In System of Logic, i.vi, Mill distinguishes between Verbal’ and ‘real’ propositions. His immediate aim is to apply the distinction to the analysis of classification and definition which follows in the next two chapters.
Mill’s attitude to traditional Aristotelian theory on these matters is two-sided—a case of his favourite Coleridgean policy of finding in old traditions valuable truths clothed in misleading forms. He wants to free the schoolmen’s Aristotelian doctrines of classification and definition from their essentialist underpinnings. But he accepts that an important truth is contained in their notion of ‘natural kinds’. ‘Kinds have a real existence in nature’ (i.vii.4) —the problem is to explain what constitutes a natural kind without relying on essences. Mill rightly sees that the notion of natural kinds plays an indispensable role in scientific—or indeed any—thinking about the world; he accepts that that fact calls for explanation from an essence-dispelling radical empiricist. His positive theory, which is that a natural kind of object, or a natural kind of stuff, is set apart by ‘an indeterminate multitude of properties not derivable from one another’ (VII 126) hardly rises to the real issues, though it does raise interesting questions of its own. We shall not pursue it,1 but we shall examine the negative side of the case, namely, his rejection of essential properties.
However the distinction between verbal and real propositions (and the corresponding distinction between real and ‘merely apparent’ inferences which Mill makes in ii.i) also has a larger, indeed, central, purpose. It will be used to show, as Mill paradoxically puts it, that ‘All deductive sciences are inductive’ (VII 252)—that is, that logical and mathematical knowledge is empirical knowledge, grounded ultimately on inductive reasoning.
The strategy involves three steps. First, to show that logic and mathematics contain real propositions and inferences. Second, to establish the cardinal Millian tenet that no real proposition or inference is a priori. Third, to explain and vindicate the inductive character of our logical and mathematical knowledge.
The strategy is implemented by Mill in a sustained argument which occupies the first three books of the System of Logic. We shall follow it in this and the next three chapters. This chapter examines and criticises Mill’s way of making the distinction. We shall also consider his remarks about essentialism and his account of definition. The next chapter considers Mill’s analysis of syllogistic reasoning, and thus begins our examination of the first step of Mill’s strategy. The examination continues in chapter 5, in which we shall also consider the second step. The final step, Mill’s analysis of the character and status of induction, will be considered in chapter 6.
…we may predicate of a name which connotes a variety of attributes, another name which connotes only one of those attributes, or some smaller number of them than all. In such cases, the universal affirmative proposition will be true; since whatever possesses the whole of any set of attributes, must possess any part of that same set. A proposition of this sort, however, conveys no information to anyone who previously understood the whole meaning of the terms. (VII 112–13)
Mill calls such propositions Verbal’ (VII 115). He mentions, in this passage, only universal affirmative propositions (All S are P), but the criterion as he states it applies to all affirmative propositions: particular (Some S are P) and singular (S is P) as well. In fact he means the distinction between Verbal’ and ‘real’ to apply exhaustively to all propositions. Extending it to negative propositions containing connotative names (‘No bachelor is married’) poses no difficulty,2 but the case of propositions containing only non-connotative names will be considered separately in the next section. They are, specifically, identity propositions whose names are proper names. Mill thinks they are verbal too, and he has separate reason for thinking them so. Leaving them aside for the moment, however, the idea is that in a verbal proposition the attributes connoted by the predicate are a subset of the attributes connoted by the subject. Let us call this the criterion of ‘connotative inclusion’.
The last sentence of the passage just quoted adds another point: a verbal proposition ‘conveys no information to anyone who previously understood the whole meaning of the terms’. A few pages later, Mill remarks that a verbal proposition is one which
asserts of a thing under a particular name, only what is asserted of it in the fact of calling it by that name; and which therefore either gives no information, or gives it respecting the name, not the thing. (VII 115)
Real propositions, on the other hand,
predicate of a thing some fact not involved in the signification of the name by which the proposition speaks of it; some attribute not connoted by that name…. When I am told that all, or even that some objects, which have certain qualities, or which stand in certain relations, have also certain other qualities, or stand in certain other relations, I learn from this proposition a new fact; a fact nor included in my knowledge of the meaning of the words, nor even of the existence of things answering to the signification of those words. It is this class of propositions only which are in themselves instructive, or from which any instructive propositions can be inferred. (VII 115–16)
Only real propositions convey information. Verbal propositions are void of genuine information content. Therein lies the point of the distinction: it gives Mill an essential tool for his epistemological analysis of logical and mathematical inference. The criterion of connotative inclusion picks out some propositions as verbal and explains just how it is that they lack content.
All of this is clear. But complications arise from the fact that the assertion of a proposition may carry existential commitments, and Mill makes pretty heavy weather of clearing them up. He believes all propositions (universal as well as particular and singular ones) normally carry an ‘implied assertion’ or ‘tacit assumption’ that there exist objects denoted by the subject name (VII 113). A speaker who assertively utters a real proposition, Mill thinks, is doing something: he is asserting of the, some, or all objects denoted by the subject name that it or they have certain attributes connoted by the predicate name. In doing so the speaker conveys his belief that there are such objects; if there are none, he fails to make an assertion. The existence of the objects is ‘really’ implied or presupposed by the assertive utterance of the proposition.
But the assertive utterance of a verbal proposition is a different story. Mill evidently thinks that someone who assertively utters a proposition like ‘All vixens are foxes’ cannot be understood to be referring to vixens, and saying of them that they are foxes. He is making no real assertion about vixens. Rather, he must be understood to be making a semantic assertion about the name ‘vixen’—giving a partial definition of its meaning.
