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Home , Bernard Bolzano, Infinitism, Logicism

Notes

Introduction

1 . Not that I am the first to discover this. The early ‘realist’ or ‘post-idealist’ Russell has been the of much first-rate scholarship in recent years: see, e.g., Hylton (1990a), Landini (1998), Makin (2000), Stevens (2005) and such collections of papers as those of Irvine and Wedeking (1993), Monk and Palmer (1996), Griffin (2003) and Griffin and Jacquette (2009). These works are examples of the scholarly interest that has arisen in ‘early analytic ’. See Floyd (2009) for an overview of this development. 2 . I borrow the term ‘analytical modernism’ from Skorupski (1993). 3 . I owe this characterization of Russell’s and McTaggart’s relationship to Nicholas Griffin. 4 . This story is told in detail in Griffin (1991). 5 . Bradley (1911, 74).

1 Russell’s Early : What Was It About?

1 . See Frege (1880/1; 1882). 2 . Cf. Tiles (1980, 158). 3 . The question became more acute with the discovery of paradoxes and the complications it necessitated in the underlying the reductionist ambi- tions. Indeed, it seems that the why Frege gave up on logicism was that he found himself unable to regard the ensuing complications as a of logic . Russell was in this respect much more flexible, and at least in the case of the theory of types, he occasionally pressed the point that the theory was really ‘plain ’ (1924, 334). However, the question has been there from the beginning of modern logic. 4 . See Russell (1944, 13; 1959, 74–5). 5 . It is good to remember that the term ‘logicism’ – or its German equivalent, ‘Logizismus’ – was introduced in something like its modern sense only in the 1920s, when Abraham Fraenkel and Carnap used it, independently of one another, to denote a certain position in the philosophy of (this derives from Grattan-Guinness 2000, 479, 501). This use became commonplace when (1931) drew the threefold distinction involving logicism, and formalism, applying the first term to Frege and . The German term ‘Logizismus’ occurs also in Theodor Ziehen (1920), but there it is used to refer to the view that there is a world of abstract objects; logicists in this sense would include such figures as , , and as well as Frege. 6 . To speak of the logical empiricists’ views on mathematics as one logicism is to commit oneself to an oversimplification that is itself yet another instance

238 Notes 239

of the one that is discussed in the text. Ayer, for instance, was interested not so much in logicism and its prospects as in maintaining the analyticity (and hence apriority) of mathematics. Indeed, the or otherwise of logicism was of little importance to him, as he claimed that on his conception of analy- ticity – the criterion for the analyticity of a that ‘its validity should follow simply from the definitions of the terms contained in it’ (Ayer 1936, 109) – the of pure mathematics will be analytic whether or not logicism in the sense of Whitehead and Russell turns out to be correct. 7 . Kant’s Critique of Pure Reason (KrV ) will be referred to in the standard manner, by citing the relevant page or pages of the appropriate edition, A (1781) or B (1787). 8 . To make this complete and convincing, one would have to explain the of this ‘cannot fail to recognize’. A familiar suggestion is to argue that the condition for possessing a certain is the accept- ance as true of a range of propositions involving the concept. If viable, this formulation would explain in a non–ad hoc manner why there can be no gap between possessing a concept and recognizing a certain truth. As we shall see in Chapter 3 , Kant applied to synthetic a priori propositions an explanation that bears some similarity to this view. 9 . The explanation was partial, because it had to be complemented by an account of the nature of language and our of it that can be deemed acceptable by the logical empiricists’ standards. But here they could refer to and behaviourism. 10 . It is significant, though, that Frege uses the concept of analyticity only in his Foundations of Arithmetic (1884). 11 . This lack of interest has been noted by a number of authors, including Taylor (1981), Coffa (1982) and Hylton (1990a, 197; 1990b, 204). 12 . This point is made very clearly by Goldfarb (1982, 693). 13 . This development is explored in some detail in Detlefsen (1996). 14 . The term ‘epistemic logicism’ is borrowed from Irvine (1989). 15 . See Wood (1959). 16 . For example, according to Kant, geometry is a synthetic science, but there are a few ‘fundamental propositions’, such as the proposition that ‘the whole is greater than its part’, which are ‘identical propositions’ and serve therefore only as ‘links in the chain of method’ (KrV , B16–7).

2 Kant and Russell on the Mathematical Method

1 . The term ‘semantic’ must be construed here in a broad sense (cf. Coffa 1991). In this sense, anyone with an interest in ‘representation’ or ‘content’ is engaged in semantics. Semantics, as here understood, is not necessarily tied to any language, even less to any formal language; such associations emerged only in the 1920s and 1930s, when certain and logi- cians effected the so-called linguistic turn. Instead of ‘semantic’, I will occa- sionally use the more neutral term ‘representation-theoretic’, introduced by Richardson (1998) for a similar purpose. 2 . The ‘external’ reading can be found, for example, in Hintikka (1967; 1969) or Brittan (1978). 240 Notes

3 . Admittedly, there is in Kant a different use of ‘content’ that relates to objec- tive validity; have contents only insofar as they can be given objects in , a claim that applies to mathematics, too ( KrV , A239/B299). Such usage by Kant, however, does not justify the omission of the semantic or representation-theoretic sense of ‘content’. In this sense the of ‘content’ is comparable, for example, to Frege’s of ‘judgeable content’ (Frege 1879) or the notion of ‘proposition’ which Russell introduced in PoM 4 . For similar passages, see KrV (B154–5) and Kant (1783, §12). 5 . A detailed treatment of this is found in Friedman (1992a). It is quite essential to Kant’s conception of geometrical that it is based on a general rule rather than an of a single figure; that it operates with a schema , a procedure for generating a figure, rather than the figure, which is nothing but an image of a geometrical concept and, as such, an empirical (KrV , A140–1/B180). One can refute Kant by ignoring the role of rules, which leads to the allegation that he confused geometrical concepts with subjective mental images (cf. Couturat 1904). On the other hand, having observed the role of rules, a critic may press the objection that this makes geometrical reasoning analytic rather than synthetic. When we prove a geometrical theorem, we do not observe the diagram and draw conclusions from this; we look at the rule to see what must hold for all objects falling under the generated concept. Thus the rule may look more conceptual than intuitive, at least in the sense that it is a general representa- tion; for this kind of objection, see Bolzano (1810, Appendix, §7); Couturat (ibid., 348–52); Kitcher (1976, 124). This, though, overlooks the that the only way to ‘look at’ a geometrical rule is to draw a figure and perform a series of further constructions on it. 6 . See Kant’s notes on Kästner in Allison (1973, Appendix B, 175–6). 7 . J. G. Schulze, Prüfung der Kantischen Critik der reinen Vernunft (‘Examination of Kant’s Critique of Pure Reason ’); quoted in Allison (1983, 94). 8 . As Friedman points out, Kant’s notion of pure intuition, understood in kinematic terms, ‘contains the seeds of its own destruction’ (2000, 202). The kinematic conception of pure intuition amounts to the view that the formal structure of spatial intuition (possible spatial ) is expressed by the conditions of free mobility and that these conditions are uniquely captured by Euclidean geometry, which is why it is a priori. After the discovery of non-Euclidean geometries, however, it became clear that these conditions do not yield the specifically Euclidean space but the three classical cases of space of constant curvature. It follows that, insofar as ‘spatial intuition’ or ‘the a priori element in geometry’ is tied up with free mobility, Euclidean geometry can no longer be regarded as a priori. This is one of the conclusions that Russell argued for in An Essay on the Foundations of Geometry , although his were quite different. 9 . Kant’s constructive semantics not only covers geometry but extends to arith- metic, analysis and algebra as well (although it is often not very clear what is going on in these extensions). The case of arithmetic will be mentioned below. For a general survey of Kant’s theory of mathematics, see Shabel (2003). She argues convincingly that Kant’s theory is best understood when it is considered against the background of what may be called ‘early modern mathematics’. Notes 241

10 . ‘Platonic atomism’ is the term used by Hylton (1990a). 11 . Some of the issues that are relevant here will be explored in Chapter 4 . 12 . Russell’s letter to Moore, 16 August 1900, quoted in Russell (1992b, 202). 13 . In the Paris Congress Russell attended a session in which Peano and his colleague Mario Pieri gave talks about definitions in mathematics. Ernst Schröder, a German logician, was present and a discussion ensued about the role of the definite article in mathematical definitions. The dispute made a huge impression on Russell. This event, though, was not Russell’s first contact with ‘symbolic logic’. He was familiar with the Boolean tradition, both in its original form and with the extensions introduced to it later by Peirce and Schröder and others, but his judgment was negative: ‘Until I got hold of Peano, it had never struck me that Symbolic Logic would be of any use for the of mathematics, because I knew the Boolian stuff and found it useless’ (letter to Jourdain, 15 April 1911, quoted in Grattan-Guinness [1977, 133]). 14 . One might have expected Russell to mention Frege. Russell had received a copy of from his teacher James Ward (Russell 1978, 65) and in autumn 1900 he was sufficiently familiar with Frege to be able to write that he has ‘acquired almost none of the great credit he deserves’ (1901b, 352, fn. 1). However, Russell did not begin a serious study of Frege until summer 1902, after he had sent the manuscript of PoM to the publisher; cf. Russell’s letter to Frege, 15 June 1902 (Frege 1980, 130). 15 . From the first edition of the Formulaire de Mathématiques ; quoted in Kennedy (1980, 49). 16 . Cf. here Kitcher (1981). This characterization of rigorous reasoning in terms of ‘elementary validity’ is the standard one. Other conceptions are possible. For example, according to Detlefsen (1992; 1994), Henri Poincaré’s conception of mathematical involves an entirely different approach to rigour in reasoning. In Poincaré’s view, Detlefsen argues, genuinely mathematical reasoning is one that reflects mathematical understanding. No doubt, one could define some notion of elementary validity on the Poincaréan approach, too, but such an approach would not seem to reflect what is germane to it, as Detlefsen develops it. 17 . We may look at the two requirements – that reasonings should be both locally and globally rigorous – from a somewhat different perspective, in such a way that, taken together, they capture the idea that a of reasonings should be rendered maximally explicit ; that is, that everything the conclu- sion is, logically speaking, dependent upon is made visible, as it were. This implies different treatments for the different ingredients that are relevant for assessing this dependence – the ultimate and intermediate for the conclusion, the rules of inference that one uses to reach the conclusion and, possibly, other elements. 18 . The necessity condition amounts to the requirement that should be independent. This feature was extensively studied by Peano and his associates, as Russell (1901b) points out. The sufficiency condition is to be understood in the sense of ‘experimental completeness’; Russell (1901b, 359) argues that the proof that the primitive propositions are sufficient is a ‘mere question of careful deduction’. In PoM , the point is formulated as follows: a given branch of mathematics begins with ‘certain axioms or 242 Notes

