The World of the uncountable:

ontology and the mathematicisation of logic

Henry Laycock  

The mathematicisation of logic represented especially by Frege – and subsequently by Russell, Wittgenstein, Tarski and others – has become a key instrument within analytical philosophy, invoked in the treatment of a great variety of problems and issues. No domain better illustrates this fact than contemporary ontology itself. As Quine quite rightly emphasizes, the ontic scope of modern logic is effectively defined by the bond between ontology and the variable : ‘Existence’, he declares, ‘is what existential quantification expresses’. There are however major problems with the ontic application of this formal framework; and it is possible to see these problems as turning on the logico-semantic status of the class of so-called mass nouns. What Davidson once called ‘the problem of mass nouns’ is, indeed, a specific problem with this framework. What is distinctive about the semantics and ontology of mass nouns – that is, in effect, the semantics and ontology of words for stuff – is that they cannot be understood in terms of countability or units, and cannot thereby be represented in a logic of quantifiers and variables.[1] Contrary, then, to Quine, existence is not what existential quantification expresses. The issue is essentially a formal one, and concerns not only mass nouns in particular, but is no less an issue over count nouns and the formal representation of commonplace, everyday talk of objects. The two issues are virtually inseparable, and I treat them here as such.

Now the work of the linguist Otto Jespersen remains substantially neglected by philosophers.[2] Yet his writings on what he calls ‘mass-words’ and ‘thing words’ – he also speaks of ‘uncountables’ and ‘countables’ – contain deeply insightful observations on matters of great interest; and not only for semantics, but even more, perhaps, for metaphysics. Jespersen contrasts thing words with mass-words, speaking of the latter group as words for substances. Jespersen writes

There are a great many words which do not call up the idea of some definite thing with a certain shape or precise limits. I call these ‘mass-words’; they may be ... material, in which case they denote some substance in itself independent of form, such as silver, quicksilver, water, butter, gas, air, etc.[3]

Elsewhere, Jespersen also suggests a far-reaching logico-semantic contrast between thing words and mass-words, forcefully asserting that

Mass-words are totally different, logically they are neither singular nor plural, because what they stand for is not countable.[4]

And in this same spirit, Jespersen speaks of the need for an ideal language ‘constructed on purely logical principles’, and tells us that a distinctive logical form would be called for

when we left the world of countables (such as houses, horses, days, miles, sounds, words, crimes, plans, mistakes, etc.) and got to the world of uncountables.

As it happens, of course, just such an ideal language had been conceived some fifty years earlier, although Jespersen himself seems to have been unaware of this. At the same time, given its origins as a system for speaking of the (eminently countable) realm of numbers, the question must be faced of the appropriateness of the symbolism of Frege’s Begriffsschrift (‘modelled upon that of arithmetic, for pure thought’) to the logical forms here at issue. In what follows, I attempt to explain the significance of Jespersen’s remarks; I also urge that what he says is true. If Jespersen is right, then indeed  the semantics and ontology of words for stuff cannot be understood in terms of units; and this itself implies, so I claim, that the ontology of stuff can only be conceived as indeterminate.

      1. Why Jespersen was right about ‘mass words’.

It is commonly remarked – by linguists and philosophers alike – that ‘mass nouns lack a plural form’, or ‘do not pluralize’. As mere comments on English syntax, such remarks are unobjectionable: it is perfectly obvious that in the usage of native speakers, then unless the contest is generic, words like ‘sugar’, ‘gold’ and ‘water’ resist the plurality morpheme ‘-s’. Typically, however, such remarks are bearers of a deeper meaning – not altogether unreasonably, they are thought to mean that mass nouns cannot but be singular, and that in a semantic sense. The view has very considerable ontic significance: it implies that references involving concrete mass nouns must be understood as references to units, discrete instances of stuff. And at first blush, this too may seem eminently reasonable: given two glasses of water, we may distinguish the water in the one glass, this water, from the water in the other, that water. How then could this water fail to be one instance of the kind or concept water, and that water another?

