Russell's Correspondence with Frege by David Bell

Reviews

Russell's correspondence with Frege by David Bell

Gottlob Frege. Philosophical andMathematical Correspondence. Abridged from the German edition by B. McGuinness. Translated by H. Kaal. Oxford: Basil Blackwell; Chicago: University of Chicago Press, 1980. Pp. xviii+214.* £17.50. US$3I.00.

I. Introduction

Russell was just twenty-eight, and had been intensively involved in the study ofmathematical logic for less than a year, when he discovered that certain intuitively plausible and apparently harmless assumptions, widely made by logicians, are in fact provably incoherent: they lead to the contradiction which has since come to be known as Russell's paradox. In its most virulent form the paradox concerns the notion ofa class and, in particular, the notion of a class of classes. Now, if classes are logically respectable entities, then there seems no good reason why we should not allow that there can be classes of such entities. If classes of classes are allowed then, trivially, a class may be a member of a class. But in order that the notion ofa class should not remain dangerously indeterminate, it must be decided, one way or the other, whether a class may be a member of itself. Now intuitively at least, this much seems clear: there are some classes which are not members ofthemselves. The class ofelephants, for example, is not itselfan elephant and is not, therefore, itselfa member of the class of elephants. And again, if the notion of a class which is not a member ofitselfis logically respectable, there seems no good reason why we should not talk about the class ofall such classes (we can call this class "C" for short). We must have gone wrong somewhere, however, because

*Originally published in German as Wissenschaftlicher Briefwechsel, ed. G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, and A. Veraart (Hamburg: Felix Meiner Verlag, 1976). Pp. xxviii+3IO.

159 160 Russell winter 1983-84 Russell's correspondence with Frege 161 class C gives rise to the following absurdity: ifC is a member ofitselfthen The contradiction posed so very great a threat to both Russell and itis not a member ofitselfwhile, conversely, ifC is not a member ofitself Frege for the same reason: both had, independently, identified cardinal then it is a member of itself. This is the contradiction which Russell numbers with classes of classes. Frege replied to Russell's letter very communicated to Frege, one year after its discovery, in his first letter, quickly, virtually by return of post, acknowledging that the latter's June 16th, 1902. discovery "seems to undermine not only the foundations of my arithme Sixty years later Russell described the reaction ofhis correspondent in tic, but the only possible foundations of arithmetic as such" (p. 132). the following terms: Although Frege spent some time investigating means by which the contradiction could be avoided, he eventually came to believe that Rus AsI think about acts ofintegrity and grace, I realise that there is nothing in my sell had undermined the only possible logical foundation of arithmetic. knowledge to compare with Frege's dedication to truth. His entire life's work As he wrote in his diary shortly before his death: "My efforts to become was on the verge of completion, much of his work had been ignored to the clear about what is meant by number have resulted in failure."3 Frege's benefit ofmen infinitely less capable, his second volume [i.e. of the Basic Laws reasons for this melancholy conclusion, and in particular his reasons for ofArithmetic] wasabout to be published, and upon finding that his fundamen rejecting all Russell's proferred solutions to the paradox, emerge clearly tal assumption was in error, he responded with intellectual pleasure clearly in the course oftheir correspondence-and it is on this topic that I shall submerging any feelings of personal disappointment. It was almost superhu concentrate in what follows. This means that I shall ignore those frequent man.... ! and often protracted passages in which Frege attempts-though to little avail-either to correct Russell's misunderstandings about the Basic While there is some truth in the portrait Russell presents here, the Laws or to get him to be logically more rigorous. In this latter respect sentimental, even fulsome tone is quite misplaced and serves only to Godel's verdict is incontestable: "It is to be regretted", he wrote about mask a number of distortions. It seems in fact likely that Russell never Principia Mathematica, "that it is so greatly lacking in formal precision in fully realized the terrible effect his first, short, almost chatty letter to the foundations that it represents in this respect a considerable step Frege had on its recipient. Frege's reaction was n~t"superhuman", and backward as compared with Frege. What is missing, above all, is a one can be sure that he took very little "intellectual pleasure" in the precise statement of the syntax of the formalism. Syntactical considera information which Russell communicated to him. On the contrary, tions are omitted even when they are necessary for the cogency ofproofs, Frege quickly abandoned completion of what was to have been his in particular in connection with the 'incomplete symbols."'4 Russell magnum opus, The Basic Laws of Arithemtic: he produced virtually no should have learned more than he did from Frege, but I shall not dwell on work of any sort for the next seven years; and he lived his last years as a the lessons that remained unlearned. broken, disillusioned, and bitter man. When Russell discovered the contradiction he was twenty-eight and had been immersed in mathemati 2. The contradiction communicated cal logic for less than a year; when Frege learned ofit he was almost twice Russell's age, and had single-mindedly devoted his intellectual efforts to This is how Frege first came to learn of the contradiction. Russell the development of mathematical logic for over twenty years. As Frege wrote: himself wrote, with admirable constraint: "Hardly anything more un welcome can befall a scientific writer than that one of the foundations of I have encountered a difficulty only on one point. You assert ([Begriffsschrift] his edifice should be shaken, after the work is finished. I have been put in p. 17)that afunction can also constitute the indefinite element. This is what I this position by a letter from Mr. Bertrand Russell."2 used to believe, but this view now seems to me to be dubious because of the following contradiction. Let w be the predicate: to be a predicate that cannot

