- Predicate functor logic
In

mathematical logic ,**predicate functor logic (PFL)**is one of several ways to expressfirst-order logic (formerly known aspredicate 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 thelogic ian andphilosopher Willard Quine .**Motivation**This 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 howBoolean algebra algebraizessentential logic . He designed PFL to have exactly the expressive power offirst-order logic withidentity . Hence themetamathematics of PFL are exactly those of first-order logic with no interpreted predicate letters: both logics are sound,complete , andundecidable . Most work Quine published on logic and mathematics in the last 30 year 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 inphilosophy andmathematical 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 byAlfred 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";

*Thepolyadic algebra ofPaul Halmos . By virtue of its economical primitives and axioms, this algebra most resembles PFL;

*Relation algebra algebraizes the fragment offirst-order logic consisting of formulas having no atomic formula lying in the scope of more than threequantifier s. That fragment suffices, however, forPeano arithmetic and theaxiomatic set theory ZFC ; hence relation algebra, unlike PFL, is incompletable. Most work on relation algebra since about 1920 has been by Tarski and his American students. The power of relation algebra did not become manifest until the monograph Tarski and Givant (1987), published after the three important papers bearing on PFL, namely Bacon (1985), Kuhn (1983), and Quine (1976);

*Combinatory logic builds oncombinator s,higher order function s whose domain is another combinator or function, and whose range is yet another combinator. Hencecombinatory logic goes beyond first-order logic by having the expressive power ofset theory , which makes combinatory logic vulnerable toparadox es. A predicate functor, on the other hand, simply maps predicates (also calledterm s) into predicates.PFL is arguably the simplest of these formalisms, yet also the one about which the least has been written.Quine had a lifelong fascination with

combinatory logic , attested to by his (1976) and his introduction to the translation in Van Heijenoort (1967) of the paper by the Russian logicianMoses Schonfinkel founding combinatory logic. When Quine began working on PFL in earnest, in 1959, combinatory logic was commonly deemed a failure for the following reasons:

* UntilDana Scott began writing on themodel theory of combinatory logic in the late 1960s, nearly all work on that logic had been byHaskell Curry , his students, or byRobert Feys in Belgium;

*Satisfactory axiomatic formulations of combinatory logic were slow in coming. In the 1930s, some formulations of combinatory logic were found to be inconsistent. Curry also discovered theCurry paradox , peculiar to combinatory logic;

*Thelambda calculus , with the same expressive power ascombinatory logic , was seen as a superior formalism.**Kuhn's formalization**The PFL

syntax , primitives, and axioms described in this section are largely Kuhn's (1983). Thesemantics of the functors are Quine's (1982). The rest of this entry incorporates some terminology from Bacon (1985).**yntax**An "atomic term" is an upper case Latin letter, "I" and "S" excepted, followed by a numerical

superscript called its "degree", or by concatenated lower case variables, collectively known as an "argument list". The degree of a term conveys the same information as the number of variables following a predicate letter. An atomic term of degree 0 denotes aBoolean variable or atruth value . The degree of "I" is invariably 2 and so is not indicated.The "combinatory" (the word is Quine's) predicate functors, all monadic and peculiar to PFL, are

**Inv**,**inv**,**∃**,**+**, and**p**. A term is either an atomic term, or constructed by the following recursive rule. If τ is a term, then**Inv**τ,**inv**τ,**∃**τ,**+**τ, and**p**τ are terms. A functor with a superscript "n", "n" anatural number > 1, denotes "n" consecutive applications (iterations) of that functor.A formula is either a term or defined by the recursive rule: if α and β are formulas, then αβ and ~(α) are likewise formulas. Hence "~" is another monadic functor, and concatenation is the sole dyadic predicate functor. Quine called these functors "alethic." The natural interpretation of "~" is

negation ; that of concatenation is anyconnective that, when combined with negation, forms a functionally complete set of connectives. Quine's preferred functionally complete set wasconjunction andnegation . Thus concatenated terms are taken as conjoined. The notation**+**is Bacon's (1985); all other notation is Quine's (1976; 1982). The alethic part of PFL is identical to the "Boolean term schemata" of Quine (1982).As is well known, the two alethic functors could be replaced by a single dyadic functor with the following