                                                

But now while it is true that in non-generic contexts, grammar rules out plural talk of ‘waters’, it does not immediately follow that mass nouns are singular. Consider, for example, the obvious fact that concrete singular constructions, e.g. ‘a water’, are also counted out. Furthermore, mass nouns reject the quantifiers appropriate to singular contexts, words like ‘each’ and ‘every’. Again, it is strikingly evident (and even commonly remarked) that in the quantifiers they do accept, mass nouns more closely resemble plural nouns. Parallel with ‘all apples’, ‘some apples’, ‘more apples’ and ‘most apples’ are the phrases ‘all water’, ‘some water’, ‘more water’ and ‘most water’. In the standard semantic model for mass nouns, these phenomena appear to be treated as syntactical anomalies, yet a quite straightforward explanation is available, and my initial aim is just to briefly lay it out.[5] From the premise that ‘mass nouns do not pluralize’ to the conclusion that they are (invariably) singular, there is a key implied assumption – that all occurrences of nouns are semantically either singular or plural. The assumption, I suggest, is simply false; and once abandoned, the otherwise anomalous behaviour of mass nouns becomes manifestly intelligible.

Mass nouns are, indeed, standardly contrasted with count nouns or CNs; and the very contrast of these groups implies that mass nouns must be among the class of nouns which are non-count – non-count nouns, or NCNs. But now occurrences of CNs are themselves semantically either singular or plural; and these two sub-categories exhaust the general category of CNs. Qua non-count, therefore, it seems that, as Jespersen claimed, mass nouns can be neither singular nor plural; and given this, the basic shape of the relationships between these groups can be neatly represented in the following tableau:

                                              Table I

	1. Singular (one)	2. Non-singular (‘not one’)
3. Plural (‘at least one’)		‘Objects’
4. Non-plural (‘not at least one’)	‘Object’	‘Stuff’

On the bold assumption, then, that the tableau has it right, it will be simply incorrect to talk about the contrast between mass and count nouns, as if there were just one – and furthermore, one which defined the entire relationship between the categories. In fact, in anything other than purely morphosyntactic terms, speaking as if a contrast is what matters is profoundly misleading. Quite apart from the point that there is not just one but three contrasts on display between these nouns or their occurrences, corresponding to the three occurrences themselves, the fact is that, quite crucially, the semantic commonalities between the groups are no less fundamental than the differences.[6] Mass nouns share with singular count nouns the feature of non-plurality, and share with plural count nouns the feature of non-singularity. Fundamental to the difference between the two, on the other hand, is the presence or the absence of a numerical unit: because ‘water’ is non-singular, there is no single water-unit. In the very nature of the case, water has no units, and there is no such object as ‘a water’.[7]

But why, it may be asked, are the referential determiners and verbs for mass nouns morphologically indistinguishable from semantically singular expressions? Why should ‘this water’ look exactly like ‘this apple’? As Leech notes, syntax ‘is much less rich in dimensions of contrast than is semantics’, and the point is illustrated in this morphosyntactic parallel.[8] There are three semantic categories here (singular, plural, and non-count) but the syntax of English demonstratives can only take two forms, and to understand their taxonomy, we need a bifold contrast merely of the plural and non-plural – a class encompassing the singular along with the non-count. Since ‘water’ is itself non-plural – but not thereby semantically singular – the water in a glass can only be referred to as ‘this water’. Quantificational differences are marked in English, but the syntax marks no contrast, in referential contexts, between singular and non-plural – potentially thereby inviting a conflation of the two.[9]

Now Table I can be expanded to incorporate the appropriate verbs and determiners, including definite and indefinite articles and quantifiers, thereby summing up the main syntactic and semantic points thus far. The resulting table provides, I suggest, a coherent, unified account of a number of key semantic features of both CNs and MNs, along with a graphic display of their relationships:

                                               Table II

	SINGULAR	NON-SINGULAR	Reference-determiners (definite articles) and verbs
PLURAL		apples	these / are
NON-PLURAL	apple	water	this/is
Quantity-determiners (quantifiers, indefinite articles)	a, an, each, every, any	some, all, any,   most, more, many/much

The table introduces an additional column, and a corresponding row, for the semantical taxonomy of a wide range of determiners. Specifically, the table displays the essentially orthogonal character of the relationship between the two types of determiners – reference and quantification are seen to be orthogonal – along with their matching functional differences. Quantifiers bear on the singular / non-singular distinction, while demonstratives and other referential determiners bear on the plural / non-plural distinction. The standard semantic model for MNs as singular object-designating expressions, on the other hand, offers no explanation whatsoever for this interrelated set of phenomena.