1 Letter from Russell to van Heijenoort, November 23rd, 1962. Published in J. van Heijenoort, ed., From Frege to Giidel (Cambridge, Mass.: Harvard Univ. Press, 1967), p. 3 Frege, Posthumous Writings, ed. H. Hermes et aI., trans. P. Long and R. White (Oxford: 12.7· Blackwell, 1979), p. 263. 2 Frege, The Basic Laws of Arithmetic, ed. and trans. M. Furth (Berkeley: Univ. of 4 K. Glidel, "Russell's Mathematical Logic" in The Philosophy of Bertrand Russell, ed. California Press, 1964), p. 127. P. A. Schilpp, 4th ed. (La Salle, Ill.: Open Court, 1971), p. 126. Russell's correspondence with Frege 163 162 Russell winter 1983-84

be predicated of itself. Can w be predicated of itself? From either answer how the contradiction might be avoided. "The contradiction only arises follows its contradictory. We must therefore conclude that w is no predicate. if the argument is a function of the function," he writes, "that is, only if Likewise there is no class (as a whole) of those classes which, as wholes, are not argument and function cannot vary independently" (p. 133)· He cites twO kinds of case in which this may happen: first, the case in which a members of themselves. From this I conclude that under certain cir 7 cumstances a definable set does not form a whole. (P. 130) concept is predicatedoOts ownextension, viz. "cf>(x(cf>x))"; and secondly the case in which a concept is predicated ofitself: "cf>(cf»". The suggestion Apart from a brief postscript which expresses the contradiction in implicit in Russell's letter is that if both forms of expression were Peano's notation, this is all the discussion the topic receives in Russell's outlawed there would be no possibility of generating the contradictions. first letter, and I have quoted it in full in order to counter a number of Frege replies (XV/4) that the cost of rejecting all expressions of the first widespread but mistaken beliefs about the extent at this time of Russell's sort is too high; for the law ofexcluded middle would also therewith have understanding of the contradiction and of how it arises within Frege's to be abandonned. If "cf>"is a significant predicate, and if"x( cf>x)"is the system. Quine, for example, says: "Russell wrote Frege announcing name of an object, then either "cf>(x(cf>x))"or its negalion must be true, Russell's paradox and showing that it could be proved in Frege's sys tertium non datur. The only way to avoid such exceptions to the law of tem."5 And van Heijenoort has claimed that Russell correctly identifies excluded middle, Frege argues, would be to deny that, properly speak the passage in the Begriffsschrift (i.e. p. 17) which is responsible for the ing, classes are objects at all. But for Frege an object is that which can be "flaw in Frege's system".6 In fact, however, Russell demonstrates the reference ofa singular term, and a singular term has reference just in neither that, nor. where, nor how Frege's work is inconsistent. The case it can participate in direct discourse which possesses a determinate passage in the Begriffsschrift to which he alludes, and which indeed truth-value. That Classes are proper objects is thus for Frege an im mentions "indeterminate elements" ina judgment, is not at all what mediate consequence of the fact that names like "x(cf>x)" can occur in gives rise to the contradiction. On the contrary, and ironically, it is at sentences possessed ofa determinate truth-value-for example, trivially, precisely this point in the Begriffsschrift that Frege introduces, for the in "x( cf>x)= x(cf>x)". As to Russell's suggestion that one avoid expres first time in the history of logic, the notion of a second-level function sions of the second, self-predicative sort, Frege has already pointed out whose arguments are always first-level functions and whose values are that a correct logical synatax will not allow them in the first place. either true or false-the notion, in other words, of a quantifier. It is precisely this syntactic distinction between proper names, first-level 4. Russell's second suggestion function-names, second-level function-names, and soon which prevents Russell's paradox of self-predication from infecting Frege's logic. As Inauspiciously, Russell begins his next letter to Frege by saying: Frege himself says in his first letter to Russell: "the expression 'a predi "Concerning the contradiction I did not express myself clearly enough. I cate is predicated of itself' does not seem exact to me. A predicate is as a believe that classes cannot always be admitted as proper names [sic]. A rule a first-level function which requires an object as argument and which class consisting of more than one object is, in the first place~not one cannot therefore have itselfas argument" (p. 132). Frege's logical syntax object but many" (p. 137). Despite his chronic failure to distinguish ruled that expressions of the form: "cf>(cf»"are simply malformed. And between names and their bearers, and despite the obscurity of his con because the Begriffsschrift employs no set-theoretic apparatus the trast between classes as one and classes as many, the gist of Russell's paradox cannot be derived within it. (Van Heijenoort is further mistaken second suggested solution to the paradox is clear enough. If we restrict in that he takes Russell to be alluding to a quite different, but equally consideration to classes with more than one member then, Russell harmless, doctrine which appears on page 19 of the Begriffsschrift.) claims, all classes are necessarily classes as many, whereas not every such class is also a class as one, i.e. a unitary whole whose members comprise 3. Russell's first suggestion its parts. The contradiction, it is claimed, arises as a result oftaking a

