2.2 Bases Teóricas
2.2.2 Bases Legales Sustantivas
In some of his earlier publications Quine alludes to Russell’s Paradox in his discussion of higher-order quantification. Russell’s Paradox arises in na¨ıve set theory in the following way. Let us say that for any open sentence there is a set that contains all and only the objects that satisfy this open sentence. Prima faciethis sounds like a reasonable suggestion, but this principle leads to Russell’s famous antinomy. Take the predicate ‘being a set that does not contain itself’. Call the set that contains all and only sets that do not contain themselves ‘r’ – the “Russell Set”. Doesrcontain itself? It seems it cannot, sincer contains only sets that do not contain themselves; for if it contained itself, it could not be amongst those. So it does not contain itself. In that case, however, it is one of those sets that do not contain themselves, and hence it has to be in the respective set, i.e. in itself. So, if r contains itself, it does not, and if it does not contain itself, it does; or formally, r ∈ r ≡ ¬r ∈ r: contradiction.
In On Universals (Quine, 1947) Quine suggests that an antinomy similar to Russell’s for na¨ıve set theory might occur if one takes second-order quantification seriously. If we allow binding predicate letters with quantifiers this means that they “acquire the status of variables”, and that we are thus “granting [them] all privileges of ‘x’, ‘y’, etc.”, i.e. of the first-order variables. This means allowing second-order variables to occur in name position, and hence allowing formulae like ‘GH’ (just as ‘Gx’) “seems a very natural way of proclaiming a realm of universals” to Quine.20 He proceeds by giving a proof in this so characterised system of an analogue of
Russell’s Paradox:21
(1) GH ≡GH logical truth
(2) ∀H(GH ≡GH) (1), universal generalisation
(3) ∀F¬∀H(F H ≡GH)⊃ ¬∀H(GH ≡GH) instance of the ‘∀’-axiom22
(4) ¬∀F¬∀H(F H ≡GH) (2), (3), modus tollens
(5) ∃F∀H(F H ≡GH) (4), quantifier conversion
(6) ∃F∀H(F H ≡ ¬HH) (5), subst. ‘¬HH’ for ‘GH’
(7) ∃F(F F ≡ ¬F F) (6), “by a few easy steps”
Quine makes the detour through line (3) and (4) as the deductive system he proposes in this paper does not have any inference rules for the existential quantifier; he treats ‘∃x’ as a mere abbreviation of ‘¬∀x¬’. Otherwise line (5) would follow immediately from line (2) by existential generalisation – if we ignore for the moment that none of the formulae is well-formed according to any standard higher-order logic or theory of types. (I will come back to the question of peculiar syntax below).
This, however, is not the only oddity about this derivation. The substitution step from line (5) to line (6) is invalid: it brings the first ‘H’ in the substituted expression ‘¬HH’ into the scope of the universal quantifier. Quine justifies this step by referring to two of the rules of inference that he introduced earlier. These are the already mentioned rule to allow the predicate letters all privileges of individual variables, and the rule for substitution of formulae: “Substitute any formulae for ‘p’, ‘q’, ‘F x’, ‘F y’,
21Compare (Quine, 1947), p. 78. The notation is changed to match the symbolisation used
throughout this thesis. Moreover, Quine’s commentary column is omitted in favour of my own annotations, as Quine’s abbreviated comments would require rather extensive introduction. Only the annotation in line (7) is verbatim.
‘Gx’, ‘F xy’, ‘Gzw’, etc. (subject to sundry provisos which need not be recounted here)”.23 The provisos that Quine skips so elegantly include that the substitution has to be uniform (so that the faulty inference from ‘∀x(F x ⊃F x)’ to ‘∀x(F x⊃ Gx)’ is barred), and that variables must not come into the scope of a quantifier that they were not in before (to avoid, e.g., the inference from ‘Gy ⊃ ∀x(F x ⊃ Gy)’ to ‘¬F x ⊃ ∀x(F x ⊃ ¬F x)’). The former restriction is obeyed in Quines “proof”; the latter, alas, is not.
In a later paper,Logic and the Reification of Universals, which is partly based on On Universals, Quine still alludes to the paradox, but ommits its “derivation”, and indeed any mention of the peculiar syntax that allows for expression like ‘¬HH’.24
What drives Quine to suggest formulae like ‘¬HH’ ? It seems that his conviction that if we allow quantification into predication position we ontologically commit ourselves to things that predicates now are meant to refer to (while at first-order we do not treat predicates as referring) is his reason to do so. It seems that for Quine the existential quantifier (of whatever order) shows us what we are ontologically committed to, and if we further assume that those entities that we are committed to are referred to by expressions that occur in name position, it is only a small step to put second-order variables into name position.
This way will indeed lead to inconsistency, whether Quine’s “proof” as such suc- ceeds or not. Russell first25formulated his Paradox in a letter that he wrote to Frege in response to Frege’sGrundgesetze der Arithmetik26 [Basic Laws of Arithmetic].27
23(Quine, 1947), p. 77. 24(Quine, 1953), p. 121.
25This was whereRussellfirst formulated the paradox. Credit where credit is due, however: Ernst
Zermelo had discovered “Russell’s Paradox” a full year before Russell, see (Rang and Thomas, 1981).
26(Frege, 1893). 27(Russell, 1902).
Frege’s system of the Grundgesetzefamously contains Basic Law V which, roughly, assigns an object to every extension of a predicate. While, strictly speaking, the formulation Russell used was not well-formed in Frege’s system, Frege saw immedi- ately how the antinomy could be formulated properly and derived from Basic Law V.28 Quine mirrors this strategy for his criticism of second-order logic. Quantifying into predicate position for him means treating predicates as referring to objects. In one way or other, a paradox analogous to Russell’s should arise. We only have to define a predicate such that the object it refers to can somehow be proven both to fall and not to fall under this very predicate, i.e. something like a predicate that expresses non-self-reference has to be found. If the formulation of the system is careless enough, this should be possible.
Quine knows, of course, that there are consistent systems of second-order logic in which no contradiction can be proven. In his Philosophy of Logic he concedes that “there is no actual risk of paradox as long as the ranges of ‘x’ and ‘G’ are kept apart”.29 In On Universals he claims, however, that these consistent systems are arrived at by making “one or another aribtrary restriction [...]. The most familiar restriction is the simple theory of types, due to Russell.”30 Quine considers the restrictions that need to be made in order to avoid the paradox as wholly artificial. It should be noted that Frege’s system of theBegriffsschrift31neither allows for a derivation of Russell’s Paradox, nor does it contain restrictions that appear artificial or arbitrary in any way. The distinction that blocks the derivation of an analogue to Russell’s Paradox is that between name and predicate (that Quine wants to make,
28(Frege, 1902).
29(Quine, 1986a), p. 68. 30(Quine, 1947), p. 78. 31(Frege, 1879).
too). In Frege’s Begriffsschrift corresponds to the distinction between object and function, that seems well motivated in Frege.32
Quine, however, takes it that the objects referred to by the predicates in second- order logic have to be classes, as in the theory of types which he explicitly mentions and which, at least for him, is a class theory. They are thus objects and belong in the (first-order) domain. A possible autobiographical reason for this was already mentioned in the introduction to this chapter. There are, however, also philosophical reasons for Quine.