Quine proposed that "to be is to be the value of a variable". However, he also devised predicate functor logic (PFL), which effectively gives a recipe for eliminating bounded variables.

How should one reconcile predicate functor logic with Quine's own ontological theory?

While one might say that insofar as PFL and first order logic (FOL) are equally expressive, PFL has the same ontological commitment as FOL, but it appears to me that the argument could also go the other way round. If they are mere "notational variants" of each other, then all we can conclude is that PFL and FOL have the same ontological commitments. It might be that, contrary to appearances, the existential quantifier in FOL is not ontologically committal after all since it is merely a cropping predicate functor, as revealed by its translation in PFL.

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    But what is the semantical interpretation of a "cropping predicate functor" ? – Mauro ALLEGRANZA Jul 8 '18 at 17:19
  • @MauroALLEGRANZA It is hard to say what is the semantics of PFL if one is asking for something like the Tarskian semantics of FOL. PFL does have a model theory, where the domain for interpretation is a set and interpretations of predicates are (infinite) strings of elements from that set. However, it is unclear that this constitutes a "semantics" of the logic, as opposed to a set-theoretical apparatus that proves the theory's consistency. Moreover, I suspect that there could be relational semantics for the logic in which functors are themselves elements of the set... – discretizer Jul 8 '18 at 17:36
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    PFL has a completeness th; see John Bacon, The completeness of a predicate-functor logic, JSL (1985). Having said that, a "universe" will be made of functors ? If so a theory developed on top of PFL will have specific axioms restricting the universe to some "collection" of fucntors satisfying the axioms. If so, they "exist" in Quine sense. – Mauro ALLEGRANZA Jul 8 '18 at 17:54
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    Maybe useful : W.V.O Quine, Variables explained away, ProcAMS (1960), page 347: "We end up with a universal algebra purely of predicates, comprising just our six operators and any arbitrary predicates as generators for them to operate on. This is a general logical notation. It is devoid of variables, yet theoretically adequate as a framework for theories generally, mathematical and otherwise. 1/2 – Mauro ALLEGRANZA Jul 8 '18 at 18:53
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    To fix it as a notation for anyspecific subject matter we merely supply the appropriate vocabulary of specific predicates, leaving the outward framework unchanged; and that framework consists of our six operators, nothing more." 2/2 – Mauro ALLEGRANZA Jul 8 '18 at 18:54

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