7 At this time, unlike Frege, Russell had no notation for classes. Here I shall use Russell's Russell's next letter (XV/3) contains his first, tentative suggestion as to later notational device: "xCcPx)"is the name of the class ofall cPs.What differences there are between sets, classes, extensions ofconcepts, and ranges ofvalues ofconcepts are not s W. V. Quine, "On Frege's Way Out", Mind, 64 (1955): 147. here relevant, and I have used these terms interchangeably-as do Russell and Frege in 6 Van Heijenoort, From Frege to Gbdel, p. 3. their correspondence. .- a

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but also a range of those values for which ¢x is decidable, or for which it has a We then have rew ."". r""ew. sense. (P. 145) Here we must consider the content of the propositions, not their meaning; and we must not take equivalent propositions to be simply identical. (P. 147) Frege reacted to this su,ggestion without enthusiasm, pointing out that the law of excluded middle will still be threatened unless (a) precise Before we can begin to extract an argument from this it is necessary to syntactic rules can be formulated which not only perspicuously determine clear away a couple of confusions on Russell's part which make it the well-formedness or otherwise of every possible function name when extremely difficult to establish exactly what he meant. In the first place, its argument-place is filled by an expr~ssionoftype-O, type-I, type-2, and Russell subscribed at this time to a faulty account ofthe logical product of so on; and unless (b) a value is specified for every function for every level two or more propositions: the definition of "logical product" which he of class as permissible argument. Frege believed,that this could not in provides at §18 of the Principles, and which he subsequently employs in practice be done. But even were it in theory possible, he believed the the discussion of the new antinomy at §soo, simply does not work. This, suggestion4was also objectionable because it would threaten the generality however, is merely a technical slip; we can assume that Russell meant by of arithmetical truths, and their fundamentally logical nature. Not only "the logical product of p and q" what we now mean by the phrase, i.e. would extraneous and largely ad hoc devices be introduced into logic, but what today would be symbolized as "p&q". In 1903, however, Russell also the numbers ~ouldneed to be defined afresh at each level in the type did' not employ any sign corresponding to "&", but represented the hierarchy. logical product of propositions by immediate concatenation: "pq". This notation no doubt encouraged him in the second, much more serious 6. A new contradiction confusion ofwhich he was guilty, namely identifying the class of proposi tions p and q with the logical product of those propositions: to someone "My proposal concerning logical types now seems to me incapable of who believed that a class was "composed of" its members, doubtless pq doing what I had hoped it would do", Russell wrQte (XV/II) a few days seemed very similar to {p, q}. In this way Russell confused a singular after receiving Frege's objections. It is clear, however, that it was not the term, a class name, with a complex proposition-and this helps to latter that caused Russell's change of mind. Rather, he had discovered, explain the peculiar opening sentence of the above quoted passage. In he believed, yet another contradiction-;one moreover whose derivation fact it was precisely this confusion of a class of propositions with the even adoption ofa theory of types for classes would fail to block. The new logical product of those propositions which enabled Russell to apply antinomy is not blocked because it concerns, not classes of classes, but Cantor's paradoxical results to the case in hand. This is surprising, for classes of propositions. the antinomy