Where the assertive utterance of a verbal proposition has point, then, the point can only be to define or elucidate the meaning of the subject name, and hence the existence of objects denoted by the subject name is not ‘really’ implied—that is, it is not a precondition of the utterance achieving its object.
Nevertheless there remains an apparent implication of existence. It ‘arises’, Mill thinks, ‘from the ambiguity of the copula’ which apart from indicating predication is also ‘a concrete word connoting existence’. The implication is no part of the point of the utterance, which is to explain the meaning of the subject name: ‘we may say, A ghost is a disembodied spirit, without believing in ghosts’. It is not, therefore, ‘really’ implied by the assertive utterance of the proposition. But Mill thinks that it is usually conveyed, even when a definition is explicitly given in metalinguistic form. The two points, that an existential implication is typically conveyed by a definition, and that it is detachable or cancellable, will be important in Mill’s discussion of geometry and arithmetic.3
Verbal propositions, then,
do not relate to any matter of fact, in the proper sense of the term, at all, but to the meaning of names. Since names and their signification are entirely arbitrary, such propositions are not, strictly speaking, susceptible of truth or falsity, but only of conformity or disconformity to usage or convention; and all the proof they are capable of, is proof of usage; proof that the words have been employed by others in the acceptation in which the speaker or writer desires to use them. (VII 109)
An instructive proposition can be true or false, in the strict sense in which truth is a matter of correspondence with facts which are not merely conventional facts about the rules of use governing the language in which we describe the world. Only such propositions, propositions to which the conception of truth as correspondence has a genuine, non-honorific application, have information content. A real proposition has genuine truth-conditions; to understand it is to grasp those rruth-conditions—to grasp what facts must obtain for it to be true.
Mill briefly mentions in i.vi.1 that propositions ‘of which the subject and predicate are proper names’ belong to ‘the class of merely verbal propositions’ (VII 110). He takes them to be verbal because, as we noticed at 2.5, he thinks they assert only that two names have been conventionally assigned to the same individual. He makes no mention of the obvious point that they cannot be accounted verbal by the eriterion of connotative inclusion, since neither the subject name nor the predicate name has connotation.
If Tully is Cicero’ did have a metalinguistic truth-condition, then it ought to be classed a real proposition conveying information about the meaning of words. It does not have; it has a truth-condition provided for it as in the account in 2.7. Semantic theory provides a truth-condition for ‘Tully is Cicero’ just as it provides a truth-condition for ‘All vixens are foxes’. Still, it does not follow that either proposition has information content. The essential point is that there is no fact in the world to which if true they correspond, nor is understanding them a matter of grasping how the world must be for them to be true.
But even if we accept that an identity proposition with non-connotative names corresponds to no fact in the world, should we say that it has no information content? As Frege pointed out, such a proposition—his example is ‘Hesperus is Phosphorus’—may be a posteriori. It cannot be seen to be true by reflection on the ‘meanings of words’ alone—as Mill several times says verbal propositions can be. Discovering that it is true certainly in some way gives us information which we did not have before.
How can a proposition which corresponds to no fact in the world be a posteriori? There are really two questions. First, why cannot we know the truth of Tully is Cicero’ just by reflecting on the meaning of the words? We have answered this question in 2.7. The second question is, in what way does finding out that it is true give one new information, and what is the new information given?
To know the semantic content of ‘Tully’ and ‘Cicero’ is to know, respectively, that ‘Tully’ denotes Tully, and that ‘Cicero’ denotes Cicero. Taken by themselves, these two pieces of semantic knowledge can yield neither the conclusion that Tully is Cicero, nor that Tully’ denotes what ‘Cicero’ denotes. Now the latter two propositions do not have the same meaning—yet given either of them, and in combination with the two pieces of semantic knowledge, the other can be deduced. It is in that way that Tully is Cicero’ conveys the information that Tully’ denotes what ‘Cicero’ denotes. Nor does it convey any information over and above that—for conversely, that Tully is Cicero can be inferred from ‘“Tully” denotes what “Cicero” denotes’: a proposition which is true simply in virtue of human decision and not of how the world is (except in so far as it requires the existence of the denotation). So in itself it contains no information content, even though asserting it may give someone new information.
As Mill says,
When we predicate of anything its proper name; when we say, pointing to a man, this is Brown. or Smith, or pointing to a city, that it is York, we do not, merely by so doing, convey to the hearer any information about them, except that those are their names. By enabling him to identify the individuals, we may connect them with information previously possessed by him; by saying, This is York, we may tell him that it contains the Minster. But this is in virtue of what he has previously heard concerning York; not by anything implied in the name. (VII 35–6)
Let us now jump to ii.i (‘Of Inference, or Reasoning in General’). In section 2 of this chapter (‘Inferences improperly so-called’) Mill makes a distinction between real and apparent inference which corresponds to the distinction between real and verbal propositions. An inference is ‘apparent, not real’ when ‘the proposition ostensibly inferred from another, appears on analysis to be merely a repetition of the same, or part of the same, assertion, which was contained in the first’ (VII 158).
