The completeness of a predicate-functor logic

Abstract

Predicate-functor logic, as founded by W. V. Quine ([1960], [1971], [1976], [1981]), is first-order predicate logic without individual variables. Instead, adverbs or predicate functors make explicit the permutations and replications of argument-places familiarly indicated by shifting variables about. For the history of this approach, see Quine [1971, 309ff.]. With the evaporation of variables, individual constants naturally assimilate to singleton predicates or adverbs, leaving no logical subjects whatever of type 0. The orphaned “predicates” may then be taken simply as terms in the sense of traditional logic: class and relational terms on model-theoretic semantics, schematic terms on Quine's denotational or truth-of semantics. Predicate-functor logic thus stands forth as the pre-eminent first-order term logic, as distinct from propositional-quantificational logic. By the same token, it might with some justification qualify as “first-order combinatory logic”, with allowance for some categorization of the sort eschewed in general combinatory logic, the ultimate term logic.Over the years, Quine has put forward various choices of primitive predicate functors for first-order logic with or without the full theory of identity. Moreover, he has provided translations of quantificational into predicate-functor notation and vice versa ([1971, 312f.], [1981, 651]). Such a translation does not of itself establish semantic completeness, however, in the absence of a proof that it preserves deducibility.An axiomatization of predicate-functor logic was first published by Kuhn [1980], using primitives rather like Quine's. As Kuhn noted, “The axioms and rules have been chosen to facilitate the completeness proof” [1980, 153]. While this expedient simplifies the proof, however, it limits the depth of analysis afforded by the axioms and rules. Mindful of this problem, Kuhn ([1981] and [1983]) boils his axiom system down considerably, correcting certain minor slips in the original paper.

Algebraic classifications for fragments of first-order logic and beyond

“…However, it was shown that on the level of sentences, the expressive power of the guarded relational algebra coincides with the guarded fragment. 3 We now provide a similar result in our setting, showing that GRA(e, p, s, \, ∩, ∃) is sententially equiexpressive with GF. We start by defining that a term guard of a term T of arity at least one is a term S with the following properties.…”

“…Predicate functor logic has been studied very little, and we are not aware of any work relating to complexity theory. The notable works within this framework include the complete axiomatizations given in [20] and [3]. Concerning further algebraic settings, Tarski's relation algebra (see [17]) is also related to our work, but focuses on binary relations.…”

A Language for Ontological Nihilism

“…For details, see Quine (1960) and Burgess (2005). Quine developed several different versions of PF-languages in his later work, and Bacon (1985) and Kuhn (1983) begin from a slightly different basic stock of functors for their axiomatizations and completeness proofs. For a presentation of natural deduction and examples of many proofs in PF, see Bacon (1985).…”

“…At least, they are unacceptable when given a flatfootedly realist construal. See Bacon (1985) and Kuhn (1983) for this presentation.…”

Logical Syntax

“…Correspondingly, the issue can be raised if some of these give us 'large' decidable fragments of predicate-logical inference in some practical sense (cf. Bacon 1985). REMARK.…”