The Southern Journal of Philosophy (1991) Volume XXIX, No. 2

ABSTRACT SINGULAR REFERENCE: A DILEMMA FOR DUMMETT Alexander Miller University of Michigan

I. Introduction

Consider a language which contains abstract singular terms ostensibly standing for the directions of lines. Then we can ask the following question: are these abstract singular terms genuine referring expressions? In other words, do they serve to pick out real, extra-linguistic, external, and independently existing features of the world? Given that such terms are susceptible to contextual definition-can be defined in such a way that sentences containing them can be transformed into equivalents which do not-two broad answers suggest themselves to our opening question. The first answer-what we might call the reductionist response-takes the contextual definitions as showing that we need not view the abstract singular terms as genuine referring expressions at all, since we can always ‘explain away’ their occurrence via those very definitions. The second answer-what we might call the Platonist response1-argues that we cannot but construe the abstract singular terms as genuine singular terms, and that this fact, combined with the fact that there are at least some true (mathematical) statements containing them, suffices to confer the status of ‘genuine referring expressions’ upon them. According to this latter view, directions-the objects denoted by the abstract singular terms ‘the direction of a,’ ‘the direction of b,’ and so on-are just as real ingredients of the world as the objects denoted by the allegedly less problematic class of concrete singular terms.2

In his (l), Michael Dummett has argued against the second of the two answers above, and in so doing has suggested a

Alexander Miller is a graduate student at the University of Michigan, Ann Arbor. He completed his undergraduate degree in philosophy and mathematics at the University of Glasgow, Scotland, before taking a masters degree in philosophy at the University of St. Andrews, Scotland. He has published in Analysis, and has some work forthcoming in Mind. He is currently working on a thesis in the philosophy of language.

257

third possible answer to our question: namely, that abstract singular terms are, contra the reductionist, to be viewed as genuine singular terms and thus-given the truth of the appropriate sentences-as having reference, but the relation of reference involved is, or must be viewed, as divested of any explicitly realistic connotations. Whereas in the case of a concrete singular term we have a relation to a n external, independently existing particular, in the case of abstract singular terms, reference becomes “a matter wholly internal to the language.”3

Dummett’s argument for this conclusion comes in roughly two stages. First, he argues that the contextual definitions cannot be read in such a way as to deny significant semantic structure-in the shape of an assignment of a referential role-to the abstract singular terms which feature on their left-hand sides. That is, he argues against what Wright calls the ‘austere’ reading of the contextual definitions, which would deny the abstract singular terms any such role. He then goes on to argue that there are sufficiently strong disanalogies between certain facts concerning abstract and concrete singular terms to suggest that they cannot both be viewed as referring in the same manner, and therefore, that given the indisputably realistic mode of reference exemplified by concrete singular terms, we have no option but to see abstract singular terms as exhibiting reference of some non-realistic genre. In what follows I shall reiterate the grounds for the first part of the Dummettian argument-for the non-austerity of the contextual definitions-and I shall then attempt to strengthen the case against the second part by suggesting tha t the Dummettian arguments for the existence of the relevant disanalogies are, prima facie, in some considerable tension with more general features of Dummett’s interpretation of Frege’s philosophical semantics.

11. Austerity and Non-Austerity

What is the austere interpretation of the contextual definitions which equate direction talk with seemingly more mundane talk of concrete lines and their properties? Let us look specifically at the second of the two relevant contextual definitions in order to get a grip on this idea:

0 k (D(a)) iff Fka, where ‘11’ is a congruence for ‘Fk’.

Intuitively, the left-hand side looks like a subject-predicate sentence: a property is being predicated of D(a), the direction of a. So the intuitive interpretation is that there is an object

258

D(a), which object has the property 0 k (all this, of course, on the assumption that 0 k (D(a)) is, in the given instance, true). The austere reading attempts to reject this intuitive interpretation by rejecting the idea that there is in fact any predication occurring in the left-hand side: rather, the expression ‘0 k(D(a))’ is to be read as a sentence equivalent to that on the right-hand side, but possessing itself no significant internal semantic structure-so that its truth, which the reductionist is happy to accept, imports no troublesome objectionable existential presuppositions into the language.

Wright provides a strong argument against this reading? namely, that if we accept it we shall be at a n utter loss to accommodate facts about some important inferential relations the sentences on the left-hand sides might have with other sentences-for example, we shall be unable to account for the validity of the inference from ‘0(D(a))’ to ‘There exists a n x such that 0(x)’-and I shall suggest briefly three other considerations which can be viewed as militating against the austere interpretation.

