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Concept
Predicate functor logic
Résumé
In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers) that operate on terms to yield terms. PFL is mostly the invention of the logician and philosopher Willard Quine.
The source for this section, as well as for much of this entry, is Quine (1976). Quine proposed PFL as a way of algebraizing first-order logic in a manner analogous to how Boolean algebra algebraizes propositional logic. He designed PFL to have exactly the expressive power of first-order logic with identity. Hence the metamathematics of PFL are exactly those of first-order logic with no interpreted predicate letters: both logics are sound, complete, and undecidable. Most work Quine published on logic and mathematics in the last 30 years of his life touched on PFL in some way.
Quine took "functor" from the writings of his friend Rudolf Carnap, the first to employ it in philosophy and mathematical logic, and defined it as follows:
"The word functor, grammatical in import but logical in habitat... is a sign that attaches to one or more expressions of given grammatical kind(s) to produce an expression of a given grammatical kind." (Quine 1982: 129)
Ways other than PFL to algebraize first-order logic include:
Cylindric algebra by Alfred Tarski and his American students. The simplified cylindric algebra proposed in Bernays (1959) led Quine to write the paper containing the first use of the phrase "predicate functor";
The polyadic algebra of Paul Halmos. By virtue of its economical primitives and axioms, this algebra most resembles PFL;
Relation algebra algebraizes the fragment of first-order logic consisting of formulas having no atomic formula lying in the scope of more than three quantifiers. That fragment suffices, however, for Peano arithmetic and the axiomatic set theory ZFC; hence relation algebra, unlike PFL, is incompletable.
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