case A completeness theorem predicate-functor logic. Predicate-functor logic, Quine's variant of algebraic (variable-free) first-order logic, has primitive predicates of determinate finite degree combined into larger predicates by means of predicate functors. My version of predicate-functor logic with identity {PFI) has a dyadic identity predicate / and five basic predicate functors: inclusion (restricted generality) <=, negation —, padding +, transposition (minor inversion) X, and cycling (major inversion) 0. These are all understood as in Quine [2], except for inclusion: PFI is, formulated in the natural-deduction format of Frederic B. Fitch with 18 rules, most of which are either the obvious thing An infmitary equational notion of forcing is denned based on Keisler's notion of a forcing property. It is shown that the closure of theories under induction with schemas, for a minimal set of functions inductively defining an initial model of the closure, is a generic set. Such closures are forcing companions consisting of strictly universal formulae. A generic model for the closure construction is exhibited which is generated by the inductive closure. Inductive closures are shown to be particular instances of Martin's axiom. This work has applications to reasoning in artificial intelligence and algebraic closure. definite descriptions which have an established semantic referent from those that do not, understanding established semantic reference relative to a speaker within a context of linguistic conventions and understandings. I argue that among definite and indefinite descriptions as well as proper names there are both situation independent and dependent referential uses, as well as existential and even universal ones. In the interest of unifying syntax and semantics, I suggest that only proper names and definite descriptions be used referentially. Uses of existential "it" could be rephrased as uses of existential "something" with no loss in meaning. Moreover, uses of existential indefinite and definite descriptions could be rephrased as special cases of existential "something". Uses of universal "it" could be prefixed with what amounts to a universal quantifier. Uses of universal "something" could be rephrased as uses of universal "it" with the added prefix. In a similar manner, universal indefinite and definite descriptions could be rephrased as special cases and the same prefix added.
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