primitive propositions, from which all ordinary results are shown to follow’ (§120). 19 . In the preface to Grundgesetze , Frege criticizes Dedekind for such a failure in explicitness. Frege observes that Dedekind had succeeded in pursuing the foundations of arithmetic much further and in a much shorter space than Grundgesetze ; but that was only because in Dedekind ‘a great deal is not really proved at all’ but merely ‘indicated’ (Frege 1964, 4). 20 . For the nineteenth-century developments in algebra, see Cavaillès (1962); for geometry, see Nagel (1939) and Freudenthal (1962). 21 . The translation is taken from Nagel (1939, 237) with slight modifications. 22 . This change is usually associated with Hilbert (1899), but it is clearly observ- able in the works of Peano and his associates. 23 . This condition is one that Euclid fails to fulfil. His proofs often depend for their validity on the intended meanings of the geometric primitives, which are made evident by corresponding figures (Pasch 1882, 43–4). Pasch illus- trates this point by the very first proposition of Elements , the construction of an equilateral triangle on a line segment as base (ibid., 44–5). Russell (1901a, 378) uses this same example. 24 . Detlefsen (1994, 278–80) argues that Pasch’s conception of rigour is in fact not available to the logicist. According to Detlefsen, that conception is built on the assumption that rigour is a matter of information control , a goal that is to be facilitated by forcing all the information contained in an axiomatized theory into explicitly stated premises; that is, by emptying all inferences of information. In this way the logical or deductive component of the theory becomes wholly informationless. But this conception of the logical is not available to the logicist, Detlefsen argues, for otherwise he could not answer the ‘Kantian challenge’ of explaining how mathematics can be more than a vast , even though its inferences are purely logical. It seems wrong to argue, however, that modern mathematics’ concern with rigour presupposes specifically an alignment of the logical with the non-informative. For one could share that concern but implement it with the help of a distinction between topic-sensitive and topic-neutral content/ information, aligning the former with premises and the latter with infer- ences; this distinction would be neutral on the question whether the logical (topic-neutral) is informationless simpliciter or just devoid of topic-sensitive information. Of course, someone who is logicist about a theory, T , holds that T has no topic-sensitive content distinct from its logical content; hence a logicist cannot draw the distinction between T ’s descriptive component and its deductive component in terms of ‘topicality’. Nevertheless, the point remains that a logicist and a champion of the mathematical notion of rigour need not differ on whether the logical is uninformative. 25 . The standard interpretation can be found, for example, in Beck (1955), Hintikka (1969a) or Brittan (1978). 26 . Here the term ‘’ is used as a general term for any kind of reasoning that is necessarily truth-preserving. Kant, of course, knew very well that not all reasoning is ‘logical’ (ordinary induction, for example, is not), but the present point is that, according to him, there are types of inferences where the conclusion is a necessary consequence of the premises but where this necessity of consequence cannot be captured by the rules of formal logic. Notes 243

27 . Richardson (1998, 116–22; 128–9) gives a detailed description of Cassirer’s criticism of the new logic. 28 . To be sure, the logicist Russell was not much interested in the conditions of possibility of anything; for him this notion was a confused admixture of psychological and logical elements. The conditions that make math- ematical reasoning possible in the Kantian sense, however, are not just conditions of grasping the content; they are as much conditions of the content itself. Even if vitiated by ‘psychologism’, the Kantian notion does have a legitimate non-psychologistic core, which is constituted by the representation-theoretic or content-related issues that we have discussed in this chapter. 29 . This corresponds, more or less, with what Kant called the ‘pure schema of magnitude’. In Kant, any class or multiplicity of objects to which a given number applies serves as an image of the number; more fundamental than this is the schema or the unity underlying any actual counting, which is a representation that unites the successive addition of unit to (homogeneous) unit’ (KrV , A142/B182). Since ‘successive addition’ is a temporal notion, the general concept of number is temporal as well. 30 . Here I follow the explanation given by Griffin (1991, 231, fn. 3). 31 . This is explicit in Russell (1897b, §56). 32 . The manuscript is in fact the first work by Russell that shows the influence of Moore; see Griffin (1991, §7.2). And yet Russell described the work to Couturat as follows: ‘I am asking the question from the Prolegomena , “Wie ist reine Mathematik möglich?” I am preparing a work of which this ques- tion could be the title, and in which the results will, I think, be for the most part purely Kantian’ (quoted in Russell 1990, 15). 33 . Bearing this in , it is easy to see why the difficulties Kantians had in accepting the actual infinite were usually formulated as reasons for resisting the postulation of entities exhibiting actual . For example, in 1897 Russell had argued in his review of Couturat’s De l’infini mathématique that ‘one would have supposed that the condition of being a completed whole, which he [Couturat] has urged as necessary to number, would have precluded the possibility of infinite number’ (Russell 1897c, 63). Russell is willing to admit the intelligibility of Couturat’s contention that a collection is given as a whole as soon as its defining condition has been given, saying that this is ‘the only hope of saving infinite number from ’ (ibid.) He clearly recognizes that this move presupposes the availability of a non-Kantian strategy for defining ‘number’: number, according to Couturat, requires not successive enumeration but simultaneous apprehension (ibid., 63). For Russell this will not do, however, for any condition that could be so used involves or presupposes an ‘addition of elements’ (ibid., 65–6). From this it follows that ‘mathematical infinity consists essentially in the absence of totality’ (ibid., 66). In this way, using Kantian , Russell is led to conclusions that directly contradict Couturat’s ‘infinitism’. 34 . The anti-Kantian suffered a serious drawback when Russell discov- ered his paradox, showing that not all propositional functions define classes and, hence, that the route from propositional functions to classes cannot be as straightforward as he after his first contact with Peano’s logic. I say a little more about this in Chapter 5 . 244 Notes

35 . For Russell the distinction between the two forms of predication is the distinction between class inclusion and class membership. He criticizes Schröder’s theory for a failure to observe this distinction (1901b, 355). Schröder’s approach, Russell explains, amounts to treating individuals as infima species and classes (collections) as ‘sums of individuals differing inter se ’. This, though, makes a consistent theory of infinite classes impossible. For example, real numbers between 0 and 1 form a class, and of any object we can say whether it is a member of the class – that is, whether it satis- fies the propositional function defining the class – but this class cannot be exhibited as a sum of individuals (ibid.). 36 . See, for instance, Beck (1955, 358–9). 37 . For instance, Hintikka’s (1967 and elsewhere) equation of intuitivity with individuality is obviously not undermined by the logicization of proof. There are also senses of ‘intuitive’ – other than ‘a priori knowledge of space and ’ – that are , apparently anyway, undermined by logicism and rigour. In particular, there is the familiar sense of intuitive as self-evident It is a common view that the point behind the rigorization/logicization of proof was to dispense with intuitive proofs in this sense. As we have seen, however, it would be more correct to say that by rigorization of proof self- is brought under proper control rather than dismissed. Kant believed that mathematical proofs were perfectly rigorous and, hence, that ‘self-evidence’, insofar as one wished to apply this notion to mathematical proofs, was under proper control, because judgments of self-evidence had a perfectly respectable source; namely, construction in pure intuition.

3 Russell on Kant and the Synthetic a priori

1 . See Prichard (1909, 72–5); cf. Allison (1983, 5–6). 2 . Of course, truth is not a privilege enjoyed only by a priori propositions. Nevertheless, it is reasonable to include it among the consequences of apri- ority, because the explanation of why the propositions are true no doubt reflects their apriority. 3 . Moore argued that confusion between, in our terminology, the psycho- logical and truth-conditional readings played a significant role in Kant’s argument for transcendental . Kant, Moore tells us, failed to distin- guish clearly the idea that the mind gives objects certain properties from the entirely different idea that ‘the nature of our mind causes us to think that one thing causes another, to think that 2 and 2 are 4’ (1903–4, 135). Clearly, it is the former idea that is more congenial to Kant – after all, if our mind does give some attributes to some objects, then these attributes do truly characterize the objects – but Moore, for one, has difficulties in compre- hending it: ‘No one, I think, has ever definitely maintained the proposition that the mind actually gives properties to things; that, e.g., it makes one thing cause another, or makes 2 and 2 = 4’ (ibid.). Hence he suggests that Kant, too, was capable of maintaining the truth-conditional reading only because he confused it with the more sensible view that our mind is so constituted as to make us think in a certain way. 4 . See Moore (1903, chapter 4; 1903–4). Notes 245