In such cases there is not really any inference; there is in the conclusion no new truth, nothing but what was already asserted in the premises, and obvious to whoever apprehends them. The fact asserted in the conclusion is either the very same fact, or part of the fact, asserted in the original proposition. This follows from our previous analysis of the Import of Propositions. (VII 160)
To simplify terminology, and bring the two distinctions more closely into line, I shall talk of real and ‘verbal’ rather than real and ‘apparent’ inference. Consider the inference, P, …, Pn, therefore C; and the conditional, if P1&, …, &P then C. We shall call the inference the corresponding inference of the conditional, and the conditional, the corresponding conditional of the inference. One might take the notion of a verbal proposition defined in terms of the criterion of connotative inclusion, as basic, and define verbal inference as the corresponding inference of a verbal conditional. Or one might go in the opposite direction, taking the notion of a verbal inference as basic, and defining a verbal proposition as the corresponding conditional of a verbal inference.
The latter option is preferable. Consider the conditional ‘If Caesar is dead then Caesar is dead’. Mill would obviously have regarded this as a purely verbal proposition. But according to his view of conditionals, it is a proposition about propositions: The proposition, “Caesar is dead”, is inferable from the proposition, “Caesar is dead’”. To apply the criterion of connotative inclusion, we would have to show that inferability from the proposition, ‘Caesar is dead’ is one of the attributes connoted by the name The proposition, “Caesar is dead’”. Instead of taking on such complications, consider the corresponding inference: ‘Caesar is dead, therefore Caesar is dead’. Obviously this is a merely apparent, or verbal, inference: the proposition ‘ostensibly inferred’ is ‘merely a repetition’ of the premise.
Let us now generalise the definition of a verbal inference. An inference is verbal if and only if the set of propositions constituting the conclusion is a subset of the set of propositions constituting the premises. Next, we define verbal propositions in terms of verbal inference. In the first place, the corresponding conditional of a verbal inference is a verbal proposition. Consider the proposition ‘If Sheba is a vixen, then Sheba is a fox’. The corresponding inference is ‘Sheba is a vixen, therefore Sheba is a fox’. Analysis of the connotation of ‘vixen’ reduces ‘Sheba is a vixen’ to the conjunction ‘Sheba is female and Sheba is a fox’. Thus, by Mill’s account of conjunction, the propositions constituting the premise are: ‘Sheba is female’, ‘Sheba is a fox’. The inference is revealed as verbal, hence also the corresponding conditional. In general, determining whether an inference is verbal or real will call for an analysis of connotations, and an analysis of ‘compound propositions’ in terms of Mill’s definition of the connectives.
Consider next the proposition, ‘All vixens are foxes’. Generality introduces new issues which will be more fully treated in the next chapter. As will be seen, Mill does not have a consistent view of general statements. However, it is comparible with one of the lines he takes to treat ‘All vixens are foxes’ as meaning ‘Any proposition of the form “x is a fox” is inferable from the corresponding proposition of the form “x is a vixen’”. The proposition “Sheba is a fox” is inferable from the proposition “Sheba is a vixen’” and is a substitution-instance of this schema. We can now stipulate that a universal proposition is verbal if and only if all its substitution instances are verbal.4
Our amended definition of verbal propositions and verbal inference still excludes propositions whose constituent names are non-connotative (‘Hesperus is Phosphorus’) from the verbal category. Nor does it include such inferences as ‘Hesperus is a planet, Hesperus is Phosphorus, therefore Phosphorus is a planet’ as verbal. Mill would rightly deny that either the proposition or the inference was real. The truth of the proposition depends on no matter of fact; and the inference advances to no proposition whose truth depends on any facts other than those on which the truth of the premises depends.
But it was Wittgenstein, in the Tractatus, who found a way of making these points stand out clearly. He shows how a language could be constructed containing no sign of identity, but in which any information content expressible in our language (and not metalinguistic —about our language) could be expressed.5 In such a reconstructed language, identity propositions would drop out, and the distinction between real and verbal propositions and inferences would successfully pick out as real the intended class of propositions and inferences—those which are genuinely ‘instructive’. In the chapters that follow we shall ignore the case of identity propositions except where there is specific reason to take it into account.
In a footnote added to the 1862 edition of the System Mill says that his distinction between verbal and real propositions
corresponds to that which is drawn by Kant and other metaphysicians between what they term analytic and synthetic, judgements; the former being those which can be evolved from the meaning of the terms used. (VII 116)6
Kant’s definition of ‘analytic’ (Critique A6–7, B10–11) is given for affirmative judgements. (He says that ‘the subsequent application to negative judgements’ is ‘easily made’.) A judgement is analytic if the concept of the predicate is ‘(covertly) contained’ in the concept of the subject; ‘adding nothing through the predicate to the concept of the subject, but merely breaking it up into those constituent concepts that have all along been thought in it’. Mill’s definition in terms of connotative inclusion is obviously analogous, though Mill, as we have seen, rejects Kant’s Conceptualist terminology, of judgements and of one concept being contained in another.
Elsewhere in the Critique however (A151, B190–1) Kant puts forward what is in fact a different account of analytic judgements:
…if the judgement is analytic, whether negative or affirmative, its truth can always be adequately known in accordance with the principle of contradiction…. The principle of contradiction must therefore be recognised as being the universal and completely sufficient principle of all analytic knowledge.
Although Kant’s meaning is not completely clear this appears to define an analytic judgement as one from whose negation a contradiction can be deduced. Obviously, ‘deduced’ here means ‘deduced in accordance with the principles of formal logic’. And so in effect we have: an analytic judgement is one which can be formally deduced from purely logical principles, with the help, where necessary, of definitions of terms. It is in this broader sense that the term ‘analytic’ has come to be most commonly used.
