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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.

An individual-concept, hereinafter “individuation”, is a function from possible worlds to individuals. Constant individuations I will call “subsistents” (the notion will presently be generalized). A “substance”, after Thomason [35], is a subsistent whose value exists for the world at hand. In the systems of quantified modal logic developed over the past twenty years, the tendency has been to restrict the range of quantifiers to substances (often represented technically by the simple individuals that would be the values of the constant individuations), while allowing constant terms (particularly descriptions) to express arbitrary individuations. One result is to invalidate unrestricted universal instantiation (and existential generalization), rather as in free logic. Such systems approximate some features of ordinary usage rather nicely, e.g. the behavior of quantifiers and definite descriptions in tensed discourse. Stalnaker and Thomason's Q3r [34], based on the latter's Q3 [35], [36], is exemplary of this approach.The suggestion has repeatedly been considered to quantify over individuations in general (Kanger [14], Kaplan [17], Hughes and Cresswell [12, p. 196], Thomason [35, p. 136], Pollock [30]).

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