5 . There are good reasons to think that Russell’s reconstruction misses the true point behind Leibniz’s reasoning. As Robert Adams (1995, chapter 7) has shown, Leibniz’s deduction of God from the of eternal is best seen as relying on the plausible and quite intuitive idea that truths (and possibilities) must be based on something in virtue of which they are true (possible). Understood in this fairly innocuous way, there is no reason to think that Leibniz’s inference implies a reduction of truth to knowledge. As Adams (ibid., 178–80) shows, Leibniz by no means neglected ‘Being’; on the contrary, he argued against the view that eternal truths are true in virtue of the existence of Platonic entities, on the grounds that the objects of math- ematics and logic are not the sort of entities that could subsist on their own. (I am grateful to Markku Roinila for drawing my attention to Adams’s work on this point.) The ontologically committal reading of ETJ is more or less forced upon Russell. He is prevented from formulating the issue in the more plausible terms of truth making, because there is no room for this notion in his . The notion of truth making presupposes that some distinction is drawn between truth makers and truth bearers, but – as we shall see in Chapter 4 – Russell’s notion of proposition effectively excludes any such distinction. 6 . ETJ is not the whole truth about Moore’s diagnoses of why Kant went wrong. Ultimately, the that underlies the reduction of truth to knowledge is not really ETJ but some more general principle creating an appropriate connection between truth and knowledge; it is not specifically proposi- tions with a ‘non-existential’ subject matter that create the problem of the synthetic a priori for Kant, although there are such propositions among the problematic cases. According to Moore’s diagnosis (1903, §78), the Copernican revolution rests on what he calls the ‘epistemological method of approaching metaphysics’ and which says that ‘by considering what is “implied in” Cognition – what is its “ideal” – we may discover what proper- ties the world must have, if it is to be true’ (ibid.). It is the theoretical coun- terpart of the principle that holds that ‘by considering what “is implied” in Willing or Feeling – what is the “ideal” that they presuppose – we may discover what properties the world must have, if it is to be good or beautiful’ (ibid.). Russell does not consider such details to explain by what principle he thinks a Kantian moves from truth to knowledge; but some principle there must be to back up the reduction of propositions about the former to ones about the latter. 7 . Russell does not always pursue the objection in terms of necessity. Indeed, his modal scepticism is well known (see, e.g., Russell 1905a), and therefore it may not have been its failure to account for necessity that most troubled him with the Kantian theory of the synthetic a priori . However, the problem with the mere matter-of-factness of our mental constitution can be formu- lated in non-modal terms, and hence the difficulty remains. The argument would then run as follows: This constitution is a ‘fact of the existing world’ (Russell 1912a, 49), and there is no assurance that it will remain constant. The subject matter of a priori propositions, on the other hand, typically does not permit such variation. If Kant’s theory were correct, our nature might change so as to make two and two become five, but this is absurd, and hence such truths as two and two are four cannot be explained in the 246 Notes

manner of Kant. I shall continue to phrase the objection in terms of neces- sity, however, as this is appropriate for the purposes of the present chapter. 8 . The difficulty that Russell points out is familiar from more modern discus- sions of the metaphysics of modality. It constitutes one-half of a quite general dilemma for any broadly truth-conditional account of necessity. As Blackburn (1986, 120–1) explains, our curiosity of why q is necessary may be satisfied by a ‘local proof’ of q from p . Such a proof may be satisfac- tory to the extent that we take ourselves to understand why p is necessary. Philosophers, however, are usually concerned not with instances but kinds of necessity. Hence, we must shift our attention to the putative source of the necessity of the kind in question. And here we are caught in a dilemma. Insofar as our explanatory is itself necessary, our proposed explanans is threatened by circularity. On the other hand, if we seek to explain a kind of necessity by ‘mere ’ (this being Russell’s point), our explanation seems to undermine, rather than elucidate, the necessity in question. 9 . For the notion of merely relative modality, see Hale (1996). 10 . As Hale (2002, 281) explains, this additional characterization of a modality as merely relative is needed because every necessity qualifies trivially as rela- tive on the definition of relative necessity given in the text. In particular, since logical truths follow from the empty set of premises, they are conse- quences of any collection of true premises. Nevertheless, though relative in this sense, logical necessity is not merely relative but is absolute , on almost any sensible characterization of logical necessity. 11 . This is more or less how Broad (1978, 7) puts it. 12 . But see Chapter 2 , fn. 28 for considerations that are relevant on this point. 13 . Brook (1992) is an illuminating discussion of this point. 14 . This point can be formulated in terms of the concept of consequence. The very definition of ‘relative necessity’ presupposes that there is available some notion of absolute necessity. A proposition q is necessary relative to another proposition, p , if and only if q follows from p in the sense that excludes the possibility of p ’s being true and q ’s being false; if this exclusion were not absolute, we would not have a well-defined notion of consequence to put to use in our definition of relative necessity (this point is made in Hale 1994, 317, fn. 29). Hence it turns out that the ‘possible worlds’ interpreta- tion of Kant does presuppose that he was in a position to invoke an absolute notion of necessity. This constitutes an important objection to considering Kant’s notion of real necessity as a species of relative modality. For there are compelling reasons to think that the notion of formal logic that was avail- able to Kant is far too weak to sustain its use in a fully general definition of consequence that is needed in a general definition of relative necessity. 15 . Martin (1955, 25; italics added). 16 . Kant makes the point in KrV , B72. 17 . Note how Schulze first explains ‘absolute necessity’ as a conception that every thinking being has (must have?) and then moves to a characteriza- tion that is closer to our previous one: space, as it is ‘originally’ given to us, is absolutely necessary, because we cannot make any sense of the idea that it might have been different. Thus, Schulze clearly thinks that the fundamental propositions characterizing space are absolutely necessary in the sense that there is no sense of possibility in which any of them could Notes 247

have been false; and this is so because every attempt to frame such deviant yields what can only be described as nonsense or, perhaps, as a ‘non-thought’. I shall return to this point below. 18 . The following paragraphs draw heavily on Brook (1992). I think that in this connection it is only fair to bracket the fact that Schulze in the above passage, like Kant in the Transcendental Aesthetic, moves quite freely from ‘representation of space’ to ‘space’ and back, as if these were interchangeable terms. 19 . Schulze’s formulation is presumably an allusion to the Transcendental Aesthetic, A24/B39: ‘We can never represent to ourselves the absence of space, though we can quite well think it as empty of objects’. Kant takes this to show that (the representation of) space cannot be derived from expe- rience but must be considered a ‘condition of possibility of appearances’ (ibid.). 20 . There is also the following characterization of transcendental proof by Kant, which is extremely revealing in the present connection; such a proof, Kant argues, ‘proceeds by showing that experience itself, and therefore the object of experience , would be impossible without a connection of this kind’ (KrV , A783/B811; emphasis added); that is, a connection which trades on the possibility of experience. 21 . Again, my discussion is indebted to Brook (1992). 22 . Admittedly, Kant himself was less than fully clear about the differences between 1. and 2. and 3. Brook (1992) makes the important observation that there is a notorious slide in how Kant formulates the central question of the first Critique . In the introduction he writes as if the question to which he is going to give an answer is, How can there be judgments which are synthetic and nevertheless necessary/a priori knowable? Once he sets out to answer the central question, however, he in fact reformulates it so that it is no longer about synthetic judgments but about us – that is, about our ability to make judgments with the relevant characteristics (Broad 1978, 5–8, makes a similar point). Clearly, insofar as our focus is on the latter question, it is not prima facie impossible that a legitimate answer to it might be along the lines suggested by 1. However, if our answer does take this form, it looks like a mere confusion to assume that it also answers the first question or that there is a legitimate transition from 1. to 2. – or even 3. However, as Brook (1992) shows, not everything that Kant has to say about the topic is attribut- able to this confusion. 23 . There is also the further point that if we insist that exactly one of several logically permissible alternatives is really possible, that may look like a brute fact about us and our psychological constitution. To take a concrete example, many neo-Kantians of the late nineteenth century advanced the view that even though non-Euclidean geometries are possible in some abstract math- ematical sense, only Euclidean geometry is ‘really possible’, because it is the only geometry that we can ‘imagine’ (see, e.g., Land 1877). Apart from all other objections to this idea, there is the obvious point that such a claim, even if it is true, is no more than a ‘psychological’, empirical and contin- gent fact about us and what we can and cannot do; and this conclusion is only enhanced by the fact that imaginability was usually understood in some fairly concrete, down-to-earth sense. Since this kind of defence 248 Notes

of Kant was fairly common in the nineteenth century and since we know that the young Russell was familiar with it (see Griffin 1991, section 4.2) it seems not unlikely that Russell’s psychologistic or subjectivist reading of Kant owes something – probably a great deal – to it. 24 . This account of necessity presupposes that p really is constitutively related to our geometrical thought; that our geometrical thought really does flow – in part at least – from a rule that commits us to accepting p . Subsequent developments in geometry (and elsewhere) have of course decisively under- mined Kant’s specific candidates for synthetic a priori propositions. What remains unclear, perhaps, is the further question of whether this shows that Kant’s explanatory strategy itself is ill conceived or whether the mistake lies only in the specific identifications of synthetic a priori principles that he made. I shall not explore this issue here. 25 . Kant’s notion of pure general logic is explored in some detail in Chapter 5 26 . This does not mean that a notion of a thing not conforming to the condi- tions of experience could not have a positive function elsewhere in Kant’s philosophy. But this presupposes that such notions are postulates and not thoughts with a truth . 27 . This, incidentally, is the most plausible reading of Kant’s famous discussion of the concept of a biangle (KrV , A220/B268). Kant says that the concept of a figure which is enclosed within two straight lines contains no contradic- tion, because the concepts of two straight lines and of two straight lines coming together ‘contain no negation of figure’. This concept, however, is ruled out by the ‘conditions of space and its determinations’ (ibid.) Some scholars have suggested that this passage shows that Kant appreciated the fact that non-Euclidean geometries are consistent; this is how Brittan (1978, 70, fn. 4) and Martin (1955, 23–5) read it. More likely, Kant’s point is just what he says: no contradiction can be derived from the concept of a biangle itself; hence, that concept is possible in the negative and minimal sense. 28 . This question is raised, for example, in Kitcher (1986). 29 . Allison (1983, 6–13) develops this line of thought. He criticizes the standard picture for failure to observe that the crucial Kantian distinctions between ideality and , on the one hand, and appearances and things in themselves on the other, can be understood empirically or transcendentally. Observing this distinction puts one in a position to argue that whether (say) a proposi- tion p about space is mind-dependent depends upon which of the two ways it is construed. As long as we speak the language of experience, p is made true by the character of space and not our mind. At the level of transcen- dental reflection, however, p becomes mind-dependent in some appropriate sense of ‘mind’. Hence the two assertions ‘ p is about space’ and ‘ p is about the mind’ turn out to be perfectly compatible. Moreover, their status cannot be decided by anything like the method of direct inspection that we found in Russell and Moore.