To understand Mill’s overall view of logic and mathematics, it is viral to see that his ‘verbal’ does not correspond to ‘analytic’ as used in this wider and commoner sense. The essential point about a verbal proposition or inference—one which is ‘analytic’ in the narrow sense—is that it is epistemologically innocuous, and if necessary can be made by analysis to be perspicuously so. There can be no problem, for the most radical empiricist, about how we can know it to be true or sound. Or rather, we cannot be said to have genuine knowledge here, since no real proposition or inference is involved. No assertion as to real facts is made, no move from one fact to another is effected. ‘A priori’ knowledge of this kind is not, strictly speaking, knowledge at all. To show, by semantic analysis, that a proposition or inference is verbal is to dissolve any perplexity about how it could be a priori—to exhibit the triviality of its claim to that status.
In the broader sense of ‘analytic’, it is a definitional truism that the principles of pure logic are ‘analytic’. But there is no epistemological moral to be drawn from that. To show that a proposition is analytic in this wide sense—that it is reducible to logic—does not eliminate perplexities about its a priori status unless the a priori status of logic itself has been shown to be unproblematic.
It cannot be assumed that the propositions and inferences of formal logic are exclusively verbal. One of the great merits of the System is that Mill sees that crucial point clearly.7 To establish whether logic is verbal requires an analysis of the import of logical propositions. For Mill that meant an ‘Analysis of the Syllogism’, to which he turns in the chapter which follows the one in which he distinguishes real from verbal inferences (ii.ii, ‘Of Ratiocination, or Syllogism’). In the remainder of Book ii he makes many false turnings, and many diversions through irrelevant, though often interesting, territory. But the path’s eventual destination is clear. Real, and not merely verbal, propositions lie at the heart of logic. The laws of excluded middle and of contradiction are real propositions: hence, since no real proposition can be a priori, the evidence for them must be inductive.
When we assess Mill’s success in carrying through his programme we should keep three elements of it separate: there are the tools Mill develops for the semantic analysis of logic (the doctrine of denotation and connotation, the semantic account of sentence connectives and the distinction between real and verbal propositions and inferences); his conception of logic, which sees logical theory in terms of the traditional theory of the syllogism; and lastly, his actual success in applying his own tools to the analysis of logic as he understood it. Mill’s conception of logical theory is pre-Fregean, which is as much as to say, pre-modern. His analysis of syllogistic theory (ii.ii-iii) badly lacks rigour in applying his own semantic concepts, and is pulled in too many directions by too many aims. But that in no way shows that essentially Millian tools— those examined in this chapter and the previous one—cannot be applied to modern logical theory, and applied more carefully and single-mindedly than Mill applied them. And it does not show that doing so will not lead to precisely the same Millian result.
For the moment, however, we must return to vi.i, where the distinction between real and verbal propositions is introduced in the context of an attack on essentialism.
Essentialism distinguishes ‘essential’ from ‘accidental’ properties of things: a distinction much stressed, says Mill, by ‘the schoolmen’ and ‘almost all metaphysicians prior to Locke’ (VII 110). The essence of a thing, according to these metaphysicians, ‘was that without which the thing could neither be, nor be conceived to be. Thus, rationality was of the essence of man, because without rationality man could not be conceived to exist’ (VII 110). Properties which were not of the essence of a thing, i.e. without which the thing could be, were called accidents. Correspondingly, an ‘essential proposition’ was one in which only essential properties of things were predicated of those things, and an ‘accidental proposition’ was one in which some of the properties predicated were accidental.
To many philosophers—until recently—Mill’s criticism of this doctrine would probably have seemed definitive. It is worth quoting at length.
…man cannot be conceived without rationality. But though man cannot, a being may be conceived exactly like a man in all points except that one quality, and those others which are the conditions or consequences of it. All therefore which is really true in the assertion that man cannot be conceived without rationality, is only, that if he had not rationality, he would not be reputed a man. There is no impossibility in conceiving the thing, nor, for aught we know, in its existing: the impossibility is in the conventions of language, which will not allow the thing, even if it exist, to be called by the name which is reserved for rational beings. Rationality, in short, is involved in the meaning of the word man: is one of the attributes connoted by the name. The essence of man, simply means the whole of the attributes connoted by the word; and any one of those attributes taken singly, is an essential property of man. (VII 110–11)
Thus objects of a given class have an essential property only inasmuch as the class is defined by a connotative name; so that the question, what are the essential properties of an object, can only be answered relative to the connotation of some term by which it is described. This, as Mill says, is in effect Locke’s doctrine of nominal essence:
it was reserved for Locke at the end of the seventeenth century, to convince philosophers that the supposed essences of classes were merely the signification of their names; nor, among the signal services which his writings rendered to philosophy, was there one more needful or more valuable. (VII 112)
‘Essential propositions’ are simply verbal propositions. Given Mill’s view that proper names have no connotation, this has a corollary: no proposition of which the subject is a proper name is an essential proposition. Or—as Mill thinks, equivalently—‘individuals have no essences’ (VII 119).