syntax andsemantics : if α and β are formulas, then (αβ) is a formula whose semantics are "not (α and/or β)" (seeNAND andNOR ).**Axioms and semantics**Quine set out neither axiomatization nor proof procedure for PFL. The following axiomatization of PFL, one of two proposed in Kuhn (1983), is concise and easy to describe, but makes extensive use of

free variable s and so does not do full justice to the spirit of PFL. Kuhn gives another axiomatization dispensing with free variables, but that is harder to describe and that makes extensive use of defined functors. Kuhn proved both of his PFL axiomatizations sound andcomplete .This section is built around the primitive predicate functors and a few defined ones. The alethic functors can be axiomatized by any set of axioms for

sentential logic whose primitives are negation and one of ∧ or ∨. Equivalently, all tautologies of sentential logic can be taken as axioms.For each combinatory functor, we give its semantics, as per Quine (1982), stated in terms of

abstraction (set builder notation), followed by either the relevant axiom from Kuhn (1983), or a definition from Quine (1976). The notation $\{x\_1...x\_n\; :\; Fx\_1...x\_n\}$ denotes the set ofn-tuple s satisfying the atomic formula $Fx\_1...x\_n.$*"Identity", "I", is defined as::$IFx\_1x\_2...x\_n\; leftrightarrow\; Fx\_1x\_1...x\_n\; leftrightarrow\; Fx\_2x\_2...x\_n.$Identity is

reflexive ("Ixx"),symmetric ("Ixy"→"Iyx"),transitive ( ("Ixy"∧"Iyz") → "Ixz"), and obeys the substitution property::$(Fx\_1...x\_n\; and\; Ix\_1y)\; ightarrow\; Fyx\_2...x\_n.$

*"Padding",**+**, adds a variable to the left of any argument list.:$+F^n\; =\_\{def\}\; \{x\_0x\_1...x\_n\; :\; F^n\; x\_1...x\_n\}.$:$+Fx\_1...x\_n\; leftrightarrow\; Fx\_2...x\_n.$

*"Cropping",**∃**, erases the leftmost variable in any argument list.:$exist\; F^n\; =\_\{def\}\; \{x\_2...x\_n\; :\; exist\; x\_1\; F^n\; x\_1...x\_n\}.$:$Fx\_1...x\_n\; ightarrow\; exist\; Fx\_2...x\_n.$"Cropping" enables two useful defined functors:

* "Reflection",**S**::$SF^n\; =\_\{def\}\; \{x\_2...x\_n\; :\; F^n\; x\_2x\_2...x\_n\}.$:$SF^n\; leftrightarrow\; exist\; IF^n.$**S**generalizes the notion ofreflexivity to all terms of any finite degree greater than 2. N.B:**S**should not be confused with the primitive combinator**S**of combinatory logic.

*"Cartesian product ", $imes$;:$F^m\; imes\; G^n\; leftrightarrow\; F^m\; exist^m\; G^n.$ Here only, Quine adopted an infix notation, because this infix notation for Cartesian product is very well established in mathematics. Cartesian product allows restating conjunction as follows::$F^mx\_1...x\_mG^nx\_1...x\_n\; leftrightarrow\; (F^m\; imes\; G^n)x\_1...x\_mx\_1...x\_n.$Reorder the concatenated argument list so as to shift a pair of duplicate variables to the far left, then invoke**S**to eliminate the duplication. Do this as many times as required, resulting is an argument list of length max{"m","n").The next three functors enable reordering argument lists at will.