On recent accounts of semantic structure, the following sorts of consideration are taken to be, if not constitutive of the existence of semantic structure, at the very least a crucial source of evidence suggestive of its existence: facts about the patterns to be found in a competent speaker’s acquisition of linguistic abilities, facts about the patterns to be found in his loss of those abilities, and facts about what changes are entailed (as a matter of fact) by his revision of certain of his semantic beliefs.5 And such evidence, suggestive of the fact that 0, (D(a)) does indeed have the semantic structure we intuitively think it has, will be readily forthcoming. Consider the following two sentences:

(1) The direction of a is that which is taken by migrating birds every autumn,

(2) The direction of b is such that if you follow it from here you will eventually reach the taj-mahal.

We can expect to find, and indeed generally would find, that exposure to, and training in the use of, (1) and (2), will suffice to impart mastery of (3) and (4), where,

(3) The direction of a is such that if you follow it from here you will eventually reach the taj-mahal,

(4) The direction of b is that which is taken by migrating birds every autumn.

259

The idea is, in a nutshell, that this fact about acquisition, and the related facts about loss and revision, of semantic competence, would be utterly inexplicable unless we were willing to read the sentences as genuinely structured, and as containing, in particular, singular terms standing for the direction of a and the direction of b. So in the face of all this evidence the reductionist would seem to be being simply obstinate in refusing to discern the requisite semantic structure in the left-hand sides of the contextual definitions: if he is not willing to countenance semantic structure in this case, when will he do SO?^

111. The disanalogies concerning explanation and description

Why then, given that he accepts tha t the contextual definitions have to be read non-austerely, does Dummett insist on refusing abstract singular terms thus defined a referential role on a par with that possessed by their concrete counterparts? The reasons for this are twofold. Firstly, Dummett argues that there a re sufficiently s t rong disanalogies between the parts played by reference in the explanation of the meanings of concrete and contextually defined abstract singular terms to sustain this conclusion. And secondly, Dummett argues that the same conclusion follows because of the asymmetry in the way in which reference features respectively in the description of the abil i t ies consti tutive of mastery of the two sor ts of expression.

What, then, is the disanalogy between the way in which the two sorts of expression are explained? Summarizing, and skipping over the detail of Wright’s painstaking unravelling of Dummett’s suggestions here,’ the disanalogy is this. Take the case in which we are attempting to explain to someone the meaning of some putative referring expression, in the case where the speaker has no other, antecedent, means of effecting reference to the object which it denotes-for example, a case in which we are attempting to impart to someone an understanding of demonstratives used to refer to persons, where that speaker has no prior means of referring to persons. As Wright puts it, there is not a great deal to say about this case other than that the speaker must be ‘plunged’ into a linguistic practice which involves the use of recognition statements consisting of the identity sign flanked, on the one hand, by a demonstrative phrase and, on the other, by a descriptive personal singular term. But crucial to Dummett’s story is the fact that:

260

‘It is quite unthinkable that such training should be successful unless the use of the relevant sentences is witnessed against a background wherein the referents of such (uses of) singular terms are perceptually confronted . . . . [and] the only coherent conception we can form of what is involved in acquiring a mastery of the most basic devices of reference requires that the trainee be capable of some sort of primitive intellectual contact with those objects . . . . I have to be capable of a primordial intellectual detachment of a person from the background in which he is presented; only thereby can I be in a position to make any sense of the linguistic practice into which I find myself “plunged”.’s

And this is where the difference between the two cases lies, for in the case of contextually defined abstract singular terms, not only is such confrontation and primitive intellectual contact unnecessary, it is not even possible. So, in summary, in the case of a concrete singular term, the referent plays an essential part in the explanation of the meaning of that term to someone completely innocent of alternative modes of reference to it, in a way in which the putative ‘referent’ of a contextually defined abstract singular term manifestly does not.