Consider now the existential significance of the orthogonal relationships between what I am calling reference-determiners and quantity-determiners. As the existence of apples consists first and foremost in that of individual apples, this apple or that apple, so, it might seem, the existence of water consists in the existence of amounts of water, this water or that water – not, indeed, of a water like an apple, but rather of some water, a certain determinate amount of water, much as some apples constitute a certain determinate number of apples. And, whether this conforms to Quine’s principle that to be is to be the value of a variable or not, it seems to be consistent with the spirit of that principle. The idea is that at the very least, to be is to be determinate – in the case of concrete material beings, concrete things or stuff, to be (as Aristotle also said) is to be this or that – this apple or that water, the potential subject of an identity-involving reference.

      

       2. Existence, determinacy, neutrality.

But now although Table II displays certain relationships between mass and count nouns and determiners, there is no suggestion that the use of these nouns requires the use of determiners. In English, concrete singular occurrences of nouns do (almost) always take determiners – either the so-called definite or indefinite article, or a singular quantifier. But in their plural (and more generally, non-singular) occurrences, nouns do not always take determiners; and this, I suggest, is a reflection of the fact that whereas the use of a singular noun is intrinsically or essentially determinate, the use of a plural or non-singular noun is not. The difference is semantically of great importance, and is manifest in the contrasting ways of describing the kinds or sorts to which the subjects belong. To the question of what a strange-looking piece of fruit on the table is, the full and complete answer could well be that it is an apple, and perhaps, a certain variety of apple – the point being that here, in the English singular, an (indefinite) article is required. If on the other hand there is more than one of the fruit, then the answer to the question of what they are will be just that they are apples; no article now required.

The use of the plural determiner ‘some’ in this context – ‘some apples’ – carries numerical information, to the effect that there is more than one; used with a CN, ‘some’ can typically be paraphrased as ‘a number of’.[10] But here however, the determiner contributes nothing to answering the question of what they are. The plural (and more generally, non-singular) description of the appropriate kind or category involves no use of a determiner. Again, much as there can be some apples on the table, there can also, of course, be some water on the table; but the answer to the question of what there is on the table – what kinds of things and stuff there are on the table – is that on the one hand there are apples, things of the apple-kind, and on the other hand there is water, stuff of the water-kind. That there is a certain determinate amount of water on the table might perhaps go without saying; but it contributes nothing to answering the question of what the stuff on the table is.

The general point is recognised, if only tacitly, by Quine, who tells us that

We commit ourselves outright to an ontology containing numbers when we say there are prime numbers between 1000 and1010; we commit ourselves to an ontology containing centaurs when we say there are centaurs.[11]

Here, appropriately, there is no use of the plural determiner ‘some’, no question of determinacy as to numbers. Conceived a la Quine, the role of the existential quantifier is quite precise. It is not that of a device for making just any kind of existential statement; its assigned role is that of an ontic device – a device essentially for giving existential information regarding categories and kinds of things, quite simply that. Having pronounced that what ‘thus confronts us as a scheme for systems of the world is that structure so well understood by present day logicians, the logic of quantification or calculus of predicates’, Quine adds

The doctrine is that all traits of reality worthy of the name can be set down in an idiom of this austere form if in any idiom. It is in spirit a philosophical doctrine of categories.[12]

In its ontic role, ironically, the quantifier is no device for providing information on quantity, on the numbers of the members of the diverse categories and kinds – on the number of pigeons, galaxies, or concrete objects there might be. What matters, from the ontic standpoint, is purely and simply whether there are things of this or that kind or not. Quine’s declaration that existence is what existential quantification expresses is followed by a bare plural form of  remark, ‘There are things of kind F if and only if (∃x)(Fx)’.

For this reason, what the operator ∃ actually represents is quantification in name only – precisely because its role is to be numerically neutral. There is a difference between the existential quantifier, so to say, and so-called numerical quantification, as in ‘There are at least five rabbits in the garden’, ‘There are exactly two prime numbers between 1 and 4’, and so on.[13] The bare plural form is the quantitatively neutral use of the quantifier, and as Quine’s remarks nicely illustrate, this is the precise way of expressing numerical neutrality in natural English.