4 Russell’s Ontological Logic

1 . This is explicit, for instance, in Russell (1904a). 2 . See Russell (1919a); Stevens (2005, chapter 5 ). Notes 249

3 . ‘Most’ rather than ‘all’, since Russell’s line of thought does not undermine the sort of view that can be found, for example, in Bolzano or Frege; that is, one where representations are regarded as objective and external to the judging mind. The early Russell does not consider such a view, except (as we saw) by misconstruing it as a variant of psychologism; however, the intro- duction, in PoM , of ‘denoting concepts’ – more about these below – does mean that his conception of proposition is not entirely free of all elements of ‘representationalism’. 4 . Robustness here consists of two conditions. First, states of affairs must be complexes of worldly entities. Second, every state of affairs is an existent entity, but only some of them actually obtain . A defender of such a concep- tion has to face a problem of unity that is analogous to that which exercised Russell: one has to explain how a state of affairs can be existent without obtaining – how states of affairs qua states of affairs differ from states of affairs qua facts – just as Russell has to offer some account of how proposi- tions qua truth-value bearers differ from propositions qua facts. Russell’s problem will be considered in this chapter and, again, in Chapter 5 . 5 . Moore uses Bradley’s theory of judgment as an example of existential theo- ries of judgment. According to Bradley, the constituents of judgments are ‘ideas’, which are not psychological entities but ideal contents or abstract universals; that is, entities which are endowed with genuine and which are, for that reason, capable of reference and truth (Bradley 1883, book 1, chapter 1 , §§3, 4, 6, 7, 10). As Moore saw it, this theory was insuf- ficiently antipsychologistic. He argued that Bradley’s explanation of how meanings or ideal contents are inferentially developed from what is given in our is in the end no different from the traditional doctrine of abstraction (1899, 176–8). 6 . This qualification is needed, because it is not clear on the Moore-Russell approach that truth and falsity really are genuine properties of properties. This question will be addressed below. 7 . This means, in fact, that Moore’s notion of composition is mereological . David Bell (1999) has shown in some detail that many of the most charac- teristic doctrines found in the early Moore are more or less direct conse- quences of a rather bold whole/part theory. According to Bell (ibid., 202–6), Moore’s is contained in the following three principles: 1) the principle of mereological : if x is a part of W , then W is necessarily such that it has x as a part (a whole and its parts are internally related); 2) the principle of mereological adequacy : all forms of complexity involve nothing but whole/part and part/part ; 3) the principle of mereolog- ical atomism : in any complex, the parts could exist independently of any whole of which they are the parts. In hindsight, we can see that Moore’s metaphysics of propositions is close to an impossible combination. On the one hand, he works with a mereological notion of complexity, while on the other hand, the entities thus constituted are ascribed features that make them propositional. 8 . The topic of Moore’s fellowship dissertation was the ‘metaphysical founda- tion of ethics’. Moore (1899) is culled from this material. 9 . No doubt, a trained in modern analytic can give such an explanation, but this question need not concern us here. 250 Notes

10 . ‘Term’ is thus the most general word in Russell’s philosophical vocabulary (PoM , §47). It excludes only classes as many, precisely because, being plurali- ties, they cannot be counted as one (§70). On the other hand, ‘classes as one’ are terms. 11 . To say that everything there is a term is not yet to say anything about what there is. A great deal has been written about the early Russell’s onto- logical commitments. Fortunately, nothing that is relevant to our discus- sion depends on this issue. For a sophisticated study, see Makin (2000, chapter 3 ). 12 . I follow the convention introduced by Griffin (1980, 119) of using slashes to indicate the mention of a proposition in Russell’s sense or one of its constit- uents. Like Griffin, I shall omit slashes when the constituent is an ‘ordinary’ entity; thus I shall say, for example, that and Socrates, rather than /Plato/ and /Socrates/, are constituents in the proposition /Plato admires Socrates/. 13 . Russell refers to the logical subject or subjects of a proposition as its terms (cf. PoM , §48). This use of ‘term’ is different from the generic one introduced above, but the connection is simple. Absolutely any term in the generic sense occurs in some proposition as a logical subject and is therefore a term of at least some propositions (although, as we shall see, not all constituents of a proposition occur in it as terms in the new sense; that is, as logical subjects). To forestall confusion, I shall reserve ‘term’ for the generic use, adding occasionally the qualification ‘in the generic sense’, and use ‘logical subject’ for Russell’s ‘term of a proposition’. 14 . But, one might ask, is the proposition / aR b / not as much about R as it is about a and b ? After all, one might argue, for example, that this proposition says about the relation R that it is had by the pair < a , b >. It may be that Russell’s ‘test’ for logical subjecthood is not purely intuitive but is, rather, a way of expressing the distinction between logical subjects and other constituents of propositions. If this is correct, the answer to the above question would be as follows: If the content of the sentence aR b is perspicuously captured by the proposition /R holds between a and b /, then we must conclude that the proposition expressed by aR b does not have three but four constituents: a , b R and a certain three-place relation. And this shows that not all constituents of a proposition can be regarded as logical subjects in Russell’s sense; that is, entities that the proposition is about. 15 . Russell, then, is committed to holding that unity is always due to a ‘relating relation’. He asserts this much in the last paragraph but one of PoM , §55. On this view, even propositions which are apparently of the subject-predicate form turn out to be relational (as Russell in fact argues in §216). On the other hand, in §53 we find him arguing that ‘subject-predicate propositions are distinguished by just this non-relational character’, which consists in the fact that a proposition like /Socrates is human/ has only one logical subject. This, of course, is flatly incompatible with his solution to the problem of unity. It is perhaps for this reason that he continues §53 with an attempt to identify a sense in which every proposition somehow does involve the assertion of a relation. Like so much else in the early chapters of PoM , the proposal remains tentative. Notes 251

16 . Nor, we might add, has unity anything to do with ‘verbs’ in the linguistic sense. For some reason, Russell often uses the terminology of ‘verbs’ and ‘adjectives’ when what he means are relations and qualities. This termino- logical oddity may be just a hangover from his idealist period; it is found in Bradley, too, for example. 17 . Klement (2004, §3) argues convincingly that Frege’s actual for his version of the saturated/unsaturated distinction are not satisfactory. In general, Frege moves very quickly from the recognition that concepts have predicative uses and that there are expressions (‘proper names’) which never occur as predicates to the conclusion that the roles of proper names and functional expressions (including concept words) are exclusive of each other. The motivation for this transition is not hard to come by, even if it does not constitute much of an independent argument; it is only by holding that the two roles are exclusive that Frege is able to solve the problem of unity: an entity which is capable of being referred to by a proper name cannot be referred to by an incomplete expression. As we shall see, Russell thought he had conclusive reasons against this type of explanatory strategy. 18 . So unlike the existential theory of judgment, which they see as reducing truth to some existent fact, Russell’s and Moore’s metaphysics of proposi- tions reduce facts to truths. 19 . Why this should be called a regress of meaning is not very clear. Most likely, Russell is just using ‘meaning’ in an extended sense in which Russellian propositions, too, can be said to have meanings. If one adopts this usage, it is then very natural to say that in order to find out the meaning of a propo- sition, one has to determine its constituents. 20 . According to Hylton (1984, 381), the problem of unity is ‘in principle unsolvable within the metaphysical framework which [Russell] establishes’, while according to Leonard Linsky (1992, 250), Russell was ‘defeated’ by the problem. 21 . I shall say a little more about this at the end of Chapter 5 . 22 . Conceivably, such a compromise is present in Bradley’s metaphysics. This would mean that Bradley did not object to facts per se, but a wrong meta- physical construal of them, one that reveals itself in attempts to explain their unity. For a discussion and defence of this reading of Bradley, see Olson (1987). 23 . How this is to be accomplished is a question which would lead to many intricate issues in the early analytic metaphysics and which, insofar as it relates to Russell, belongs to a later phase of his philosophy. The primacy of complexes over their constituents is in some sense the ontological coun- terpart of Frege’s context principle. It was advocated by Wittgenstein in the Tractatus and, following him, Frank Ramsey (1925); which is not to say that the Wittgenstein-Ramsey line is the only conceivable way to implement the idea. Evidently, even if the early Russell anticipated such developments, this does not yet tell us anything about how the issue of facts vs. their constit- uents was conceived in his eventual metaphysics of facts – as in Russell (1918a) and elsewhere. 24 . An early version of this argument, or something similar to it, is given in Russell (1899b, 285). Something similar to Russell’s argument is found in 252 Notes

Frege as well; see, for example, Frege (1918, 352–3). Truth primitivism is also present in Moore (1899) and, more explicitly, in Moore (1902). 25 . For Russell, non-propositional functions are therefore derivative from suit- able propositional functions. In PoM this is explained as follows: ‘If f (x ) is not a propositional function, its value for a given value of x (f (x ) being assumed to be one-valued) is the term y satisfying the propositional function y = f (x ), i.e. satisfying, for the given value of x , some relational proposition; this relational proposition is involved in the definition of f (x ), and some such propositional function is required in the definition of any function which is not propositional’ (§482). Russell holds, that is to say, that non-propositional functions – in Principia they are called ‘descriptive functions’; see Whitehead and Russell (1910, 31, *30) – are less fundamental than propositional func- tions, since every n -ary function is, logically speaking, derived from some + 1–place relation (or propositional function). For example, starting from the proposition /6 is the immediate successor of 5/, which contains a binary rela- tion as a constituent, we ‘derive’ from it the binary propositional function ‘ is the immediate successor of x ’, and this in turn can be used to define the non-propositional function whose value for a given argument is the unique entity satisfying the propositional function for that argument. 26 . How the propositional functions of PoM should be referred to is not an entirely straightforward matter. They are not linguistic expressions. On the other hand, despite the fact that each particular propositional function is to be regarded as being derived from a proposition through variation, proposi- tional functions are not constituents of propositions either; every proposi- tion can be regarded as a value of several propositional functions, and it is not to be thought that propositional functions should literally be constitu- ents of propositions. Nevertheless, as Klement (2004–5, 108) argues, propo- sitional functions are best seen as ontological correlates of open formulas, just as propositions are the ontological correlates of closed formulas. Hence, I will refer to them using the slash notation introduced originally for propo- sitions and their constituents. 27 . Infinity is not the only case that forces a modification in the simple formula for ‘aboutness’, according to which a proposition is about x by virtue of having x among its constituents. Another case involves definite descrip- tions, where we have thoughts about one definite entity without having that entity literally before our mind (‘the person who did this’). This case is important for Russell, because it occurs in mathematical definitions, which always involve propositional functions: an entity is defined in the math- ematical sense when it is identified as the unique satisfier of some propo- sitional function (PoM , §31). And we can have thoughts purporting to be about some entity or entities, even though there are no such entities (‘the gnome that stole my precious gold is such-and-such’). In each such case our thought has as its object a proposition which has a new kind of entity as a constituent; cf. PoM , §§63–4. 28 . As is well known, Russell returned to the problems posed by denoting concepts immediately after the publication of PoM in 1903. Between 1903 and 1905 he sought to work out, in a number of manuscripts, a workable semantics for denoting concepts, a semantics that would, at the same time, provide a foundation for logic and logicism – the relevant texts are printed in Notes 253