The second source of the disanalogy concerns the way in which competence with the two sorts of expression is to be described. In the case of concrete singular terms the ability constitutive of mastery with ‘X’ is naturally construed as an ability to determine the truth of the recognition statements of the form ‘This is X,’ and as Wright puts it ‘it is simply of the essence of the central use of demonstratives that their referents be (literally) open to view simultaneously with their use.’9 The ability here must therefore be viewed as being based on a response to the referent. Again, matters stand differently in the abstract case: although in this latter case the ability is to be construed as an ability to recognize the truth of identity statements featuring both demonstratives referring to abstract objects and some other abstract singular term ‘grasp of the meaning of any such statement will always be parasitic on an understanding of an identity statement in which both flanking terms are non- demonstrative.’ lo It follows that in the abstract case, unlike the concrete case, we simply cannot view the appropriate ability as based on a response to a referent. Thus, in summary, in the case of a concrete singular term, the referent plays an essential role in the description of the ability constitutive of understanding that term, in a way in which the putative ‘referent’ of a contextually defined singular term manifestly cannot.

26 1

IV. Sense, reference, and ingredients in meaning

These, then, are Dummett’s main reasons for denying ‘real’ reference to the contextually defined abstract singular terms under discussion. There appear to be two general ways in which Dummett’s position here can be criticized. Firstly, we might simply accept that the disanalogies exist, and argue on behalf of the Platonist that this does nothing to vitiate the ascription of ‘real’ reference to abstract singular terms. I will not pursue this line of argument here.“ Secondly, we might attempt, somehow, to cast some doubt on the disanalogies themselves. In the present section I attempt to do just this, albeit in a somewhat indirect manner.

To do this, let us backtrack some four hundred pages to chapter five of Dummett (1).12 Here Dummett himself notes that there is, at least prima facie, some internal tension within the account he give of Frege’s distinction between sense and reference. As he puts it, the problem is to see how, given his characterization of what sense and reference are, any room can be left for the former notion over and above what is already captured via the latter.

Why does Dummett think this? According to Dummett there are two clearly separable constituents on the Fregean notion of ‘bedeutung.’ On the one hand, that of semantic value (or semantic import or semantic power), which is defined to be that feature, or maybe those features, of an expression in virtue of which it has an impact upon the truth-values of those sentences in which it appears. On the other hand, reference is construed as that relation which obtains between the expression and some independent, external constituent of the world. Then, according to Frege, in the case of proper names these two notions of reference happen to coincide-the semantic value of a proper name simply is its reference in the latter of the two senses distinguished. Once the reference (in this latter sense) is in place, the idea seems to be, the truth-values of the appropriate class of sentences will take care of themselves-

‘Once the reference of each word in a sentence has been determined, the truth-value of the sentence is thereby determined,’13

‘On Frege’s view, it is precisely via the reference of the words in a sentence that its truth-value is determined,’14

‘The semantic value of any expression is . . . . that feature of it which must be ascribed to it if every sentence in which it occurs is to be determined as true or false . . . . the semantic value of a singular term is the object to which it refers.’ls

262

‘Frege was distinctive in supposing that the semantic power of an expression was determined by that expression being associated with some extra-linguistic reality.’ 16

But now consider Dummett’s characterization of the notion of sense:

‘The sense of an expression is . . . that part of its meaning which is relevant to the determination of the truth-value of sentences in which the expression occurs.’17

and the tension I mentioned above should be readily apparent: reference determines truth-values, sense is that part of a n expression’s meaning germane to the determination of the truth-values of the sentences in which it figures, therefore-it would seem-sense simply is reference: the sense of a n expression is its Fregean reference, and any scope for a substantial distinction between sense and reference would appear to have vanished.

Dummett’s attempted way out of this impasse is as follows: he sticks to his characterizations of sense and reference, but denies that Fregean reference is a part of meaning-he denies that it is a n ingredient in meaning. From this, and the characterization of sense, it follows that whatever sense and reference are they cannot be identical, for sense is a n ingredient in meaning while reference is not. And this is where the notion of ingredientship in meaning is to be spelled out as follows:

‘What we are going to understand as a possible ingredient in meaning will be something which it is plausible to say constitutes part of what someone who understands the word or expression implicitly grasps, and in his grasp of which his understanding in part consists.’ l8

Now what sort of content are we to give to the notion of ‘implicit grasp’ which features in this quotation? How can we give a n acceptable account, by Dummettian lights, of what it is for something to form part of a speakers ‘implicit grasp’? Dummett himself gives us little in the way of guidance as to how these questions are to be answered, but the following quotations might provide us with some clues:

‘Implicit knowledge cannot meaningfully be ascribed to someone unless it is possible to say in what the manifestation of that knowledge consists: there must be an observable difference between the behavior or capacities of someone who is said to have that knowledge and someone who is said to lack it.’19

‘We learn the meaning by learning the use, and our knowledge of its meaning is a knowledge which we must be capable of manifesting by the use we make of it.’20