bears a striking resemblance to the Paradox of the Liar and The text of Russell's letter is dense and extremely obscure, and al was, indeed, later handled by Russell in that guise.9 It seems, however, though some light is shed on it by §sooofthe Principle.s where the same that at this time Russell had yet to see any link between the modern, contradiction is also mentioned, this section too is very far.from clear. Camorian mathematical contradictions and the ancient antinomies: there For this reason, then, and also because a number of subsequent letters is, for example, no mention of the Liar either in the correspondence with are devoted to the new antinomy, it will be worthwhile attempting to" Frege, or in the first edition of the Principles. spell out what Russell had in mind. This is how Russell expressed the Putting Russell's confusions to one side, then, his argument would problem to Frege: ' seem to be as follows. Let mbe a class of true propositons,i.e. such that

If m is a class of propositions, then "pem .:J". p" represents their logical a: pem ,~p.p product. This proposition itselfcan either be a member ofclass m or not. Let w be the class ofall propositions of the above form which are not members of the but, also, such that pertinent class m, i.e., b: ,y (aem);

w =p3(3m3{p .=: qem .:JQ • q:.p_em});

9 Russell, "Mathematical Logic as Based on the Theory of Types" in R. C. Marsh, ed., and let r be the proposition pe~.:J". p. Logic and Knowledge(London: Allen & Unwin, 1956), pp. 59, 62, etc.

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.-v(ceu);

rv(dev).

the

that

truth-value

raise

not

the

merely

two

the

generated.

classes

which

examine

ofthoughts,

the only

proposition all

between

rather

is,

winter

is,

treated names,

is,

truth-value, nature

does

remaining

is taken

nature

the

verison

(rew).

member

identical.

now say,

distinguish

besides.

its

class

validly objection

the

from

them

a the

contain

are

that

words

that

the

thoughts

exact

for

can

form Russell

proper

his

else

can

.=.

antinomy

also

thought,

itself

long,

other

The

as 168

same

not as

will

appropriate

given,

rew

much

during jects,

contradiction

example,

prompts

much

new

sentence,

tions failure

effect

stands contradiction

for

particular

general between

antinomy

cannot

class difference, elephants. 170 Russell winter 1983- 84

8. Conclusions

The twenty letters which passed during this period of just over two years between Frege and Russell together comprise a rich Source of historical information and philosophical stimulation-by no means every aspect of which has been touched upon in this review. In particular it is fascinating to witness the speed and inventiveness with which Russell was able to suggest radically new solutions to the contradictions. Indeed, we have in these letters some of the very earliest, tentative, and often naIve formulations ofdoctrines which were later to become central tenets of Russell's mature philosophy: the theory of types, for example, the doctrine that the meaning of a proper name is the object it refers to and hence that names do not express a sense, and also the "no-class theory". In this respect the collection of letters will be of greater use to those studying Russell than it will be to those studying Frege: throughout the correspondence Frege steadfastly continued to defend the doctrines which he had already published. Indeed it appears that as a result oftheir exchange of ideas Russell managed to change Frege's mind about only one major point. This was no small thing, however; for what Frege came to see was thathis life's work was in ruins. Theletters together provide us with a portrait of a genius who, though he was capable of winning virtually every battle, finally and tragically lost the war.

Department of Philosophy Sheffield University