Cast in symbolic, formal terms, the operator tells us that there is at least one thing of such and such a kind – a numerically neutral form of statement, plainly – and the use of the bare plural is the uniquely appropriate natural-language mode for the expression of this fact. The role of numerical quantifiers is indeed in the proper sense to quantify – to specify numbers, whether definitely or indefinitely – whereas pace Quine, the role of the existential quantifier is merely to assert existence. On the one hand, there is the purely taxonomic, kind- or category-specifying form of information; on the other, there is information with an empirically quantitative form. But thanks in part to the misleading terminology of quantification – the fact that what is called ‘the existential quantifier’ is so called – this numerical irrelevance or quantitative neutrality is not always clearly understood.[14]

But now, there is a certain tension in Quine’s formulation here, that there are things of kind F if and only if (∃x)(Fx). There can be no objection to the bare plural formulation, ‘there are things of kind F’; however, it is not well-matched to the other half, the formal portion of the statement.  The problem lies in the unwarranted assumption that ‘there are Fs’ can be paraphrased as ‘there is at least one F’, which has, among other things, an indirectly referential character, whereas the use of the bare plural, as such, does not. Ontological assertions, in Quine’s sense and mine, are quantitatively neutral; and that is certainly the principle underlying the existential quantifier, conceived as an ontic operator or device. However, for existence to be expressed by the existential quantifier, severe restrictions must be imposed on what in general can be said to exist; Quine’s principle is not a general truth. It must be possible to recast the sentence in an indirectly referential form – a form, that is, that requires the possibility of reference to something of what is said to be. And this can only be done in a way that preserves the neutrality of the original sentence where the predicate at issue is distributively singular in a strong sense. The problem lies in the potential mismatch of the semantically singular variable (a familiar feature of the standard first-order predicate calculus) with the semantically unrestricted character of the predicate. The trouble is that the informal half of the statement only counts as quantified in virtue of its equivalence with the formal half; and this latter portion itself only works for a certain highly restricted range of predicates – ones which are distributively singular in a strong sense. And if we attempt to broaden the scope, to produce a comprehensive form of sentence, we find that our sentences can no longer be quantified (into the form of ‘there is something such that F’) or are no longer indirectly referential, and must be understood semantically as indeterminate. In effect, they become sentences without logical subjects – and that is precisely what many ontic assertions involving bare mass nouns are like .

Many common count noun predicates call for there to be more than one object of which they are true. For example, where ‘F’ = ‘family member’, ‘classmate’, ‘piece of gravel’, or ‘member of the team’, there is nothing wrong with a sentence having the form ‘There are things of kind F’ – ‘There are classmates, family members, pieces of gravel, members of the team, and so forth’; but no things of these kinds can live a logically solitary existence. In one good sense, even though they are true of each member of the corresponding kind, the predicates are essentially collective. For predicates of this type, ‘There is exactly one F’ and ‘There is at least one F’ are not well-formed formulae. For there to be classmates, there must be some of them. Nevertheless, the role of the plural determiner ‘some’ is to carry numerical information to the effect that there is more than one, which in this case goes without saying (and applied across the board, would be incorrect; in most cases, it is not necessary for there to be more than one F). Used with a CN, ‘some’ can typically be paraphrased as ‘a number of’, but it contributes nothing to answering the question of what they are. Here, if existential quantification involves the assertion of ‘at least one’, then existence is not what existential quantification expresses. There must be at least two; but this is non-neutral, and does not go into the expression of existence.

But the problem is more serious than this. Quantification requires that there be (real and not just grammatical) subjects for the predicates; the point goes back to Frege. This, functions require arguments; whereas concepts do not, it seems, need objects. In one respect,, the problem is latent even with the use of distributive predicates: the contrast of a pair of bare sentence and non-bare sentences will illustrate the point. Contrast

       (Q) Some dogs have been barking all night

with

       (B1) Dogs have been barking all night.