263

These two passages would seem to suggest, or at least lend plausibility to, the view that we determine the content of a speaker’s possession of a piece of implicit knowledge by checking on what figures in the description of that speaker’s correct use of the expression in question, and by checking on what figures in the explanation of that correct use to that speaker. And this seems to chime well with Dummett’s conception of what a semantic theory actually amounts to-

‘Its function is solely to present an analysis of the complex skill which constitutes mastery of a language, to display, in terms of what he may be said to know, just what it is that someone who possesses that mastery is able to do.’21

-a theory of meaning for a language is a ‘theoretical representation’ of the ability that constitutes mastery or understanding of that language, and to say that a speaker implicitly knows ‘X’ is to say that ‘X’ features in some appropriate such theoretical representation. Of course, as it stands, this characterization of the possible constituents of implicit knowledge is probably insufficiently discriminating: we do not, presumably, want to allow anything which features in the relevant description as forming part of the speakers implicit grasp-we will refuse, that is, to include merely accidental features of the theoretical description, and restrict our a t tent ion t o those characterist ics which essentially so feature. So the suggestion is, on a very crude first approximation, that: X forms part of a speaker’s implicit grasp of the meaning of an expression ‘Y’ just in case X essentially features i n the description of t he abil i ty consti tutive of understanding ‘Y,’ or jus t in case X essentially features in the process whereby the ability to correctly use ‘Y’ is imparted to that speaker as a novice.22 This gives us a clear-cut, respectable, way (barring any worries concerning the notion of a n essential feature) of determining what forms part of a speaker’s implicit grasp of the meaning of an expression, and it does so in such a way that the content of any implicit knowledge ascribed does not transcend the use which is made of the relevant expression. And given that the notion of ingredientship in meaning is to be spelled out via the notion of implicit grasp or knowledge, we thereby also attain a clear-cut perspective on the former.

The problem which I have been setting the scene for here should now be clear: is not the conclusion of the argument I rehearsed in the previous section just the claim, that in the case of a concrete singular terms, each of the two conditions

264

on implicit grasp, and hence on ingredientship on meaning, happen to be satisfied by the reference of that name? Does it not then follow, contrary to Dummett’s express intentions, that reference in this case cannot but be construed as a n ingredient in meaning?

How, then, do matters stand with the alleged disanalogies between concrete and abstract singular terms? Given our filling out of his notion of implicit knowledge, Dummett would now seem to be faced by the following dilemma. On the one hand, he could simply swallow the conclusion that in the case of concrete singular terms the reference is a n ingredient in the meaning-but this is hardly appealing, as the whole thrust of Frege’s famous argument concerning the possibility of informative identity statements is, according to Dummett, towards denying this very f ac t the argument is, if reference were such an ingredient then the possibility of such statements would thereby be precluded.23 On the other hand, he could look for some motivation for the claim that although, in the concrete case, reference features in the relevant explanations and descriptions, this is not an essential feature of those explanations and descriptions. This, I think, is the direction in which Dummett must ultimately move, although it is not initially obvious what independent motivation there could be for such a move. But a move there has to be-and the upshot of this move will be that there are now no significant disanalogies of the sort initially adumbrated. Just as the requisite facts concerning the explanation and description of competence with abstract singular terms can be given without involving the notion of reference (in the second of the two ways distinguished earlier), so must the correlative facts in the case of concrete singular terms, if we are to have a distinction between sense and reference at all. And if this is true there seems to be no reason for attributing a type of reference-that imbued with ‘realistic connotations’-to one class of singular terms, the concrete ones, whilst refusing to attribute the same type of reference to their abstract counterparts.

V. Grasping the second horn: an independent motivation?

In the previous section I argued that reference cannot play the parts in the explanation and description in the use of concrete proper names which Dummett requires in order to sustain the alleged disanalogies between concrete and abstract singular terms, unless Dummett jettisons his interpretation of how and why Frege draws the sense/ reference distinction in the first place. I also said that there

265

appeared, on the face of it, to be no independent motivation for denying reference those parts. In this final section I want briefly to suggest that such independent motivation for this conclusion might be gleaned from some of the remarks which Bob Hale makes in his discussion of the possibility of identifying thought about abstract objects.24