*"Major inversion",**Inv**, rotates the variables in an argument list to the right, so that the last variable becomes the first.:$InvF^n\; =\_\{def\}\; \{x\_1...x\_n\; :\; F^nx\_nx\_1...x\_\{n-1\}\}.$:$Inv\; Fx\_1...x\_n\; leftrightarrow\; Fx\_nx\_1...x\_\{n-1\}.$

*"Minor inversion",**inv**, swaps the first two variables in an argument list.:$invF^n\; =\_\{def\}\; \{x\_1...x\_n\; :\; F^n\; x\_2x\_1...x\_n\}.$:$invFx\_1...x\_n\; leftrightarrow\; Fx\_2x\_1...x\_n.$

*"Permutation",**p**, rotates the second through last variables in an argument list to the left, so that the second variable becomes the last.:$pF^n\; =\_\{def\}\; \{x\_1...x\_n\; :\; F^n\; x\_1x\_3...x\_nx\_2\}.$:$pFx\_1...x\_n\; leftrightarrow\; Inv\; inv\; Fx\_1x\_3...x\_nx\_2.$Given an argument list consisting of "n" variables,**p**implicitly treats the last "n"-1 variables like a bicycle chain, with each variable constituting a link in the chain. One application of**p**advances the chain by one link. "k" consecutive applications of**p**to "F"^{n}moves the "k"+1 variable to the second argument position in "F".When "n"=2,

**Inv**and**inv**merely interchange "x"_{1}and "x"_{2}. When "n"=1, they have no effect. Hence**p**has no effect when "n"<3.Kuhn (1983) takes "Major inversion" and "Minor inversion" as primitive. The notation

**p**in Kuhn corresponds to**inv**; he has no analog to "Permutation" and hence has no axioms for it. If, following Quine (1976),**p**is taken as primitive,**Inv**and**inv**can be defined as nontrivial combinations of**+**,**∃**, and iterated**p**.The following table summarizes how the functors affect the degrees of their arguments.

**Rules**All instances of a predicate letter may be replaced by another predicate letter of the same degree, without affecting validity. The rules are:

*Modus ponens ;

* Let α and β be PFL formulas in which $x\_1$ does not appear. Then if $(alpha\; and\; Fx\_1...x\_n)\; ightarrow\; eta$ is a PFL theorem, then $(alpha\; and\; exist\; Fx\_2...x\_n)\; ightarrow\; eta$ is likewise a PFL theorem.**ome useful results**Instead axiomatizing PFL, Quine (1976) proposed the following conjectures as candidate axioms.

*$exist\; I$"n"-1 consecutive iterations of

**p**restores the "status quo ante":

*$F^n\; leftrightarrow\; p^\{n-1\}F^n$**+**and**∃**annihilate each other:

*$F^n\; ightarrow\; +exist\; F^n$

*$F^n\; leftrightarrow\; exist\; +F^n$Negation distributes over**+**,**∃**, and**p**:

*$+lnot\; F^n\; leftrightarrow\; lnot\; +F^n$

*$lnotexist\; F^n\; ightarrow\; exist\; lnot\; F^n$

*$plnot\; F^n\; leftrightarrow\; lnot\; pF^n$**+**and**p**distributes over conjunction:

*$+(F^nG^m)\; leftrightarrow\; (+F^n+G^m)$

*$p(F^nG^m)\; leftrightarrow\; (pF^npG^m)$Identity has the interesting implication:

*$IF^n\; ightarrow\; p^\{n-2\}\; exist\; p+F^n$Quine also conjectured the rule: If $alpha$ is a PFL theorem, then so are $palpha,$ $+alpha,$ and $lnot\; exist\; lnot\; alpha$.

**Bacon's work**Bacon (1985) takes the

conditional ,negation , "Identity", "Padding", and "Major" and "Minor inversion" as primitive, and "Cropping" as defined. Employing terminology and notation differing somewhat from the above, Bacon (1985) sets out two formulations of PFL:

* Anatural deduction formulation in the style ofFrederick Fitch . Bacon proves this formulation sound andcomplete in full detail.

