Quine's Logic and the Class Paradox

Abstract

so that both 'xXx' and '{x: Fx } 0 {x: Fx }' are notationally well formed in SL-even though the two predicates are not the same. As Quine puts it, 'the two sorts of 'E are prima facie the merest accidental homonyms' (p. 40). We can of course resolve the ambiguity contextually, by inspection of what follows 'e' (p. 33), but a difficulty emerges in expressions of the type 'Foe' wbere predicate letters are employed and where there is no immediate indication of the exact context in which the hidden predicate 'e' is used. rhe difficulty is that 'Foe' is not logically well formed if 'F' is the predicate '(c)e(J)' with 'e' primitive and if oc is a class abstract that cannot be replaced by an individual constant. Indicating the primitive or derived status of 'e' by a subscript, we must say that 'F{x:xOx}' is ill formed if it means '{x:xXx} 0 {x:xXx}', since any 'e' before a class abstract must be derived. But if 'F {x:xox}' is construed to mean the well formed expression '{x::xOx } d {x:xx }', then F cannot be the same predicate as F in 'xXx' where the 'e' is primitive. In this connection it is significant that Quine introduces the 'Foc' locution (pp. 40-4I) with no explanatory rubric apart from the designation of 'oc' and 'P' as ranging over class abstracts as well as variables. Although the rules of derivation (SL2.i) and (SL5.5) justify the use of expressions in which class abstracts are preceded by a derived 'e', they do not legitimatize such expressions in which the 'e' is primitive. If such rules were provided, then (4) would yield not a contradiction but an expression such as (5). (5) {x:xXx} Ed {X:X?X} *{X:X0X} Op {X:Xf} In (5) a clarification of the difference between the use of class abstracts after a derived 'e' and their use after a primitive 'e' would be sufficient to resolve the apparent contradiction. But if no definition is provided for the use of class abstracts after a primitive 'e', then expressions of the type 'F {x :Fx }', when {x:Fx } is an ultimate class and F the predicate which determines it, will be ill formed-and the naive abstraction rule (4) will hold. The above argument does not claim that 'Foc' expressions as such are ill formed in SL, since the predicate letter 'F' can be viewed as standing for one place predicates in which 'e' may be either primitive or derived. For example, the derivation of 'Foc' might be defined as either (6) or (7). (6) 'Foc' for 'oce {x:Fx}' (7) 'Foc' for '(3y) (y-oc. Foc)' This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 06:11:46 UTC All use subject to http://about.jstor.org/terms