Why did ‘these reflections, so easy to us’ not occur to the schoolmen? Because, Mill thinks, the doctrine of essences rested on a ‘theory’ according to which there exist ‘general substances’ named by some, but not all, general terms—for example by the term ‘gold’. Gold, the general substance, was thought to have properties, and to inhere in any chunk of gold; its presence, together with its properties, constituting the chunk a chunk of gold (VII 111). The theory of general substances, as Mill recognises in the next chapter (i.vii), does contain an insight; namely, that some classes of things constitute ‘real kinds’, and others do not. The pieces of gold scattered about the universe constitute a kind, the objects I picked up on the beach last Sunday do not. Mill undertakes to explain without metaphysical notions what makes a class of objects a ‘kind’, along the lines we noted briefly in section 1. But the idea of real essence he brusquely dismisses as verbal sorcery: ‘Metaphysics, that fertile field of delusion propagated by language, does not afford a more signal instance of such delusion’ (VII 127).
He is too brusque. Allow, for the sake of argument, that ‘Man is rational’ is a verbal proposition. Then that certainly shows one sense in which a man cannot be conceived who is not rational. But Mill’s point— that a being may be conceived exactly like a man in everything except rationality—does not show that no other inconceivability is involved.
There is a sense—the one noted by Mill—in which one cannot conceive that the mother of Casanova should have been childless. Since ‘The mother of Casanova had a child’ is a verbal proposition, no circumstances can be conceived which would make it false. Suppose however that someone says, The mother of Casanova might have had no children’. We would not take him to mean that circumstances may be imagined which would make the proposition, The mother of Casanova had a child’, false. He means that the individual—the person who in fact was the mother of Casanova—might have had no children, and thus not been a mother at all. (She might have died before child-bearing age, become a nun, etc.) In schoolmen’s language, we would take him to be saying that a certain property of Casanova’s mother—that of being a mother—is an accidental and not an essential property; and that is quite compatible with holding the proposition ‘The mother of Casanova had a child’ to be verbal.
It is obvious that Casanova’s mother might not have had children. But it is not so obvious that, for example, she might not have been human, or might never have been conscious, or might have had different parents. So these (unlike ‘being a mother’) look like essential properties of Casanova’s mother.
To identify the notion of an essential property we must, then, distinguish between saying that a proposition is necessarily true, and saying, of an individual, that that individual necessarily has a property: that is, that the property is an essential property of the individual. The essentialist need not believe that the proposition, ‘man is rational’, is verbal—he need not be interested in that question at all. What he is saying is that each individual man has the essential property of being rational: that it is necessarily true, of each individual man, that that individual is rational. Mill says that there could, for all we know, be a being exactly like a man in all points except rationality. But this does not in itself show, of any actually existing man, that he could have existed without rationality.
To say that rationality is an essential property of man is thus not at all the same as saying that ‘Man is rational’ is a verbal proposition. Attempting to ‘make sense’ of the former claim by interpreting it in terms of the latter only befogs a philosophical issue.
Of course, if Mill accepted these points about what is meant by an ‘essential property’ he could still deny that there are any. Compare the supposed metaphysical distinction between ‘necessarily’ and ‘contingently’ true propositions. Mill specifically asserts that the distinction is quite empty and groundless; he uses the word ‘necessary’ not ‘in its metaphysical but in its popular sense’ (see p. 130). But he grasps well enough what some philosophers understand by the alleged metaphysical distinction, and he does not at all deny the psychological fact, concerning what we do or do not find conceivable, which leads them to believe that certain propositions (for example ‘No two straight lines can enclose a space’) are ‘necessarily’ true. He thinks himself obliged to give an account of the psychological fact which will prevent it from being used in support of the metaphysical claim.
Essentialism requires a similar response. Mill’s reinterpretation of essence as ‘nominal essence’ stops him from seeing that by concealing the real strength of the essentialist case.8 But how the empiricist’s response should run in detail is no easy question. The distinction between essence and accident in some way turns on our general criteria of identity for objects, kinds and stuffs. These criteria however are not in any normal sense of the word conventional; a deep-seated psychological inevitability resides in our conceptualising the world as we do. The Millian empiricist must identify these psychological facts and then somehow cut off the support they seem to give to the essentialist’s metaphysical claim. A much more detailed inquiry than Mill provides would be necessary— into our conceptual scheme of substance, individual, kind and attribute —before one could say anything sensible about how that might be done.
We turn to i.viii, ‘Of Definition’. ‘A Definition is, a proposition declaratory of the meaning of a word’ (VII 133). So proper names, which have no meaning, ‘cannot be defined’; on the other hand, ‘In the case of connotative names, the meaning… is the connotation; and the definition of a connotative name, the proposition which declares its connotation’ (VII 133).9 A connotative name may connote a plurality of attributes, or just one. Where it connotes a plurality, the definition spells out the attributes connoted, and may be called (Mill takes the term from Condillac) an analysis;
To resolve any complex whole into the elements of which it is compounded, is the meaning of analysis; and this we do when we replace one word which connotes a set of attributes collectively, by two or more which connote the same attributes singly, or in smaller groups. (VII 134)
Analysable names are semantically complex. 10 Obviously there must also be semantically simple connotative names, names which admit of no semantic analysis. These, one might assume, would be the ones which connote a single attribute, such as ‘white’. Confusingly, Mill does not say that. ‘It might seem’, he says (VII 135), that the meaning of such names can be declared only by giving a synonym, where one exists, or in the form ‘“white” connotes the attribute whiteness’. But he thinks that their meaning can be further analysed: one can ‘analyse’ or ‘define’ the attribute which they connote.
The rest of i.viii.2 compounds the confusion. What does Mill mean by analysing the attribute itself? If the ‘single’ attribute connoted is genuinely simple, any proposed further ‘definition’ of it cannot be a matter of semantic analysis. If connotation is of attributes, the analysis of connotations can go no further than a specification of the simple attributes connoted. Is it then some other kind of analysis that is involved? The shift from ‘definition’ as a matter of declaring the attributes connoted by a name (i.viii.1) to talk of defining or analysing the attributes themselves (i.viii.2) suggests as much.