Russell (1994). These developments culminated in ‘On Denoting’, Russell’s most famous essay. It is significant that the theory of denoting advocated there starts from a of variable and of propositional func- tion (cf. Russell 1905d, 416). When Moore asked Russell what sort of entity the variable is, he received the following reply: ‘I only profess to reduce the problem of denoting to the problem of the variable. This latter is horribly difficult, and there seem equally strong objections to all the views I have been able to think of’ (Russell to Moore, 25 October 1905; as quoted in Hylton 1990a, 256). The difficulties relating to denoting concepts cannot be explored here, but the basic point deserves to be mentioned. As Russell explains in PoM (§65), when a denoting concept occurs in a proposition, the proposition is as a rule not about the concept but is about whatever, if anything, the concept denotes. The qualification is needed, because denoting concepts are terms and ought therefore to occur as logical subjects in some proposi- tions. It turns out, however, that this feature of denoting concepts resists consistent formulation, a point that Russell eventually reached in the infa- mous ‘Gray’s Elegy’ argument of ‘On Denoting’. 29 . The notion of unrestricted generality will be discussed in Chapter 5 . 30 . See Grattan-Guinness (1977, 112–13), where the general point is made very forcefully. 31 . Since Russell’s ‘⊃’ indicates a relation between terms and is not a statement connective, it should be read as ‘implies’. Observing this, however, it may look as if propositions are not appropriate entities to flank it; a statement like ‘four is greater than three implies three is greater than two’ looks ill formed. In Russell’s logical grammar, however, when propositions stand in the rela- tion of implication, they occur as propositional concepts. These are propositions considered merely as complex concepts and distinct from assertions ( PoM , §38). Grammatically, this distinction corresponds to that between ‘verbs in the form they have as verbs’ and ‘verbal nouns’ (§52). This is tolerably clear, but for some reason Russell fails to adhere consistently to the point that the distinction really is between occurrences of entities and not between entities. He does recognize this in §38, but then in §52 he first notes that an assertion cannot be made the subject of a proposition, and draws from this the conclu- sion that here the contradiction of an entity which cannot be made into a logical subject appears to have become ‘inevitable’. 32 . As Landini (1998, 44) points out, Russell’s actual practice in the PoM tends to obscure the true nature of his variables. In presenting his ‘’, Russell uses such letters as p , q , etc. (cf. PoM , §18). A present-day reader, unless cautious, is practically guaranteed to read these as proposi- tional letters. Nevertheless, Russell’s practice and some of the things he says make it quite clear that this is not the intended reading. 33 . In his (1906a) Russell found a way of avoiding conditionalization while retaining the doctrine of the term variable. He now defined p ⊃ q as equiva- lent with ‘p is not true or q is true’. On this interpretation of implication, the relation can hold between entities that are not propositions. The proposi- tion ‘Augustine ⊃ ’ comes out true, for example, because Augustine is not a proposition and is therefore not true. It follows from this account that a tautology like p ⊃ p can stand on its own, without any further condition 254 Notes

being added to it, for it is true whether or not p is a proposition. Russell does not recommend this procedure merely for reasons of . He points out that the truth of p ⊃ ( q ⊃ p ) should not be made dependent upon q being a proposition; if p is true, then, ‘even if q is not a proposition, it must be added that if p and q were both true, p would be true’ (1906a, *1.2). 34 . For details of the substitutional theory, see Landini (1998; 2003). 35 . The matrices of the substitutional theory are thus ‘incomplete symbols’ in the sense of ‘On Denoting’. A matrix p /a has ‘no meaning in isolation’ in the Russellian sense of that phrase; it does not refer to any single entity and acquires meaning only when it is embedded in a sentence that is concerned with the result of the substitution. This is the sense in which the theory avoids commitment to classes and propositional functions, whose roles are now taken up by matrices. 36 . The term ‘schematic conception’ is borrowed from Goldfarb (2001). 37 . The availability of formulation (1) is in some sense dependent on the idea of logical connectives as devices with which compound formulas are built from simple ones. Such a view is present, if only implicitly, in the tech- nical machinery that Russell adopted from Peano, and it is very far from the notion of connective that is implicit in the Moorean conception of proposi- tions. It is clear that in PoM there is nothing like a clear separation between these Peanist and Moorean elements in Russell’s thinking. 38 . The notion of interpretation by replacement comes from Dummett (1993, 23–4). He calls it a ‘presemantic’ notion of interpretation, primarily because it eschews the task of systematically describing how the semantic value of a sentence is determined on the basis of the semantic values of its semanti- cally significant parts. For Russell, this feature of ‘truth theory’ is concealed by the fact that he regards ‘having the same form’ as a primitive relation for propositions. 39 . Note that in standard such talk about ‘absolutely everything’ does not even make sense. Model theory relies on , which admits no ‘universal set’. On this point, the distance between Russell and model theory appears to be very great indeed. Of course, we may be inclined to think of this particular feature of Russell’s universalist conception of logic as nothing but logical naiveté, expecting that it must have been thoroughly shaken, if not completely disappeared, by the discovery of the paradoxes. Russell himself acknowledges, in the penultimate sentence of PoM , that the ‘totality of all logical objects, or of all propositions, involves, it would seem, a fundamental logical difficulty’ (§500). And in §344 we find the following reflection on Cantor’s power set theorem: ‘The difficulty arises whenever we try to deal with the class of all entities absolutely, or with any equally numerous class; but for the difficulty of such a view, one would be tempted to say that the conception of the totality of things, or of the whole universe of entities and existents, is in some way illegitimate and inherently contrary to logic’. Russell eventually fixed upon a version of the theory of types as the proper way out of this fundamental difficulty, and it certainly looks like the introduction of types does undermine unrestricted generality; after all, the notion of ‘unrestrictedly everything’ seems oblivious to distinc- tions of type. The issues here are nevertheless quite complex; Stevens (2005) provides some useful information. Notes 255

40 . Versions of this argument are also found in Russell (1906c, 205) and (1908, 71–3). 41 . Thomas Ricketts (1986, §II; 1996; 1997) applies a version of this interpreta- tive argument to Frege, whose views on truth are well known, if not particu- larly well understood. According to Ricketts, Frege’s conception of judgment as the acknowledgement of the truth of a thought commits him to the view that truth is not a at all. Ricketts derives this conclusion from a dissection of Frege’s famous against truth definitions (which, we may note, is not so dissimilar from Russell’s indefinability argu- ment). This conception of truth, Ricketts argues, ‘precludes any serious metalogical perspective and hence anything properly labeled a semantic theory’ (1986, 76). 42 . The distinction between ‘implies’ and ‘therefore’, Russell argues in §38 of PoM , is the lesson that we ought to derive from Lewis Carroll’s puzzle about Achilles and the tortoise; this matter will be taken up in Chapter 5 . We should note that Russell’s symbolic practice tends to obscure the distinc- tion. For example, in ‘The Theory of Implication’, *1.1, he gives an illustra- tion of how dots are used to indicate the scope (‘range’) of the assertion sign: ‘“├ : p . ⊃. q ” means “it is true that p implies q ”, whereas “├. p . ⊃ ├. q ” means “p is true; therefore q is true”’, the difference between the two formulas being that the first ‘does not necessarily involve the truth of either p or q , while the second involves the truth of both’. In letting ‘⊃’ stand for both ‘implies’ and ‘therefore’, he is ignoring the lesson of §38 of PoM . This prac- tice is also followed in Principia . There is, however, no real confusion here. 43 . Whitehead and Russell (1910, 9). 44 . The extreme version of the universalist conception, one that combines the impossibility of semantics with a similar impossibility of syntax, is found in the Tractatus . According to Wittgenstein, logic is ‘transcendental’ (1922, 6.13); logic, that is, is a necessary condition of describing the world, a condi- tion which cannot itself be made subject to a theoretical description but which shows itself in logically correct pictures of reality. This unsayability applies to conventionally syntactic notions as well. For instance, that one proposition follows from another is something that ‘expresses itself’ in the relation in which the forms of the propositions stand to one another, this being something that we recognize from the structure of the relevant prop- ositions. There can therefore be no ‘laws of inference’ (as in Frege or Russell, according to Wittgenstein); only the propositions themselves can justify an inference (ibid., 5.132). 45 . In Goldfarb’s formulation, the contrast is between the universalist concep- tion and one ‘that is more common today’. In van Heijenoort’s original formulation, it is between universalists like Frege and Russell-Whitehead, on the one hand, and the algebraic approach to logic, where free use is made of conceptualizations and methods not available to universalists – van Heijenoort mentions Boole, Schröder and incipient model theory, as in Löwenheim. 46 . Tappenden (1997) treats many of the general interpretative questions relating to the van Heijenoort tradition. His focus is mostly on Frege; for the case of Russell, compare Hylton (1990b) with the criticisms in Landini (1998, chapter 1 ). 256 Notes

47 . I use a terminology that is somewhat different from Landini’s. I prefer the term ‘logic as theory’ to Landini’s ‘logic as calculus’. The reason is that such logical realists as Russell and Frege saw a logical system more as a theory with its own subject matter than a calculus. This aspect is brought out nicely by Russell’s essay on the ‘regressive method of discovering the premises of mathematics’ (1907a); for the case of Frege, see Korte (2010). It should be emphasized, though, that ‘theory’ and ‘calculus’ are not exclusive of one another. 48 . Another point where Hylton’s argument is vulnerable to criticism is his claim that for Russell the metaphysical notions of a proposition, of a propositional constituent and of truth are ‘available independently of logic’. Whatever this means exactly, this view is quite doubtful. The criterion of independ- ence that Hylton attributes to Russell – these notions are ‘ones to which we have direct and immediate access, through a non-sensuous analogue of ’ (1990b, 216) – plays no real role in Russell’s metaphysical logic. We can, nevertheless, extract a sound point from this; namely, that the metaphysics of propositions and, with it, the metaphysical framework for logic was in fact available to Russell independently of the more ‘syntactic’ or ‘formal’ framework that he derived from Peano. There are many points of details where the two come into conflict – for instance, they differ over how they conceptualize logical connectives, a point that was mentioned in passing above. And one might go further, arguing that due to certain points relating to its grand design, the metaphysical framework in fact excludes a properly syntactic-cum-formal development of logic. The official view of PoM is that the approach inspired by Peano is useful for a ‘symbolic’ devel- opment of logic, whereas the metaphysical framework spells out the - sophical truth about the matter. But the fact is that the two are not so easily reconciled. 49 . Later, Russell gave a similar reply to Philip Jourdain’s question about inde- pendence and logical axioms; see Grattan-Guinness (1977, 117). 50 . See, for example, Ricketts (1996, 136), Kemp (1998, 222). 51 . I do not reproduce Frege’s notation. 52 . ‘Comprehensiveness’ is used in this sense by Ricketts (1997, 148). Another term would be van Heijenoort’s ‘experimental completeness’ (1967, 327).

