Attributes vs. Classes in Principia Nicholas Rescher Mind, New Series, Vol. 67, No. 266. (Apr., 1958), pp. 254-257. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28195804%292%3A67%3A266%3C254%3AAVCIP%3E2.0.CO%3B2-B Mind is currently published by Oxford University Press.
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ATTRIBUTES vs. CLASSES I N PRZNCZPIA
I QUINE has shown l that the distinction between properties in extension (classes) and properties in intension (attributes or propositional functions, in the appropriate sense) breaks down in the system of Principia if the following rule of inference is adjoined as a principle of individuation of attributes : (Q) If I- +x = +x, then I- +=$. Quine's demonstraticn runs essentially as fcllows : (1) [*20.18] I- [&(4z)]= &($z)]3 [F{&(~z)} = F{&($z))].
Letting F = ,,f 2 E &(xi)= 4, we have,
(2) [(I)] I- [&(+z)= &($z)] 3 [(iZ E &(4z)= 4) = (2 E &($z) = +)I. (3) [*20.3] I- x E &(+z)= 4x. (4) [(3), (&)I t- Ei. E &(4z)= 4. (5) [(2), (4)l I- [&(4z)= &($z)l 3 [g i. &($z)= 41. (6) [(4), (5)l I-E&(dz)= &($z)l 3 (4 = $1. Thus the addition of (Q) to Principia leads to the conclusion that properties of common extension are identical, removing the warrant for distinguishing between classes and attributes in the system of
Princi~ia.~ I1 I n an effort to reconstitute the class vs. attribute distinction in the face of a n acceptance of (Q), Smullyan has proposed a twopoint programme. His first proposal is : (Sl) To interpret the Principia symbol "$! " to represent the extensional counterpart of the property 4, so that 4 ! is 4 taken in extension : 4 ! = 2 E &(4z).* " On Frege's Way Out ", M ~ N D , lxiv (1955), 145-159. Only the
,,,
Appendix, given on the last two pages of the paper, will be under discussion here. I n the modified system of Principia discussed in the Introduction, to the second edition (p. xxxix) all functions are extensional by explicit stipulation. I n this system there is, of course, little point in a discussion of the question of attributes vs. classes. " A Note on an Argument of Quine ", M ~ Dlxv , (1956), 255-258. Smullyan does not explicitly give this definition of " 4 ",but takes it as (+!x E ,+x), undefined, subject to satisfaction of the two axioms ; (A) (3+) and (B) +!x = ,$!x 3 +! = $! (this is misprinted in Smullyan's paper). However, the stated definition is the logical outcome of Smullyan's discussion, since (1)it adequately formalizes the intended meaning of " ! " as discussed by Smullyan, and (2) it satisfies the axioms (A) and (B) in virtue of *20.3 and *20.13, respectively. This explicit definition of " ! ", as based on ideas of Smullyan jnduces a fundamental change in the interpretation of the " ! " symbol in Principia, where this is undefined. For example, the axiom of reducibility, "12.1, now states not that every function (or predicate) has a first-order (formal) equivalent, but that every predicate has an extensional (formally equivalent) counterpart. 254
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Smullyan's second point is to insist that a n explicit notation be introduced in Principia for the scope of abstracts, analogous with the scope-notation for descriptions in "14.01 : (52) To re-adapt "20.01 as definition for class abstraction : [;r($x)lF{;E($x)) = D~(RX)[#X XX !x & F { !~)I.' When these innovations are introduced, Quine's line of argument is upset. For (1)now becomes, (7) I- [&($z)= &(4z)]> {gX)[$x = x~ ! & Fix !)I =
(EX)[*. =xx !X Fix Letting F{A} = ~f A = 4, we have, (8) [(7)1 I- [&($el = &(#z)l { ( ~ x ) [ $ x= xx !X& x ! = $1 = (3x)[*x = x x !x 85 x ! = $1). Since the formal equivalence of identical properties is an inevitable consequence of any acceptable definition of property equality, (9) I-(3x)[$x = x x !x x ! = $1 '(3x)(x! = $1. (10) ["12.1ja (&)I I- (3x)(x! = $1. (11) [(a), (9), (1011 W($Z) = w4l :, (3x)[*x 'xx !x 85 x ! = $1' But we cannot infer $=# from the right hand side of (11). Thus, adoption of Smullyan's proposals pulls the teeth of Quine's proof that co-extensive properties are identical.