There is something important in the idea that a discontinuity is involved here, a shift from one kind or level of analysis to another. We will come to it in the next section. But at first sight Mill himself appears to see no such discontinuity. He does think a further semantic analysis of ‘white’ can be given: ‘Whiteness may be defined, the power or property of exciting the sensation of white’ (VII 136).
The only names’, he goes on,
which are unsusceptible of definition, because their meaning is unsusceptible of analysis, are the names of the single feelings themselves. These are in the same condition as proper names. They are not, indeed, like proper names, unmeaning; for the words sensation of white signify, that the sensation which I so denominate, resembles other sensations which I remember to have had before, and to have called by that name. But as we have no words by which to recall those former sensations, except the very word which we seek to define, or some other which, being exactly synonymous with it, requires definition as much, words cannot unfold the signification of this class of names; and we are obliged to make a direct appeal to the personal experience of the individual whom we address. (VII 136)
It turns out that we only arrive at a truly unanalysable, or semantically simple, class of names when we get to the names of sensations. But what immediately looks implausible here is the direction of semantic analysis: names denoting physical objects in terms of names denoting private experience. Even if one accepts that the proposition ‘An object is white if and only if it has the power of exciting the sensation of white’ is verbal, it still does not follow that ‘white’ should be defined in terms of ‘sensation of white’. It may be the other way round: ‘sensation of white’ may be definable as ‘sensation which white objects characteristically have the power of exciting’. That ‘sensation of white’, rather than ‘white’, is the semantically complex name, is evident enough from its surface structure. It is only because he endorses an epistemological doctrine—that we are immediately conscious only of our experience—that Mill takes the opposite view.
But we also learn something here about Mill’s model of how a semantically simple connotative name is understood. Such a name has meaning, unlike a proper name, because it is predicable of objects in virtue of some resemblance between them. Mill does not mean that This is a sensation of white’ literally means This resembles sensations S1, S2,.—.’. The point is that the resemblance is what justifies predicating the name in new cases. In learning a simple connotative name one is introduced to a sample of items which are all denoted by the name, one grasps the criterion, or directly ascertainable common feature, on the basis of which the name is applicable to all of them, and one is then in a position to predicate the name of new cases. The resemblance must be a directly recognisable one—that is why it must involve a resemblance of sensations. To grasp the meaning of a simple name is to grasp that phenomenal resemblance or common feature in virtue of which it is applicable to new cases.
So far we have discovered no genuine difference of level or kind between the semantic analysis of names and the ‘analysis of attributes’. Analysing the attribute whiteness seems to mean nothing more than giving a further semantic analysis of ‘white’. But we cannot leave matters there. Deeper Issues are raised in Mill’s discussion; they lead into borderlands of psychology, metaphysics and the analysis of language which are still being explored. Mill says,
we must remember that every attribute is grounded on some fact or phenomenon, from which, and which alone, it derives its meaning. To that fact or phenomenon…the foundation of the attribute, we must, therefore, have recourse for its definition. (VII 135)
The ‘definition’ will involve ‘dissecting the fact or phenomenon (whether of perception or of internal consciousness) which is the foundation of the attribute’ (VII 136).
In an earlier chapter (i.iii), Mill distinguishes substances and attributes, and recognises—not as a final metaphysical verdict, but as representing the common view—two kinds of substances: bodies and minds. He takes it for granted that ‘knowledge is phenomenal’—that the only knowledge of substances and their attributes which we can have is knowledge of how they appear to us. Moreover he takes the doctrine in a subjective vein—as referring not to appearances in the objective sense (‘the look of the table as one comes into the room’), but to subjective experience. Our immediate knowledge is only of our own state of consciousness. The significance of this doctrine, and its tenability, will be considered further in chapter 7; Mill took it to be uncontroversial—as indeed it was when he wrote.
A consequence of the doctrine, in the subjectivist version in which Mill holds it, is that any assertion about substances and attributes must be epistemologically grounded on the states of experience which are all I directly know. That being so, ‘when we ascribe whiteness to any substance, as, for instance, snow; when we say that snow has the quality whiteness, what do we really assert?’ (VII 65) Mill considers two possible answers. Each of them accepts that I am justified in saying that snow is present if and only if I experience ‘a certain assemblage or series of sensations’; and am justified in saying that it is white, if and only if the sensation of white is in a certain way interposed in that series. But the first answer infers from this that the content of the assertion, ‘snow is white’, can be no more than that the sensation of white is structurally interposed in that way, in that kind of assemblage, or series of sensations. The other answer does not:
It may be said, that it is true we know nothing of sensible objects, except the sensations they excite in us; that the fact of our receiving from snow the particular sensation which is called a sensation of white, is the ground on which we ascribe to that substance the quality whiteness; the sole proof of its possessing that quality. But because one thing may be the sole evidence of the existence of another thing, it does not follow that the two are one and the same. The attribute whiteness (it may be said) is not the fact of receiving the sensation, but something in the object itself; a power inherent in it; something in virtue of which the object produces the sensation. And when we affirm that snow possesses the attribute whiteness, we do not merely assert that the presence of snow produces in us that sensation, but that it does so through, and by reason of, that power or quality. (VII 65)
A little later (VII 67)—
I shall say…that the quality of whiteness ascribed to the object snow, is grounded on its exciting in us the sensation of white; and… shall term the sensation of white the foundation of the quality whiteness.
