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.("xn)[F(x1,.x,n ) ”! G(x1,.x,n )]is provable in »HST", independently of (Ext"), which was Frege s Basic LawVb.Given the consistency of »HST"+(Ext") (relative to Zermelo set theory),which includes Leibniz s law, (LL"), as a theorem schema, it is not Frege s BasicLaw V that was the problem for Frege so much as the way his hierarchy ofuniversals was reflected downward into the first- and second-order levels.Thiswas because Frege had heterogeneous, and not just homogeneous, relations in hislogic, including heterogenous relations between universals and objects, such asthat of predication, and these were included as part of the reflection downwardof his hierarchy.9 The hierarchy consistently represented in »HST", on thehand, consists only of homogeneous relations.The representation of heterogenous relations can be retained, however, byturning to an alternative reconstruction of Frege s logic that is closely related to»HST".This alternative involves replacing the standard first-order logic that ispart of »HST" with a logic that is free of existential presuppositions regardingobjectual terms, including nominalized predicates such as that corresponding tothe complex predicate involved in Russell s paradox.Now it is significant that in an appendix to his Grundgesetze Frege consideredresolving Russell s paradox by allowing that there are cases where an unexcep-tional concept has no extension .10 Here, by an unexceptional concept Fregehad Russell s rather exceptional concept, or property, in mind.After all, whatis exceptional about the Russell concept, or property, in Frege s logic is thatit leads to a contradiction, unless, that is, we allow that it has no extension.But allowing that the Russell property has no extension in Frege s logic requires9See Cocchiarella 1987, section 9, for a more detailed discussion of this point.10Frege 1893, p.128. 106 CHAPTER 5.FORMAL THEORIES OF PREDICATION PART IIallowing the nominalized form of the Russell predicate to denote nothing.Thatis, it requires a shift from standard first-order logic to a logic free of existen-tial presuppositions regarding objectual terms, including especially nominalizedpredicates.In other words, instead of axiom (A8) of the logic of possible objectsin chapter 2, namely,("x)(a = x),where x does not occur in a, we nowhave("x)("y)(x = y),where x, y are distinct object variables.In other words, the logic of possibleobjects is now a free logic, just as the logic of actual objects is a free logic.We want to use this logic as our free logic because although abstract objecthave being as values of the bound object variables, nevertheless, they do not exist in the concrete sense of existence, a restricted notion of being that wewant to retain in an extended development in chapter 6 of the second-order logicwith nominalized predicates that we are now considering.In fact, this strategy works.By adopting a free first-order logic and yetretaining the unrestricted comprehension principle (CP" ), all that follows by»the argument for Russell s paradox is that there is no object corresponding tothe Russell property, i.e.,¬("y)([»x("G)(x = G '"¬G(x))] = y)is provable, even though, by (CP"), the Russell concept, or property, exists »as a concept, or property; that is, even though("F)([»x("G)(x = G '"¬G(x))] = F)is also provable.Because the first-order logic is now a free logic, the originalrule of »-conversion must be modified as follows:[»x1.xnÕ](a1,., an) ”! ("x1).("xn)(a1 = x1 '".'" an = xn '" Õ),("/»-Conv")where, for all i,j d" n, xi does not occur in aj.Revised in this way, our original second-order logic with nominalized pred-icates can be easily shown to be consistent.But that is because, without anyfurther assumptions, we can no longer prove that any property or relation hasan extension.That is, because the logic is free of existential presuppositions,all nominalized predicates might be denotationless, a position that a nominalistmight well adopt.But in Frege s ontological logicism some properties and relations must haveextensions, and, indeed, it would be appropriate to assume that all of the prop-erties and relations that can be represented in »HST" have extensions in thisalternative logic.That in fact is exactly what we allow in our alternative recon-struction of Frege s logic. 5.2.FREGE S LOGIC RECONSTRUCTED 107The added assumption can be stipulated in the form of an axiom schema.But to do so we need to first define the key notion of when an expression of ourlogical grammar can be said to be bound to objects.Definition: If ¾ is a meaningful expression of our logical grammar, i.e.,¾ " MEn, for some natural number n, then ¾ is bound to objects if, andonly if, for all predicate variables F, and all formulas Õ " ME1, if ("F)Õ is aformula occurring in ¾, then for some object variable x and some formula È, Õis the formula [("x)(F = x) ’! È].To be bound to objects, in other words, every predicate quantifier occurringin an expression ¾ must refer only to those properties and relations (or concepts)that have objects corresponding to them, which in Frege s logic are classes asthe extensions of the properties or relations in question.The axiom schema weneed for this is given as follows:("y)(a1 = y) '".'" ("y)(ak = y) ’! ("y)([»x1.xnÕ] =y), ("/HSCP")»where," (1) [»x1.xnÕ] is h-stratified," (2) Õ is bound to objects," (3) y is an object variable not occurring in Õ, and" (4) a1,., ak are all of the object or predicate variables or nonlogical con-stants occurring free in [»x1.xnÕ].11Because of it close similarity to our first reconstructed system, »HST", wewill refer to this alternative logic as HST".12 As we have shown elsewhere,»HST" is equiconsistent with »HST", and therefore with the theory of simple»types as well.It is of course also consistent relative to Zermelo set theory
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