Let us remind ourselves briefly why reference appeared essentially to enter the picture in the case of competence with concrete singular terms. Roughly, reference entered the picture because demonstrative identification and recognition statements did so. In the argument concerning explanation, demonstrative identification entered the picture because it was viewed as providing a ‘prototypical’ mode of reference in terms of which less fundamental, non-demonstrative modes of reference could be accounted for. And in the argument concerning the description of competence it entered in the following way-competence with a concrete proper name was construed as possession of a n ability to determine the truth-values of recognition statements in which that proper name features. This was because Dummett construes understanding a concrete singular term as being able ‘at least in principle, to identify the referent of the term as such if it is appropriately presented,’ and ‘with concrete terms the natural form of expression for one’s recognition of the referent, actually confronted, will be a n identity statement in which one of the flanking terms is that singular term and the other is some sort of demonstrative phrase.’25

So, one way to loosen the role played by reference in the explanation and description of competence with concrete singular terms would be to challenge head-on the construal of tha t competence as being basically recognitional in character: we get the disanalogies discussed only if we buy into the recognitional conception of knowledge of sense and view recognition statements as basic-so if we can show that th i s la t ter conception is flawed we will obtain some independent motivation for rejecting the disanalogies.

As a n example of a case in which the recognitional construal proves to be inadequate, we might consider the name of some temporally distant object-Hale uses the example of ‘General Gordon.’26 On the Dumrnettian conception, competence with ‘General Gordon’ consists in the ability to recognize the object denoted by it, if presented. We should thus expect that ability to be exercised in any case in which a speaker manifests an understanding of a sentence i n which it is contained. But, we can ask, how is a recognitional capacity exercised in the determination of the

266

truth-value of, e.g., ‘General Gordan drank heavily’? In effecting this determination I do not appear to be exercising the recognitional capacity supposedly constitutive of competence with the name-it seems that I do not determine its truth-value via determining the truth-value of a recognition statement such as ‘This is General Getdon.’ However, and as Hale notes, this observation is not yet sufficient to discredit the Dummettian conception:

‘It will be enough for Dummett’s anti-Platonist purposes if the model describes a verification procedure that is always, at least in principle, available (i.e., for any kind of atomic statement which we can properly regard as true in virtue of states of affairs involving elements of an external reality).’27

That is, all Dummett requires is the possibility of our so determining the truth-value-but even granting this, no such possibility exists in the case of, e.g., ‘General Gordon was cut to pieces by the Mahdi’s dervishes’ or ‘General Gordon died some time before the sun finally set on the British empire.’ So it would appear that I can determine the truth- values of certain sentences containing names when there is no possibility of my doing so via verification of an appropriate recognition statement: we do not take this as signalling that the objects denoted by those names are not genuine features of an external reality, so why should the fact that I determine the truth-values of sentences containing abstract s ingular terms i n the absence of the same possibility be taken as signalling that the species of denotation exemplified by those names has to be shorn of any overtly realistic connotations? As Hale summarizes his conclusion:

‘If, in regard to perfectly respectable statements about concrete particulars, we may not always demand that determination of their truth-value shall proceed-or at least be such that it could proceed-via demonstrative identification of the objects they concern, the absence of this possibility in the case of statements about (pure) abstract objects can be taken as casting doubt upon their external reality only at the price of acknowledging that it casts a parallel doubt upon that of those spatially or temporally remote or otherwise relatively inaccessible concrete objects.’28

Even if the determination of the truth-values of recognition statements (to do with concrete objects) must proceed via demonstrative identification of those objects, if we can show that the ability to effect such determination is not always necessary for possession of t he relevant species of competence, then i n effect we will have provided a motivation for grasping the second horn of the dilemma

267

which I claim Dummett finds himself facing. As this is precisely what Hale attempts to do, his argument can perhaps be viewed as reinforcing mine. In any case, even if Hale’s argument is ultimately unsuccessful, this would not absolve the Dummettian from facing square-on the argument that I sketched in the previous section.29

NOTES

1 The notion of platonism in play here should be distinguished from that of the Godelian variety-there is no commitment here to the claim that our knowledge of mathematical truths is to be accounted for via some super- perceptual encounter with mathematical objects. See Wright (1) for more on why the two notions of platonism need not be run together.

2 I am of course taking the concrete/abstract distinction for granted here-but it is itself a substantial philosophical issue as to how this distinction is actually to be drawn. See in particular Dummett (l), pp. 470- 494.

3 Dummett (l), p. 499.

5 For a summary of these ideas, see Davies (1). ti Obviously, similar sorts of evidence will show that D(a) is semantically

structured too, being the product of the operation of the term-forming operator ‘D’ to the singular term.