Mill thinks that he does not need to choose between the two answers ‘for the purposes of logic’ (VII 65). The question whether physical attributes in some way reduce to their foundations, or are merely epistemically grounded on them, is one of ‘metaphysics’. But he makes it clear that his own answer is the first. He thinks, that is to say, that what we ‘really assert’ when we make a statement about the physical world can be analysed phenomenalistically. And yet we have just seen him, in i.viii.2, defining whiteness as the property or power of producing the sensation of white.
There need be no contradiction here, so long as what we have in mind when we talk of ‘what is really asserted’ by using a sentence diverges from, or goes deeper than, a merely semantic account of its meaning—as determined by the denotation and connotation of its names in the manner of chapter 2. A familiar problem about philosophical reduction arises here. The strict meaning of what we say often seems to transcend what there is for us to mean. On the one hand, the initially obvious semantic analysis of a given area of discourse—it might be mathematics or ethics or statements about the physical world or causal statements and counterfactuals—has us referring to certain types of objects and attributing to them certain properties and relations. On the other hand, philosophical reflection suggests that we could not know that any such objects and attributes exist. Or it suggests that they could not exist, or even that the very idea of them is unintelligible.
If the plausible semantic analysis of what we mean by our assertions is correct, we are presented with a paradox—for it then looks as if none of our assertions in the particular area of discourse being considered could, strictly speaking, be known to be true, or be true, or even be intelligible. (Hume’s analysis of causation is the classic case.) And yet we appear to get along, and succeed in conveying something by our assertions, something which can be unproblematically assessed as true or false.
In this situation there are three basic reactions. One is to revise the language game. Another is to rewrite the semantic analysis. Another is to leave both language game and semantic analysis alone, but to go to another level—a non-reductive philosophical analysis which nevertheless brings out the ‘real content’ of talking about such objects and attributing properties to them. In favour of this last option one might say that the task of semantic analysis is to represent accurately the rules which do in fact govern our language and which determine what we strictly mean. Further inquiry into the metaphysical tenability of what we actually say or the foundations or ultimate groundings of our discourse belongs to another department of philosophy. This view is the one Mill tries to hold to in the System of Logic. He tries to insulate philosophical analysis of the ‘foundation’ of attributes from the strictly semantic business proper to a treatise on logic.
Is such insulation possible? And if it is, then what is one doing when one analyses the foundations or grounds of attribution? Mill certainly finds it hard to stick to the insulating strategy. There is a constant undertow in his thinking towards direct semantic phenomenalism. One factor which pushes him in this direction can be removed without too much dislocation. It is the dual role played by his notion of an attribute, in metaphysics and in semantic analysis. His real metaphysical view (chapter 7) is that there are only states of consciousness. We have, he thinks, no ground whatsoever to postulate that there are substances which causally affect our states of consciousness, but exist independently of them. But if there are no substances, or powers inhering in substances, and if a name which has meaning has it by dint of connoting attributes, attributes cannot be identified with powers inhering in substances. Either language has no meaning, or attributes must be identified in some way with their experiential foundation. Thus the metaphysical view, that there are only states of consciouness, entails, via the doctrine that the meaning of a name is the attributes it connotes, a phenomenalistic reduction of the meaning of all propositions.
We have however already seen reason to relieve the concept of an attribute of the semantic role which it plays in Mill’s analysis of language, and a way of doing so. (See 2.6.) So some of the undertow can be removed. But if Mill’s analysis of the experiential foundations of attributes and the propositions in which they feature is not to be regarded as semantic analysis, then what is its point?
His question is epistemological: what information can a proposition — ‘Snow is white’—convey to us: what knowledge can it really give us? He thinks that the information it gives us can at bottom only be, that any total experiential state which includes a foundation for attribution of ‘snow’ (‘This is snow’) will also include a foundation for attribution of ‘white’ (This is white’).
It is not that the meaning of ‘snow is white’ is literally expressible in some such form as ‘Whenever an assemblage of actual and possible sensations S1, …, Sn occurs, the sensation of white is interposed in the senses, actually or potentially’. But there is a conception of meaning which would still allow one to see Mill’s probing into the experiential foundations of attribution as analysis of meaning in another sense—an analysis, as one might say, of the cognitive role of connotative names and sentences, rather than of their strict semantic content.
I refer to this conception of meaning in 1.2 and 7.4 as the ‘epistemic conception’. It is foreshadowed at a number of points in Mill’s philosophy (which is by no means to say that he formulates it or even works with it inexplicitly—he certainly does not). All of them are points connected with his marked pragmatic interest in analysing what assertions do—what difference they make to the habits of inference with which we encounter new experience. Mill often argues from the grounds which justify one in asserting a proposition to its ‘real’ content. One can see this as a matter of clarifying the proposition’s cognitive role; clarifying, that is, what understanding of the proposition consists in. For what, after all, is it to understand the semantic content of a sentence? I understand it just in so far as it has a potential role in my thinking. Or in other words: I understand a sentence when I grasp its cognitive role—when I grasp what states of experience warrant its assertion. This line of thought, generalised, leads to a conception of meaning in which understanding a sentence consists in grasping its assertion conditions. And that opens up a perspective from which Mill’s strategy of insulating the semantic analysis of propositions from an investigation of their experiential foundations makes sense.