*An axiomatic formulation which Bacon asserts, but does not prove, equivalent to the preceding one. Some of these axioms are simply Quine conjectures restated in Bacon's notation.Bacon also:

*Discusses the relation of PFL to theterm logic of Sommers (1982), and argues that recasting PFL using a syntax proposed in Lockwood's appendix to Sommers, should make PFL easier to "read, use, and teach";

*Touches on the group theoretic structure of**Inv**and**inv**;

*Mentions thatsentential logic ,monadic predicate logic , themodal logic S5, and the Boolean logic of (un)permuted relations, are all fragments of PFL.**From first-order logic to PFL**The following

algorithm is adapted from Quine (1976: 300-2). Given a closed formula offirst-order logic , first do the following:

* Attach a numerical subscript to every predicate letter, stating its degree;

* Translate alluniversal quantifier s intoexistential quantifier s and negation;

* Restate allatomic formula s of the form "x"="y" as "Ixy".Now apply the following algorithm to the preceding result:

**1.**Translate the matrices of the most deeply nested quantifiers intodisjunctive normal form , consisting of disjuncts of conjuncts of terms, negating atomic terms as required. The resulting subformula contains only negation, conjunction, disjunction, and existential quantification.**2.**Distribute the existential quantifiers over the disjuncts in the matrix using the rule of passage (Quine 1982: 119)::$exist\; x\; [alpha\; (x)\; or\; gamma\; (x)]\; leftrightarrow\; (exist\; x\; alpha\; (x)\; or\; exist\; x\; gamma\; (x)).$**3.**Replace conjunction byCartesian product , by invoking the fact::$(F^m\; and\; G^n)\; leftrightarrow\; (F^m\; imes\; G^n)\; leftrightarrow\; (F^m\; exist^m\; G^n);\; mmath>$**4.**Concatenate the argument lists of all atomic terms, and move the concatenated list to the far right of the subformula.**5.**Use**Inv**and**inv**to move all instances of the quantified variable (call it "y") to the left of the argument list.**6.**Invoke**S**as many times as required to eliminate all but the last instance of "y". Eliminate "y" by prefixing the subformula with one instance of**∃**.**7.**Repeat (1)-(6) until all quantified variables have been eliminated. Eliminate any disjunctions falling within the scope of a quantifier by invoking the equivalence::$(alpha\; or\; eta\; or\; ...)\; leftrightarrow\; lnot(lnotalpha\; and\; lnoteta\; and\; ...).$The reverse translation, from PFL to first-order logic, is discussed in Quine (1976: 302-4).

The canonical

foundation of mathematics isaxiomatic set theory , with a background logic consisting offirst-order logic withidentity , with auniverse of discourse consisting entirely of sets. There is a single predicate letter of degree 2, interpreted as set membership. The PFL translation of the canonicalaxiomatic set theory ZFC is not difficult, as noZFC axiom requires more than 6 quantified variables. [*[*]*http://us.metamath.org/mpegif/mmset.html#staxioms Metamath axioms.*]**ee also***

algebraic logic

*combinatory logic

*cylindric algebra

*relation algebra

*term logic **Footnotes****References***Bacon, John, 1985, "The completeness of a predicate-functor logic," "Journal of Symbolic Logic 50": 903-26.

*Paul Bernays , 1959, "Uber eine naturliche Erweiterung des Relationenkalkuls" in Heyting, A., ed., "Constructivity in Mathematics". North Holland: 1-14.

*Kuhn, Stephen T., 1983, " [*http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093870313 An Axiomatization of Predicate Functor Logic,*] " "Notre Dame Journal of Formal Logic 24": 233-41.

*Willard Quine , 1976, "Algebraic Logic and Predicate Functors" in "Ways of Paradox and Other Essays", enlarged ed. Harvard Univ. Press: 283-307.

*--------, 1982. "Methods of Logic", 4th ed. Harvard Univ. Press. Chpt. 45.

*Sommers, Fred, 1982. "The Logic of Natural Language". Oxford Univ. Press.

*Alfred Tarski and Givant, Steven, 1987. "A Formalization of Set Theory Without Variables". AMS.

*Jean Van Heijenoort , 1967. "From Frege to Godel: A Source Book on Mathematical Logic". Harvard Univ. Press.

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