4 Wright (l), pp. 67-70.

Wright (l), pp. 70-81. 8 Zbid., pp. 76-77. 9 Zbid., p. 76. 10 Zbid., p. 80. 11 For a convincing pursual of this line, see Wright (l), pp. 81-84. 12 In particular, pp. 90-96. l3 Dummett (l), p. 91. 14 Zbid., p. 93. 15 Dummett (Z), p. 120. l6 Evans (l), p. 8. l7 Dummett (l), p. 89. l8 Zbid., p. 93. 19 Dummett (3), p. 99. 20 Zbid., p. 107. 21 Dummett (4), p. 70. 22 It might be argued that we should take the condition as only necessary

but not as sufficient. But then suppose that p and q both satisfy the condition. How are we to decide which, if any, of p and q, are to be counted as forming part of the speaker’s implicit grasp? Ex hypothesi, not by recourse to that speaker’s explicit verbal utterances, and not by reference to his behavior (since they both, ex hypothesi, feature essentially in the relevant descriptions). So in order to obtain a workable notion of implicit knowledge, the conditions have to be taken as sufficient, as well as necessary.

23 Although Dummett does express himself rather unclearly in this regard. Speaking of the argument about informative identity statements he says that ‘Frege now argues that the sense of a proper name cannot merely consist in its having the reference that it has’ (p. 94). The suggestion of this sentence seems to be that Frege is going to go on to argue that the sense of a proper name must consist in its having the reference that it has together with some other, at present unknown quantity, which we can call

268

X. (Why else would he have used the word ‘merely’?) But if this is indeed what Dummett means then I think he is in grave danger of lapsing into inconsistency. To see this, consider the following rerun of the argument concerning the informativeness of identity statements which he attributes to Frege: ‘Suppose that the sense of a proper name consists in its having the reference that it has, together with some other (unspecified) semantic property X. Then, anyone who grasps the sense of “a=b” will, ~ n c e he grasps the sense of “a” and the sense of “b,” know that “a” has the reference it has together with an additional semantic feature, and also that “b” has the reference it has together with an additional semantic feature. Then, by a modified version of the principle of the transparency of sense (see below), he will know that “a” and “b” have indeed got the same reference; and so the possibility of there being identity statements which are nevertheless informative will again be precluded.’ I think the transparency principle that we require here is something like: ‘If F forms part of the sense of P, and G forms part of the sense of Q, and if F=G, then anyone who knows the sense of P and the sense of Q, thereby knows that F and G are identical.’ I will not attempt to defend this modified version here-it suffices to say that it is at least as plausible as Dummett’s original transparency principle (see p. 95). Now where does this leave Dummett’s contention that Frege was in fact arguing that ‘the sense of a proper name cannot merely consist in its having the reference it has’? I think that Dummett has to drop this contention, or drop his account as to how Frege’s argument proceeds. Given that he wants to keep his story about the informativeness of identity statements and the principle of the transparency of sense, I think he really meant to interpret Frege as arguing that ‘having the reference that it has cannot form any part of the sense of a proper name’-i.e., that reference is never an ingredient in meaning.

24 See Hale (l), Chapter 7, sections I1 and IV.

26 Hale (l), pp. 178-180. z7 Ibid.. D. 178.

25 Wright (l), pp. 78-79.

28 Ibid.; i. 179-180. 29 I would like to thank both Crispin Wright and Bob Hale for their

comments on an earlier draft of this paper.

REFERENCES

Davies, M. (1) ‘Tacit knowledge and semantic structure: can a 5% difference

Dummett, M. (1) Frege: Philosophy of Language (Duckworth 1981). Dummett, M. (2) ‘Frege’s distinction between sense and reference,’ in his

Truth and Other Enigmas (Duckworth 1978). Dummett, M. (3) ‘The philosophical basis of intuitionistic logic,’ in Putnam

and Benacerraf (eds.) Philosophy of Mathematics: Selected readings (2nd Edition, Cambridge, 1983).

Dummett, M. (4) ‘What is a theory of meaning? (II),’ in Evans and McDowell (eds.) Truth and Meaning: Essays in Semantics (Oxford 1976).

Evans, G. (1) The Varieties of Reference (Oxford 1982). Hale, B. (1) Abstract Objects (Basil Blackwell 1987). Wright, C. (1) Frege’s Conception of Numbers as Objects (Aberdeen

matter?,’ in Mind, 1987.

University Press, 1983).

269