Whereas (Q) implies that the same dogs have been barking all night, (B) does not.  (Q) is determinate, whereas (B) is not. There is nothing of which the predicate in (B) need be true; it is enough if, at any time during the night, at least one dog was barking. Here, then, a certain reduction is possible. But when a collective predicate becomes involved, as with

        (B2) Classmates have been partying upstairs all evening,

neither singular reduction nor plural is possible. The singular is of course ruled out by the collective nature of the predicates; and the plural is ruled out by the fact that there is no requirement that at least two classmates were partying upstairs all evening; there might be a constant rotation. A reduction is nevertheless still possible; it must be the case that at any time during the evening, at least two classmates were partying. But finally, mass nouns can be used in a cognate statements involving temporally extended and essentially unbounded episodes, truth values must be assigned to sentences relative to intervals of time, rather than to points in time or moments, thus:

       Oil is pouring over the barriers,

       Cheap imported furniture flooded into the domestic market,

       Smoke is billowing from the chimneys,

       Mud is oozing through the wall.

And here, the subjects of the predicates cannot be open to identity-involving reductions, since pace Heraclitus, the process-predications themselves demand continuous change.

( Draft - texte de la conférence prononcée par l'auteur le 21 Janvier à Aix-en-Provence )

---

[1] It is with the application of more general semantic considerations on mass nouns to the specific case of words for what it is natural to call materials – very roughly, substances, kinds of stuff or matter – that I am centrally concerned. It is among such words that the potential for a substantial ontic contrast with words for things emerges. The ‘substances’ of chemistry are more narrowly defined, constituting a relatively narrow sub-class of the broad class of materials, being confined, in effect, to elements and compounds alone. Here, my contextual default is to speak of concrete mass nouns as words for either stuff or kinds of stuff.

[2] Quine might here – just – be counted as an exception.

[3] The Philosophy of Grammar, p.198.

[4] Selected Writings of Otto Jespersen, p. 272, Routledge Revivals (2010).

[5] The standard semantic model for mass nouns is mereological, often invoking semi-lattice theory; and it is always premised on the singularity assumption. The argument which follows is derivative from others; see Laycock (1998, 2005, 2006).

[6] For the sole reason that they are concrete mass nouns, I count words like ‘clothing’ and ‘furniture’ – which I call atomic mass nouns – as words for stuff. Ontically, nonetheless, there is no difference between these words and the cognate plurals ‘clothes’ and ‘pieces of furniture’. Hence the tableau is essentially semantic, and not ontic. (With ‘clothing’ and ‘furniture’, there is no such thing as ‘one’, ‘an’ or ‘at least one’; in this sense there is no unit. No less than a entire heap, pile, or roomful, a single item of clothing or piece of furniture counts only as clothing or furniture).

[7] Hence the so-called mass / count contrast over ‘cumulative reference’ is by no means the contrast it is said to be, since Quine’s cumulative principle applies to both: adding water to water results only in more water; just so, adding apples to apples results only in more apples.

[8] Semantics, Penguin.

[9] Corresponding to the threefold semantic contrast of singular, plural and non-count, there are two pairs of semantic contrasts among determiners which are syntactically marked in English. There is the referential contrast of plural and non-plural in the form of ‘these’ and ‘this’; on the other hand there is the quantifier-contrast of singular and non-singular – ‘each’, ‘every’ and ‘a’, versus ‘all’, ‘some’ and ‘most’ (‘the’ and ‘any’ are all-purpose determiners).

[10] The claim that I have written some books, or a number of books, may be misleading although it is not false, if I have written only two; but the claim is outright false, and not merely misleading, if I have written only one. As Strawson notes in Introduction to Logical Theory, it is a distinctive feature of ‘some’ that it ‘carries an implication of plurality’ (178).

[11] ‘On What There Is’, Willard V. Quine, The Review of Metaphysics , Vol. 2, No. 5 (Sep., 1948), pp. 21-38; p. 28.

[12] Word and object, 237.

[13] See, for example, ‘In so many possible worlds’, K. Fine, Notre Dame Journal of formal logic, 1972; SC Shapiro (1979): ‘Numerical quantifiers and their use in reasoning with negative information’, in: Proceedings of the 6th International Joint Conference on Artificial Intelligence, Tokyo, 791-796).

[14] As one linguist avers, ‘A quantifier is a word or expression that specifies which or how many of some kind of things have some property’. Everything that Linguists have always wanted to know about Logic, Chicago 1993, p.23.