5 Russell and the Bolzanian Conception of Logic

1 . The following discussion on formality relies on recent work by John MacFarlane; see MacFarlane (2000), (2002). Wolff (1995) is another useful discussion. 2 . Or even that logic is the distinctively formal discipline, a view that has recently come to be known as logical hylomorphism , a term introduced by MacFarlane. 3 . MacFarlane (2000, 50); the labels in the text are mine, not MacFarlane’s. 4 . Here it is useful to remember Manley Thompson’s point about Kant on predication (see this volume page 75): general logic represents predication as a relation between concepts, whereas in ‘-with-content’ it is a rela- tion between objects and concepts. Notes 257

5 . This conclusion is also reached by MacFarlane (2000, section 4 .4; 2002, §3). He uses it to undermine the common view that when Kant holds logic to be formal, he is merely repeating an essentially traditional character- ization. According to MacFarlane, when Kant argues, on the basis of the generality-cum-normativity of logic, that logic is also formal in the sense of non-substantiality and non-particularity, he is in fact arguing against a well-entrenched neo-Leibnizian tradition; therefore, Kant should be seen not as a follower of an existing tradition but as an originator of a new one; namely, logical hylomorphism . This is the view that logic is the distinctively formal discipline, in one sense or another of formality. If MacFarlane is correct, it means, from our point of view, that the advocates of the Bolzanian conception of logic were not in fact suggesting anything radically new but were just returning to pre-Kantianism also in logical theory. Of course, this is no big surprise when the point is applied to Russell, who is explicit about his indebtedness to Leibniz. MacFarlane makes the further point that logical hylomorphism became rapidly ubiquitous in the course of the nineteenth century and that our position with respect to it is not entirely dissimilar: we are inclined to take formality as the criterion of logicality, a fact that reflects the continuing influence of a broadly Kantian conception of logic. 6 . There is also another problem for Kant’s model of the analytic a priori: what is its necessity (and truth) grounded in? From a Russellian point of view at least, a straightforward resort to the allegedly constitutive-normative character of the relevant laws would be decidedly unsatisfactory and is so for reasons that are strictly analogous to those operative in the case of the synthetic a priori indeed, Kant’s account of the analytic a priori is from this point of view nothing but a thinly disguised version of the relative model of the a priori. 7 . Russell, though, arrived at his views on logic independently of Bolzano. Bolzano was known to him only through the posthumous work Paradoxien des Unendlichen (‘The Paradoxes of the Infinite’), which is referred to in PoM , but Bolzano’s theory of logic plays no role there. 8 . For example, in his Treatise on Consequences , Jean Buridan gives the following formulation of the distinction between form and content (or matter): ‘I say that in a sentence in which we speak of matter and form we understand the matter of the consequence or sentence to be the purely categorematic terms, namely the subject and the predicate. The syncategorematic terms added to it, by which the subject and the predicate are connected or denied or distributed or taken to supposit in a certain way are not included. And we say the entire remaining part of the sentence pertains to the form’ (Buridan 1985, 194). 9 . The connection is very clearly explained by Buridan. According to him, formal consequence is one that is valid (true) for uniform substitution of categorematic terms: ‘A consequence which is acceptable in any terms is called formal , keeping the form same’ (Buridan 1985, 184). 10 . ‘Idea’ is Bolzano’s word for propositional constituents. Ideas are objective entities, independent of actual thought processes. 11 . This is pointed out by MacFarlane, too (2000, 41). 12 . For a different attitude towards the questions of demarcation, see Tarski (1987) and Quine (1936, 324–5; 1970, chapter 14 ). 258 Notes

13 . ‘Almost’ seems to be just a slip here. Russell does not really think that there are cases where the validity of deduction depends on something else than its ‘form’. Indeed, on the very next page he says that ‘the validity of any valid deduction depends on its form’ (1911a, 36). Note Russell’s use of ‘form’ here. He does say that validity depends on form, but this use of the notion is a non-substantial one. The quoted passage is followed by an explanation that the propositions of pure mathematics or of are obtained by just this process of generalization, where constants are replaced by vari- ables and through which ‘we finally reach a proposition of pure logic, that is to say a proposition which does not contain any other constants than logical constants’ (1911a, 35). 14 . Here I will put variables aside. Whichever way they are treated, they are indicators of form. 15 . An outline of some of the developments that are relevant is given at the end of this chapter. 16 . Indeed, since variables are denoting concepts, according to the Russell of PoM , it would be natural to classify variables as logical constants. 17 . I put aside the vexing question as to why Russell does not feel the need to consider alternative but logically equivalent formulations of definitions; this matter is briefly discussed by Byrd (1989, 347–8). 18 . Frege (1893, 12; 1897b, 128; 1918, 351). 19 . This does not mean that all prescriptions founded on the laws of logic would be prohibitive; rules of inference, for example, are permissions : once you have recognized such-and-such thoughts as true, you are permitted to recognize such-and-such a thought as true. 20 . See MacFarlane (2002, 35–43). 21 . Frege’s later discussion of truth, judgment and related matters in his ‘Thoughts’ elaborates on this. He defines judgment as the acknowledge- ment or recognition ( Anerkennung ) of the truth of a thought (1918, 356). This is at two removes from more customary accounts of judgment and assertion. It is not just that, for Frege, judgment involves the truth of a thought rather than just a presentation of a thought as true; as he sees it, judgment and assertion (the public manifestation of judgment) are, quite literally, knowledge , as they are conceptualized as the end points of a process of (ibid.). This begins with a formulation of a yes-no question, ‘Is A true?’ which involves grasp of the thought A , as distinct from judgment and assertion. ‘Appropriate investigations’ then put the inquirer in a posi- tion to acknowledge the truth of either A or its negation, not-A . Thus, Frege characterizes judgment in terms of a strongly idealized process of inquiry, constituting in fact a justification . Once we take into account that when he thinks about logic, he has this sort of epistemic function, the establishment of truths, in mind, it is not difficult to see why he holds that logical laws, or norms derived from them, possess a constitutive role. It may or may not be feasible to think that logical laws are constitutive of a general process of knowledge seeking, but for Frege the of justification is deduc- tive argument – indeed, proof. This, incidentally, explains his otherwise quite counterintuitive claim that we can draw inferences only from true premises. As Sundholm (2002, 572) points out, Frege is perfectly correct about this, given his adherence to an essentially Aristotelian conception Notes 259

of demonstrative science. As Sundholm observes, Frege’s logic is essentially logic-in-use: ‘[Frege] is not concerned with the so-called logical truth of propositions, but with how we obtain further knowledge by proceeding from theorem to theorem. We do that by proceeding from known truths to a novel insight by drawing a valid inference’. 22 . Not everyone would agree that Frege’s case shows this to be so. In particular, this would be denied by those who argue that Frege in fact accepted, rather than rejected, a broadly Kantian account of ; that is, it would be denied by those who claim to detect an implicit idealist or anti-realist quali- fication in his pronouncements on the status of non-spatio-temporal objects. As the Kantian interpretation sees it, Frege does not regard the objectivity of logic as consisting in the fact that it is concerned with an independent realm of logical objects (or, more generally, with an independent realm of logical structures); his view is rather that objectivity consists in the fact that ‘the rules of logic make the notion of objective judgment or assertion possible in the first place’ (Friedman 1992b, 536). In other words, logic is objective, or has an ‘objectifying function’, because without our acknowl- edging that logic has a normative bearing upon our assertions, inferences and suchlike, there would be no practice of judgment as an essentially inter- subjective enterprise. All the details aside, I would be inclined to argue that this interpretation of Frege misconstrues his actual order of explanation. It is clear that the so-called Platonic reading of Frege is quite compatible with seeing him as ascribing an ‘objectifying function’ of sorts to logic (Burge 1998 develops this theme in some detail). It is, indeed, perfectly correct to say that, for Frege, ‘the func- tion of the rules of logic is ... to regulate our (human) discourse – the possi- bility of agreement or disagreement, the possibility of reasoning’ (Friedman 1992b, 536); this is so because there are segments of human discourse which are answerable to ‘the most general laws of truth’. But to this it must be added that these laws can have ‘constitutive and normative force’ only if they are of a certain kind (a point very much in the spirit of the early Russell and Moore); namely, only if they really are laws of truth in the full-blown sense of truth, a sense which essentially involves independence. In Frege’s view, then, the constitutive-regulative function of logic presupposes objec- tivity qua independence simply because otherwise that which gets consti- tuted and regulated would not deserve to be called judgment in the first place. The weightiest consideration against the Kantian reading of Frege may well be that it seems to be committed to regarding Frege’s constitutive-regulative rules as essentially normative. As we have seen, this was not Frege’s view. This theme should not be pursued here, though. 23 . Of course, Russell’s propositions are ‘ordinary entities’ – that is, terms in the sense of PoM – but a protosemantic generalization over propositions is still different from a syntactic generalization over all terms, including proposi- tions. It really is a pity that the Russell of PoM was not more observant of the differences between syntactic generalization in the style of Peano and the protosemantic one that is involved in his own metaphysical logic. 24 . Russell’s letter to Meinong, 15 December 1904; English translation in Smith (1985, Appendix, 347). Makin is, of course, well aware of Russell’s self-proclaimed agreement with Meinong (Makin 2000, 195–6). He uses it 260 Notes

