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In my lecture notes for Discrete Structures, the professor introduced a definition on functors in the "Syntax of Predicate Logic" section.

Definition of functors: Let us consider a collection of symbols called functors (each functor is associated to a natural number n, called its valence or arity, we say that the functor is n-ary). The 0-ary functors are called constants. Let us consider a collection of symbols called variables. Let us consider the two parenthesis symbols ( and ) and the comma symbol ,.

Can anyone tell me what is the intrinsic difference between functors and predicates?

Thanks.

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2 Answers 2

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An n-ary functor is an object which accepts as input an n-tuple from the domain in question and delivers as output a 1-tuple of the domain in question. An n-ary predicate is an object which accepts as input an n-tuple from the domain in question and delivers as output a 1 or 0 (true or false respectively).

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That seems to confuse functions with functions, and predicates with relations. It is relations which "accept objects", while predicates are symbols which get concatenated with terms to form wffs: predicates have relations as their semantic values. Likewise functions are symbol entities, certainly in the usual of the quoted notes, which have functions as their semantic values. –  Peter Smith Sep 2 '12 at 17:47

Functors correspond with functions and predicates correspond with relations. A n-ary functor supplied with n arguments is a name. A n-ary predicate supplied with n arguments is a sentence.

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