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However, even if we adopt S1 and 52 to resolve the particular point of difficulty adduced by Quine, other, related difficulties continue to stand in the way of maintaining in Principia the class vs. attribute distinction, in the face of acceptance of (Q). Three considerations are especially relevant here : ' No. 1 : Acceptance of (Q) renders it impossible to distinguish between a property $ and its extensional counterpart !. This is shown by the argument :
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No. 2 : Acceptance of (Q) renders it mandatory to identify (qua attributes) demonstrably co-extensive properties (e.g. self-identity and non-self-difference). For if I- &($z)= &(#z), then by "20.14, I-$z =, #z, and so, by (Q), I-$=#. No. 3 : Acceptance of (Q) forces upon us awkward and unnatural consequences regarding the interpretation of Principia symbolism, if we wish a t the same time to avoid the undesirable consequence, In Smullyan's paper, there is a typographical error in that the last occurrence of the symbol " ! " is omitted. Here, and in various instances below, the starred numbers refer to propositions of Principia., not as they stand, but as reinterpreted under Smullya~n'sproposa,ls.
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N. RESCHER
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I-[.k(+z) = &(#z)]3 (+ = #). Por example, in view of the interpretation of ! " embodied in (Sl), we should like to be able to assert : (A) F(+x 5 .$x) > (+! = $!), i.e. that formally equivalent properties have the same extensional counterpart. But by *20.14, &(+z)= &(#z)implies the antecedent - of (A), and by No. 1 above, the consequent implies = #.
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IV A resolution of these additional difficulties entailed by acceptance of (Q) calls for a more drastic revision of Principia than is embodied in Smullyan's proposals. What is required is a modification in the definition of class abstraction *20.01-and correspondingly (S2) -in the following way : (D) [g(+x)lF{fi(+x)}= .f(3~)[+ !X = x x !X8~ F{x !}I. It is clear that any consequences regarding the property that are derived from considerations regarding its class abstract as defined by (D) will relate to its extensional counterpart !. In particular, if "20.01 is replaced by (D), two important theorems of "20 now read as follows : *20.14 I- [&(+z)= 6(#z)] > (4 !x = #, !x) "20.3 I- x e &(4z)= 4 !x. Note that if (Q) is applied to the second of these we merely obtain (81); and that from the first theorem and F&(+z)= &(#z), (Q) empowers us to infer no more than I-+ ! = # !. Thus, adopting (D) in place of (S2) immediately does away with all of the problems raised in section 111. And it does so without itself entailing any difficulties for the interpretation of class abstraction in Principia. Given the interpretation of " ! " embodied in (Sl), (D) appears in consonance with the express intention of the authors of Principia (p. 188) that a function of a class abstract be always an extensional function of the property that is involved, since the definiens of (D) contains only the extensional counterpart of this property.
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V Of course, another and more straightforward way of avoiding difficulties for the class vs. attribute distinction in Principia induced by the addition of (Q), is simply to reject it. After all, the authors of Principia nowhere hint a t subscription to (Q), and-as Smullyan has justly pointed out-acceptance of (Q) is indeed incompatible with positions held by them. Thus from the standpoint of the Principia the simplest way of keeping from trouble would be to reject (Q), or rather, to accept it only in the weaker form : (Q') If I-4x E X#x,then I-+! = # ! .
(Q') leads to no difficulties in interpretation (if we accept (Sl)), and does not in any way interfere with the class us. attribute distinction, either via Quine's argument or by the difficulties raised in section 111.
VJ In entertaining the proposal of section V, we abandon (Q) in favour of some other, narrower principle of individuation for attributes (properties in intension). What is its nature ? I t must surely be, in the classical manner : (1) [F{+} =?$B{$}l = (+ = 4). The properties 4 and $ are identical if, and only if they are intersubstitutable in all contexts (including intensional contexts). I t is assured by (I) that identical properties are formally equivalent. For if 4 = $, let us take F as defined by : B{X} = D f ~ x +x. Then (+x =, +x) = (#x = , +x), and therefore +x = ,#x. Moreover, (I) permits us to distinguish between attributes and classes, and specifically between 4 and !. In rejecting (Q) and accepting (Q') we take a step that makes sense only relative to the thesis, (T) (3x)(3*)[x ! = ! X $1, that is, that several distinct attributes (properties in intension) can correspond to one and the same class (property in extension). By (T) there is some property x that has the characteristic B defined : F { ~ )= D~(R#)[x! = # ! & X $1. By (T), then, ( ~ x ) ~ { x ) . Now the extensional counterpart of X,viz. x !, either has the characteristic B or it does not. Assume, on the one hand, that !}.
'Then, ( 3 $ ) [ ~! = # ! & x ! + $1, and therefore (3$)($ ! $). On
the other hand, if x ! does not possess B, then x ! + x by (I),and
indeed by any acceptable criterion of identity. Thus (I) does
not compel us to identify 4 and 4 !-to the contrary, if we are willing
to postulate (T) we are compelled by (I) not to make this
identification.
To be sure, we do not by accepting (I) as definition of identity for attributes solve any of the really complex problems that can arise in this domain. But we do at least absolve ourselves of the charge that we do not even know what we mean by the term.
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NICHOLAS RESOHER
Lehigh University