to illustrate the nature of the theoretical enterprise that Russell saw himself as being engaged in. My only complaint about this reading is that Makin’s use of ‘generic metaphysics’ is potentially misleading on Russell’s inten- tions. To repeat, the conclusion that the most general science is concerned with being and is in that sense metaphysics is not built into the concept of that science but results from a reflection on the nature of its key concepts. 25 . For more on this point, see Levine (2001, 220–1). 26 . A note on ‘absolutely everything’ is in order here. Russell says that ‘term’ – which, it will be recalled, is the name that applies to any entity – is the most general word of his philosophical vocabulary. This leads one to expect that ‘term’ covers everything. This, however, is not the case. As Russell explains in PoM , §47, a term is anything that can be counted as one . This criterion excludes ‘classes as many’; that is, classes considered as aggregates or plurali- ties rather than as single objects or unities – that is, wholes composed of their members (ibid., §70). Not being single entities, classes as many cannot be logical subjects and are therefore not values of variables, either. It turns out, then, that even for Russell, ‘absolutely everything’ does not include absolutely everything, although it certainly includes absolutely every thing (every entity, term). 27 . The two notions of inference – the purely syntactic one and that of valid inference – are not unrelated. We may say, recalling from Chapter 4 the distinction between logic as a theory and logic as a science, that the syntactic notion of inference is meant to give a theory (or model) of valid inference. 28 . Even if we are entitled to ignore the case of false premises by having our focus on inference, there is still the notion of deducibility , which involves no such restriction. And we do find Russell arguing on one occasion that ‘from a false hypothesis anything can be deduced’ (1907b, 41). As Moore (1919, 100) observes, this identification of deducibility with material implication is an ‘enormous howler’. As we shall see in the text, however, it must be regarded as a howler even in the light of Russell’s own considered views on valid inference: even if ‘ q is a consequence of p ’ amounts to the same as ‘ implies q ’, this does not yet show that q can be validly inferred or deduced from p . 29 . As we have seen, the Russell of PoM is not willing to reduce material impli- cation to a distribution of truth values or define the relation in terms of such a distribution. But this does not affect the point that a certain relation between propositions can be singled out using a particular such distribu- tion, a relation which plays an essential role in valid inference. On one occa- sion, though, Russell does argue that material implication holds ‘without any reference to the truth or falsehood of the propositions involved’ (PoM §38). I do not know what, if anything, ought to be made of this enigmatic remark. 30 . For later statements of the same point, see (Whitehead & Russell 1910, 20–1; Russell 1910b, 356; 1919b, 153). In the latter two passages Russell explains that although there is more than just material implication to the practice of inference – namely, our recognizing that p implies q independently of the actual truth values of these propositions – what we thus recognize is still just the fact that either p is false or q is true. Hence, Russell , he is able to retain the view that material implication is the relation that grounds valid Notes 261

inference. This is important for him partly for the reason that it enables him to resist the conception of deductive reasoning which holds that ‘what can be inferred is always in some sense already contained in the premiss’ (ibid.). 31 . For a later use of ‘relations of form’, see Russell (1919b, 153–4). 32 . Indeed, we may declare any truth-preserving sequence whatsoever to be ‘formally valid’ in the Bolzanian sense simply by keeping all the constitu- ents of the relevant propositions fixed. ‘All the constituents’, however, does not seem to single out any interesting class of constants, and hence the point just serves to highlight the purely schematic nature of the Bolzanian notion of form. 33 . Rusnock and Burke (2010) apply this point to Bolzano. They argue that the well-known criticisms that John Etchemendy has levelled against Tarski’s theory of consequence do not apply to Bolzano’s ostensibly similar account. According to Etchemendy, an account of consequence ought to capture the core content of our ordinary concept of consequence, which assigns this concept a variety of semantic, epistemic and informational features. As against this, Rusnock and Burke point out that Bolzano’s strategy is quite different: Bolzano first gives an account of ‘deducibility’ that is purely formal and defines that relation for propositions in the objective sense of ‘Sätze an sich’; this puts him in a position where he can refer to a variety of features of these propositions to explain differences between different cases of deducibility in the formal sense. 34 . Or, at any rate, thinking that the logicist reduction, which shows that mathematical constants are logical constants, when combined with the Bolzanian notion of form, which supplies a tractable notion of logical constant, together show where the philosophical questions are to be found. A similar strategy can be found in Bolzano, I believe, although the context is not the same. 35 . This construal of Lewis Carroll’s puzzle follows Smiley (1995), who refers to Thomson (1960). Others have made the point as well; see, e.g., Theodore de Laguna’s review of The Problems of Philosophy (de Laguna 1913, 331–2). 36 . A minor issue concerns the notion of assertion that figures in Russell’s rule. The PoM concept of assertion is quite confused, involving as it does several incompatible elements. In discussing the notion, Russell first makes the solid observation that the proposition ‘ p implies q ’ asserts an implication, though it does not assert p or q (§38) – he adds that ‘assertion’, as he uses it here, is meant in a non-psychological sense, but this is something better left aside. Unfortunately, he then continues by arguing, for reasons that are not made clear, that the p and the q which enter the proposition ‘ p implies q ’ ‘are not strictly the same as the p and the q which are separate propositions, at least, if they are true’ (ibid.). This, of course, violates the ‘Frege point’ and will not do, as it would render an invalid mode of inference. Fortunately, Russell corrects the mistake in §52, although the correction is not noted as such. 37 . Russell treats as a special case of the rule of detach- ment: ‘Another form in which the principle is constantly employed is the substitution of a constant, satisfying the hypothesis, in the consequent of a formal implication. If ϕx implies ψx for all values of x , and if a is a constant 262 Notes

satisfying ϕx , we can assert ψ a , dropping the true hypothesis ϕa . This occurs, for example, whenever any of those rules of inference which employ the hypothesis that the variables involved are propositions, are applied to partic- ular propositions. The principle in question is, therefore, quite vital to any kind of demonstration’ (PoM , §38). 38 . To be sure, Russell’s solution to the puzzle fails in the task that the Tortoise assigned Achilles; namely, it fails to force the Tortoise, logically, to accept the conclusion. Here, though, the problem lies in the task and not Russell’s solution. 39 . Russell, of course, will also need formal implications for a quite different purpose; namely, in a theory of implication, which enumerates general truths about the implication relation in the form of an axiomatic-deductive theory. 40 . Of course, logic is also about the structure of thought, but only in the deriv- ative sense that the structure of thought (in a non-psychological sense) is the structure of reality. 41 . See, e.g., Goldfarb (1988) and Bernard Linsky (1999, chapter 6). 42 . I say ‘direct’, because the Tractatus certainly had indirect influence on Russell’s logical theory, via Ramsey. The Russell-Ramsey-Wittgenstein connection cannot be explored here, however. 43 . Nevertheless, in My Philosophical Development Russell continues to profess agreement with Wittgenstein on the subject of ‘propositions of logic’: ‘Wittgenstein maintains that logic consists wholly of tautologies. I think he is right in this, although I did not think so until I read what he had to say on the subject’ (1959, 116). This combination of views is quite puzzling, unless we assume either that his agreement with Wittgenstein was purely termino- logical or else that, by the 1950s, he had lost his faith in mathematical logic (whatever that might mean). 44 . ‘[T]he propositions of logic – and only they – have the property that their truth or falsity, as the case may be, finds its expression in the very sign for the proposition’ (Wittgenstein 1995, 60). 45 . But we should also note that Russell’s further elucidation of the concept of tautology in An Analysis of Matter (1927b, 172) in fact relies on the old idea of ‘true in virtue of form’; since tautologies are ‘true in virtue of form’, they retain their tautological character even when their non-logical constants are turned into variables. In logic, therefore, it is just a waste of time to ‘deal with particular examples of general tautologies’, and no constants should ever occur in logical propositions except such as are formal in the sense of ‘maximally general’. We may also note Russell’s quite un-Tractarian explana- tion of why two tautologies ‘say the same thing’; that is because there is only one fact which makes both of them true (or false). At least in the Tractatus Wittgenstein has no room for the idea of truth making for the propositions of logic, which, indeed, are Sinnlos and, therefore, cannot really say anything. 46 . This suggestion is made by Stevens (2005, 106). 47 . See, for example, Russell (1919b, 203). 48 . The details of the development are recounted in Landini (1998). 49 . See Russell (1910b, 119; 1912a, 72; 1913, 153; 1918a, 223). 50 . In fact, a little more can and should be said here. It would clearly be advantageous to double primitivism if non-propositional complexity could Notes 263

