomain type of <i>G</i>, so this is required for the composition to be defined.

One also gets certain function symbols automatically.

GivenIn anyuntyped type <b>T</b>logic, there is an ''identity predicate'' id_<b>T</b> with ___domain and codomain type <b>T</b>; itthat satisfies id_<b>T</b>(<i>''X</i>'') = <i>''X</i>'' for all <i>''X</i> of type <b>T</b>''.

Similarly,In iftyped <b>T</b>logic, isgiven aany [[subtype]]type of <b>U</b>'''T''', then there is an inclusionidentity predicate of ___domain type id<bsub>'''T'''</bsub> with ___domain and codomain type '''T'''; it satisfies id<bsub>U'''T'''</bsub>(''X'') that= satisfies''X'' thefor sameall equation; there are additional function symbols associated with other ways''X'' of constructingtype new types out of old ones'''T'''.

Similarly, if '''T''' is a [[Subtyping|subtype]] 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 a [[formal 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 <i>''X</i>'' (or every <i>''X</i>'' of a certain type), [[there exists]] a [[unique (mathematics)|unique]] <i>''Y</i>'' satisfying some condition <i>''P</i>'', then you can introduce a function symbol <i>''F</i>'' to indicate this.

Note that <i>''P</i>'' will itself be a relational [[predicate (logic)|predicate]] involving both <i>''X</i>'' and <i>''Y</i>''.

So if there is such a predicate <i>''P</i>'' and a theorem:

: For all <i>''X</i>'' of type <b>'''T</b>''', for some unique <i>''Y</i>'' of type <b>'''U</b>''', <i>''P</i>''(<i>''X</i>'',<i>''Y</i>''),

then you can introduce a function symbol <i>''F</i>'' of ___domain type <b>'''T</b>''' and codomain type <b>'''U</b>''' that satisfies:

: For all <i>''X</i>'' of type <b>'''T</b>''', for all <i>''Y</i>'' of type <b>'''U</b>''', <i>''P</i>''(<i>''X</i>'',<i>''Y</i>'') [[if and only if]] <i>''Y</i>'' = <i>''F</i>''(<i>''X</i>'').

== Doing without functional predicates ==

== Doing without functional predicates ==

Many treatments of predicate logic don't allow functional predicates, only relational [[predicate (logic)|predicate]]s.

This is useful, for example, in the context of proving [[metalogic]]al theorems (such as [[GoedelGödel'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 sybmolssymbols wherever the former may occur; furthermore, this is algorithmic and thus suitable for applying most metalogical theorems to the result.

Specifically, if <i>''F</i>'' has ___domain type <b>'''T</b>''' and [[codomain]] type <b>'''U</b>''', then it can be replaced with a predicate <i>''P</i>'' of type (<b>'''T</b>''',<b>'''U</b>''').

Intuitively, <i>''P</i>''(<i>''X</i>'',<i>''Y</i>'') means <i>''F</i>''(<i>''X</i>'') = <i>''Y</i>''.

Then whenever <i>''F</i>''(<i>''X</i>'') would appear in a statmentstatement, you can replace it with a new symbol <i>''Y</i>'' of type <b>'''U</b>''' and include another statement <i>''P</i>''(<i>''X</i>'',<i>''Y</i>'').

To be able to make the same deductions, you need an additional proposition:

: [[For all]] <i>''X</i>'' of type <b>'''T</b>''', for some [[unique (mathematics)|unique]] <i>''Y</i>'' of type <b>'''U</b>''', <i>''P</i>''(<i>''X</i>'',<i>''Y</i>'').

(Of course, this is the same proposition that had to be provedproven 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 (logic)|schema]] that applies only to functional predicates <i>''F</i>''; 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 <i>''F</i>''(<i>''X</i>'') with a new variable <i>''Y</i>''.

Then [[universally quantify]] over each <i>''Y</i>'' immediately after the corresponding <i>''X</i>'' is introduced (that is, after <i>''X</i>'' is quantified over, or at the beginning of the statmentstatement if <i>''X</i>'' is free), and guard the quantification with <i>''P</i>''(<i>''X</i>'',<i>''Y</i>'').

Finally, make the entire statmentstatement a [[material implicationconditional|material consequence]] of the uniqueness condition for a functional predicate above.

Let us take as an example the [[axiom schema of replacement]] in [[Zermelo-FraenkelZermelo–Fraenkel set theory]].

(This example uses [[mathematical symbols]].)

This schema states (in one form), for any functional predicate <i>''F</i>'' in one variable:

:&forall; <imath>\forall A</i>, &exist;\exists <i>B</i>, &\forall; <i>C</i>, <i>C</i> &isin;\in <i>A</i> &rarr;\rightarrow <i>F</i>(<i>C</i>)\in &isin; <i>B.</imath>.

First, we must replace <i>''F</i>''(<i>''C</i>'') with some other variable <i>''D</i>'':

:&<math>\forall; <i>A</i>, &exist;\exists <i>B</i>, &\forall; <i>C</i>, <i>C</i> &isin;\in <i>A</i>\rightarrow &rarr; <i>D</i> &isin;\in <i>B.</imath>.

Of course, this statmentstatement isn't correct; <i>''D</i>'' must be quantified over just after <i>''C</i>'':

:&forall; <imath>\forall A</i>, &exist;\exists <i>B</i>, &\forall; <i>C</i>, &\forall; <i>D</i>, <i>C</i> &isin;\in <i>A</i> &rarr;\rightarrow <i>D</i>\in &isin; <i>B.</imath>.

We still must introduce <i>''P</i>'' to guard this quantification:

:&<math>\forall; <i>A</i>, &exist;\exists <i>B</i>, &\forall; <i>C</i>, &\forall; <i>D</i>, <i>P</i>(<i>C</i>,<i>D</i>) &rarr;\rightarrow (<i>C</i> &isin;\in <i>A</i> &rarr;\rightarrow <i>D</i> &isin;\in <i>B).</imath>).

This is almost correct, but it applyingapplies to too many predicates; what we actually want is:

:<math>(&\forall; <i>X</i>, &exist;\exists ! <i>Y</i>, <i>P</i>(<i>X</i>,<i>Y</i>)) &rarr;\rightarrow (&\forall; <i>A</i>, &exist;\exists <i>B</i>, &\forall; <i>C</i>, &\forall; <i>D</i>, <i>P</i>(<i>C</i>,<i>D</i>) &rarr;\rightarrow (<i>C</i> &isin;\in <i>A</i> &rarr;\rightarrow <i>D</i> &isin;\in <i>B)).</imath>)).

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.

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==

*[[Function symbol (logic)]]

*[[Logical connective]]

*[[Logical constant]]

{{Mathematical logic}}

{{DEFAULTSORT:Functional Predicate}}

[[Category:Model theory]]