The opposition, and also the interplay, between what Mill calls ‘Nominalism’ and his own views on language and logic have been a recurrent theme of the last two chapters; we have reached a convenient point for taking stock of what he understands by it. It involves, in fact, two views, which Mill thinks related. The first is a view of names which seeks ‘for their meaning exclusively in what they denote’ (VII 91); the second is the view that the propositions and inferences of logic and mathematics are purely verbal. Mill’s rejection of the first view has been studied in chapter 2. His rejection of the second will be considered in the next two chapters. But why does he think them connected?
If we took seriously the Nominalist’s compositional rules as stated in (c) on p.60, then Nominalism would entail that all propositions are ‘verbal’: not however in the sense defined in the present chapter, but in the sense that they would all be metalinguistic. Their truth or otherwise would rest on linguistic convention. And of course if that was true of all propositions it would be true of logical and mathematical propositions in particular. On the other hand once we are clear that compositional rules do not state sentence-meanings but show them (p. 60), the consequence that all propositions are verbal in the sense of being about language will no longer flow from the thesis that the semantic content of names is determined by their denotation. But it would still be true, on that thesis, that propositions were empty of content and true by convention in the sense in which identity propositions containing only proper names are.
Nominalism about mathematics would then be the doctrine that all mathematical propositions are non-connotative identities. This empties mathematical propositions of content, but only because it makes all propositions empty of content. Nor does it explain the aprioricity of logic and mathematics. ‘Tully is Cicero’ is, as we saw, void of cognitive content. It is ‘true by convention’ in the sense that it is true by nothing else—it corresponds to no fact. But it is not a priori.
A third form of Nominalism is more plausible than either of these. It holds that logical and mathematical propositions are verbal (or ‘analytic’): not in the way non-connotative identities are, but in the sense defined by the criterion of connotative inclusion. This would explain the aprioricity of logic and mathematics, and it too would hold that logical and mathematical propositions are true by convention—in the sense of being true in virtue of nothing else. Mill opposes this third view by arguing that analysis shows that logic and mathematics contain real propositions.
In the current sense of the word, however, Mill was himself a nominalist. He rejects real essences and he rejects abstract entities. But we must be careful not to give his nominalism too up-to-date a flavour.
In the Examination of Sir William Hamilton’s Philosophy, ‘Nominalism’ and ‘Conceptualism’ reappear as views about the signification of general names. ‘Realism’ holds that ‘General Names are the names of General Things’ (IX 301) which have an immaterial existence outside the mind. General Things or Universals are existents, but they are not individuals; they are denoted, not connoted by general names, so that the copula stands for an obscure relation of ‘participation’ which is like, but not like, the relation of identity. Mill wastes no time on this view: ‘Realism being no longer extant, nor likely to be revived, the contest at present is between Nominalism and Conceptualism’ (IX 302). According to the Nominalists, a name ‘is general, if it is applied in the same acceptation to a plurality of things; but every one of the things is individual’, while the Conceptualists held that
generality is not an attribute solely of names, but also of thoughts. External objects indeed are all individual, but to every general name corresponds a General Notion, called by Locke and others an Abstract Idea. General Names are the names of these Abstract Ideas. (IX 302)
Mill’s discussion of Abstract Ideas in the Examination slavishly follows Berkeley’s, contains all Berkeley’s errors, and vitiates his analysis of judgement, reasoning and thought. But this is irrelevant at present. He clearly sees that whether or not the idea of a ‘General Notion’ is required in the psychology of thought, it is not required in the semantic analysis of language. General Names do not denote General Notions, any more than they denote General Things. As the positions are stated in the Examination, then, Mill’s view is Nominalist. His improvement on traditional Nominalism is his recognition that most names do not merely denote individual things, but also connote attributes, and that this connotation constitutes their meaning.
We have seen however (2.6) that Mill’s concept of an attribute cannot play the role he wants to give it in his analysis of language. So the analysis must be amended in one of two ways. ‘Attributes’ may be treated as abstract intensional entities. This would mean abandoning Mill’s nominalism for a form of ‘Realism’, though it would not be the Realism Mill considers, according to which general terms denote abstract universals. Or one can try to show how the essential insight of Mill’s doctrine of connotation can be captured, without treating connotation as a relation between names and something else. This was the approach taken in chapter 2.
But even then another kind of abstract individual seems indispensable in the semantic analysis of mathematics and science, and indeed of all relational predications (2.10), namely, classes or sets. What Mill says about classes is partly incoherent,11 but one thing is clear: the idea that classes might be conceived of as abstract individuals simply does not occur to him. Not that that is surprising—classes as extensionally determined abstract objects had not, when he wrote, been dreamt of. The development of set theory, the elucidation of the central place of the concept of a set in the foundations of mathematics, the discovery of set-theoretic paradoxes, the resulting idea that not every predicate determines a set, all of these things lay a long way in the future. Before them, modern ‘platonism’, which differs from Plato’s in treating its abstract entities as particulars, could hardly have made an impact.
Reading Mill’s account of geometry or arithmetic today, it becomes evident (chapter 5) that its weaknesses arise not from his empiricism but from his rather primitive nominalism, and from his account of inductive logic. I am not saying that nominalism cannot be defended, and I am not endorsing the Quinean view that one can be a naturalistic empiricist and at the same time a realist about sets. But Mill had no opportunity to get to know the real strength of his enemy. I am sure that he would have stuck to his nominalism if he had. Various strategies would have been available to him.12 But we cannot hope to learn from Mill which one a nominalist should choose, or with what prospects of success.