be dispensed with. This, moreover, would be very much in line with the bottom-up approach to the composition of propositions that is a key element in Russell’s Moorean metaphysics of propositions. Russell took one step in this direction when he showed, in ‘On Denoting’, how all ‘denoting complexes’ are eliminable in favour of quantified propositions. And he took another step when he showed, in the substitutional theory, how classes can be eliminated in favour of the method of substitution. Of course, even if these are steps in the right direction, they will not eliminate the problem of unity; if all complexity is propositional, we no longer have to bother about how to distinguish propositional from non-propositional complexity, but we still have to explain what distinguishes complexity from simplicity . 51 . The different versions of the multiple-relation theory are discussed in Griffin (1985). 52 . Russell thus retains his old division of entities into terms and concepts: ‘[I]n every complex [fact] there are two kinds of constituents: there are terms and the relation which relates them: or there might be (perhaps) a term qualified by a predicate. ... But there are some terms which appear only as terms and can never appear as predicates or relations. These terms are what I call particulars . The other terms found in a complex, those which can appear as predicates or relations, I call universals ’ (1911b, 135; cf. 1912c, 182). (Incidentally, the fact that concepts/universals can occur as subjects in elementary propositions does not reproduce the paradox of self-predication. As Klement (2004–5, 22) points out, Russell can well have a theory of sparse universals, and even if he needs abundant proposi- tional functions, he is not committed to there being a universal for every propositional function.) On the other hand, drawing the distinction does not yet explain how fact unity comes about, for Russell must still explain how concepts differ from terms. More precisely, he has to explain how a universal in the predicative capacity differs from that same universal when it lacks that capacity. Indeed, contact with Wittgenstein forced Russell eventually to reconsider the distinction. 53 . Thus, Landini (1998 and elsewhere) interprets Principia as advocating a kind of nominalist semantics. In it, only individual variables are interpreted objectually. They range over particulars and universals, there being thus no distinctions of types among entities . Predicate variables, on the other hand, are interpreted nominalistically: they are in fact schematic letters whose values are well-formed formulas. According to Landini, this interpretation of Principia helps us see how Russell managed to retain the doctrine of the unrestricted variable. To be sure, Landini’s reconstruction of the semantics of Principia would not be to everyone’s taste as an interpretation of Russell. 54 . Russell, I believe, was quite firmly committed to the principle that if we are acquainted with x , then x is an actually existing object; and hence that if x is not an object, then we cannot be acquainted with x . Thus, Russell’s doubts about acquaintance with logical forms stem from doubts about whether logical forms really are objects. This means that compelling reasons against logical forms qua objects would undermine the 1913 version of the multiple-relation theory. As we shall see below, Russell did in fact believe he possessed such reasons. 55 . Cf. here Griffin (1980, 152). 264 Notes

56 . Russell (1914, 168) makes this claim. No reasons for it are given there, although the wording suggests considerations similar to those given in Theory of Knowledge . In the first edition of the book the relevant passage is accompanied by a footnote alluding to ‘unpublished work by my friend ’. Although Wittgenstein’s influence may have been important, it seems clear enough that Russell’s own thinking was moving in the direction of the ‘no logical constants’ view quite independently of Wittgenstein.

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Index

Adams, Robert M, 88n5 deductive rigour, 50–1 Allison, Henry E., 80 Pasch’s notion of, 51–5 analyticity Russell’s notion of, 55–8 Bolzano, 168 denoting concepts, 108n3, 128–9, Frege, 23 172n16, 186–7, 189–90 Kant, 21–2 Detlefsen, Michael, 55n24 logical empiricism, 22–3 Dreben, Burton, 146 Russell, 23–5, 174–5, 198 Dummett, Michael, 123, 153, 254n38 Ayer, Alfred J., 20, 21n6 Eberhard, Johann Augustus, 96, 98 Berkeley, George, 80 Euclid’s Elements, 41, 61 Bolzano, Bernard, 2, 21n5, 42n5, 58, Etchemendy, John, 200n33, 201–2 140, 147, 159, 166, 167–9, 175, existential theory of judgment, 11, 191, 194, 200, 211 87–9, 88n5, 89n6, 117n18 Boole, George, 2, 179 Bradley, Francis Herbert, 2, 14, 110n4, facts 111n5, 114n10, 118, 122, 122n22, as true propositions, 14, 110, 222–3 117–18, 124 Brentano, Franz, 21n5 form Brittan, Gordon, 91 logical, see logical form Buridan, Jean, 167n8, n9 schematic notion of, 10, 166, Byrd, Michael, 175, 178 169–70, 173, 174–5, 200–1 formal logic calculus of logic (Russell), 12–13, Kant on, see Kant, Immanuel, 144–6, 148–50, 152, 155–7, 192 formal logic Cantor, Georg, 26, 49, 56, 254n3 formality, criteria for, 161–4 Carnap, Rudolf, 21n5 Frege, Gottlob, 2, 18, 21, 50, 51n19, Cassirer, Ernst, 38, 63 58, 74, 108, 144, 147, 202, 208–9 consequence analyticity, 23, 25 formal, 167–70 conception of logic, 146–9, 155–6, logical, 147, 154–5, 175–6, 194–6, 176, 178 198, 201–2 normativity of logic, 180–3 content propositional unity, 116–18, 134 form and content, 166–7, 172–3 truth, 123, 124 particular vs. general, 10, 158, Friedman, Michael, 43, 182n22 170–1 semantic, 158, 170–1, 219 generality, topic-sensitive and topic neutral, descriptive notion of, 179, 54–5 185–90 Couturat, Louis, 74n33, 78 normative notion of, 178–85 unrestricted, see unrestricted De Laguna, Theodor, 78 generality Dedekind, Richard, 49, 51n19 see also maximal generality

275 276 Index geometry Lewis Carroll, 146n42, 204–5 Euclid’s, 32, 35, 61 logic Kant’s theory, 39–43, 94–5, 100–1 demarcation of, 160, 169–70, 184 proof, 51–4 formal, see Kant, Immanuel, formal and pure mathematics, 44–5 logic Goldfarb, Warren D., 138, 146–7 justification of, 148, 153–6 Griffin, Nicholas, 3n3, 118–9, 235 transcendental, 163 logical complexes, 229–31 Heijenoort, Jean van, 7, 146 logical consequence, see consequence Hintikka, Jaakko, 77n37 logical constants, 10, 16, 172, 176–7, Husserl, Edmund, 21n5 190–3, 201, 236–7 Hylton, Peter, 7, 142–3, 148–9, 206–7, logical form 228 as determined by logical constants, 169–70, 175, 185, 193 implication in multiple relation theory of formal, 128, 129–33, 156–7, 195–7, judgment, 226–7, 229–30, 232–3, 198–9, 205–8 235–6 material, 195–7, 201 in traditional logic, 6, 38, 75, 164–6 independence proofs, 150–1 logical truth, 20, 25 infinite regress, 113, 118, 120–4, 230, as maximally general truth, 10, 131, 234–5 137–9, 175, 186, 193–4, 210, 215 Lotze, Hermann, 85–6 Jourdain, Philip, 151n49 MacFarlane, John, 161, 164n5 Kant, Immanuel, 180 McTaggart, John M. E., 2 analytic and synthetic judgments, Makin, Gideon, 187–8 21–2, 37, 218 Martin, Gottfried, 94–5 analyticity, 21–2, 164–5 maximal generality as a hallmark of applicability of mathematics, 44–6, logic, 10, 159, 185–90, 193 63 Meinong, Alexius, 21n5, 188–9 concepts and intuitions, 38–9 metalanguage, 150 conditions of objective thought, 10, and universality of logic, 8, 142–3, 100–1, 162–3 157 criticism of Leibniz, 5–6, 37 Mill, John Stuart, 36, 218 formal logic, 9–10, 38, 40, 61–2, modalities 94n14, 160–4 absolute vs. relative, 91–3 geometrical constructions, 40–3 see also necessity, Russell’s rejection intuitions in geometry, 62–3 of (pure) general logic, 9–10, 75, Moore, George Edward, 2, 14, 46, 100–1, 157–8, 162–4, 165, 180 69n32, 85n3, 103, 126, 184, 220 real necessity and logical necessity, constitution of propositions, 90–102 110–14 criticism of Kant, 10–11, 87–9 Lakatos, Imre, 29 multiple-relation theory of judgment, Landini, Gregory, 77, 132, 148, 221–2, 107, 121, 172, 223, 225–9, 234–6 229n53 Leibniz, Gottfried Wilhelm von, 5, necessity 24, 35, 55, 58, 165 psychological notion of, 83–4, criticized by Kant, 37 85–6, 89–91, 180 criticized by Russell, 47–8, 87–8 Russell’s rejection of, 174, 202, 215 Index 277

Newton, Isaac, 71–2 rules of inference, 130–1, 145–7, 152, non-Euclidean geometry, 27 156, 195–8, 204, 206–9 Russellian propositions, 8, 10–12, paradoxes 107–10, 117–18, 219–20 Cantorian, 221–2 as complexes of constituents, of self-predication, 211–3, 229n52 14–15, 109, 111–12, 135–6, 225 Pasch, Moritz, 51–5 Passmore, John, 1 Schröder, Ernst, 18, 46n13 Peano, Giuseppe, 1, 46–7, 48–9, Schulze, Johann, 42, 95–6, 99 50n18, 55, 58, 129–30, 142 self-evidence Pieri, Mario, 46n13 Russell on, 55–8, 156 Poincaré, Henri, 50n16, 178 Sheffer, Henry M., 150, 156 criticism of formal logic, 19 Stout, George Frederick, 2 mathematical induction, 59–60 Strawson, Peter, 80–1 Prawitz, Dag, 154–5 substitutional theory of propositions, Prichard, Harold Arthur, 81 135–6, 214, 220–2 principles of inference, see rules of syntheticity inference Russell’s use of, 7, 23–5, 77–9, propositional functions, 139, 174, 228 209–11 explained in Principles, 126n25, 126–7 tautology, 15, 215–18, 237 and Kant, 73–5, 211 Thompson, Manley, 75, 163n4 in substitutional theory, 135–6 truth propositional unity, 14, 115–23, definition in the multiple-relation 132–6, 224–5, 226 theory, 223–4, 227 distinguished from fact unity, 121 see also logical truth propositions truth primitivism, 11, 117–8, 144–5 distinguished from facts, 224 Russell’s argument for, 123–5, in multiple-relation theory of 224–5 judgment, 227–9 types, 142n39 Russellian, see Russellian in Principles, 213–4 propositions in the substitutional theory, 221–2 psychologism, 80, 84, 108, 110n4, 180, 183–4, 228 unity, see propositional unity universality of logic, 137–9, 192–3 Ramsey, Frank Plumpton, 122n23, unrestricted generality, 131–2, 186, 216–7 189–90, 214 relations primacy of, 12–14, 142–3 internal and external, 120 positional, 120–1 validity of inference, 50, 52, 59, 61–2, relating, 118–21 154–6, 171, 175, 184–5, 193–6, Ricketts, Thomas, 144n41, 175 198–209 rigour variables in Principles, 128–9 epistemic notion of, 28–30 mathematical, 25–7, see also Weierstrass, Karl, 49 deductive rigour Wittgenstein, Ludwig, 2, 122n23, semantic notion of, 30–3 147n24, 202, 207, 216–8, 237n53