Abstract
Anything specified in response to the question, ‘How much?’ may, I suppose, be properly called an amount. But the amounts I want to discuss are those only which are amounts of such things as water or gold, objects designated by what Terence Parsons1 has called ‘applied amount terms’ — phrases like ‘ten pounds of water’ or ‘three teaspoons of gold’, the result of concatenating a numeral, the name of a unit of measurement, and a word or phrase with the grammatical syntax of a mass noun.
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Notes
In ‘An Analysis of Mass Terms and Amount Terms’, Foundations of Language 5 (1970). It will be obvious to anyone familiar with that paper that much of what I have to say especially in Sections 8 and 9, was inspired by his treatment of applied amount terms. Sentence (8), in particular, is a variation on an example of his. I have profited as well from discussion with a number of people, in particular Richard L. Cartwright and C. Wade Savage.
For an interesting theory which emphasizes these similarities, see Samuel C. Wheeler, ‘Attributives and their Modifiers’, Nous 6, No. 4 (1972). I think Wheeler’s treatment of mass nouns in that article is at least consistent with what follows, though I am not entirely sure.
Thus Russell’s discussion of quantity in Principles of Mathematics (1903), Part III (especially, Sections 151-158 and 163), is much better suited to applied amount terms which are not the names of properties than it is to his professed subject; and that is why I have appropriated his word ‘quantity’ and contrasted it with ‘amount’ instead of his ‘magnitude’. Norman Campbell couches his discussion of measurement in terms of properties, but in speaking of the ‘physical addition’ of weights, he says, “We say that the body C is ‘added to’ the body A,when A and C are placed in the same pan of the balance;…” (Physics the Elements (1928), reprinted by Dover as Foundations of Science (1957), p. 279, my emphasis). Since we apparently do not add weights, it is at least tempting to suppose that, according to Campbell, we add amounts of something having weight. And this way of talking is perpetuated in, e.g., Morris Cohen and Ernest Nagel, Introduction to Logic and Scientific Method (1934), pp. 297-8. See also the quotation from Carnap below.
For example, Brian Ellis, Basic Concepts of Measurements (1968); “… generally speaking, quantities are thought to be ‘properties’ of objects — characteristics which things must possess to some specific degree or other, even if we have no way of measuring or determining this degree” (p. 2). Ellis attributes this view to positivists and realists alike, and it is the general target of attack in his book. Parsons is a realist with respect to quantities in this sense, at any rate he gives a realistic account of ‘isolated amount terms,’ expressions which designate such things as one pound, and he seems not to be a realist with respect to applied amounts.
Principles of Mathematics,Section 152.
Heraclitus and the Bath Water’ (1965) and ‘Quantities’ (1970), both in The Philosophical Review.
Thus Wxy’ represents an amount function for ‘water’ iff there is a function f whose values are applied amounts of water, and which is such that for every x and yin a set of quantities of water, (i) xis as much water asy ifff(x) f(y) (ii) a linearly orders the values of any function which satisfies (i). It is to be understood here and throughout that in a context like this means ‘as large an amount of water as’ and that the field of q is restricted to some set of quantities of water.
Paul Halmos, Measure Theory (1950), p. 30, p. 73. m is a ‘set function’. I am beginning with the notion of a measure on a set of sets because of the way additivity is defined; more generally, I am beginning here to emphasize the similarity, as well as the differences, between applied amounts and what might be called ‘applied numbers’ (objects designated by, e.g., ‘ten rocks’). Though I shall not pursue the matter here, if (7) were ‘Every bucket contains as many rocks as every other,’ the second of the atomistic variants of m discussed below could reasonably be said to be a measure of the number of rocks in one or more of the buckets.
The relation in question has the formal properties of the ‘part-whole’ relation in Lesniewski’s mereology (A. Tarski, ‘Foundations of the Geometry of Solids’, Logic, Semantics, Metamathematics,trans. by J. H. Woodger (1956)), an algebraic structure equivalent to the calculus of individuals of Henry Leonard and Nelson Goodman (‘The Calculus of Individuals and Its Uses’, Journal of Symbolic Logic 5 (1940)), in which the notion of the fusion of a set is introduced.
Here I am relying in part on D. H. Krantz, R. D. Luce, P. Suppes, A. Tversky, Foundations of Measurement 1 (1971). I am defining an abstract mereology as follows: Where M is a non-empty set which is the field of a relation R that is transitive, anti-symmetric and reflexive in M,the ordered pair (M,R) is a mereology just in case (i) if A is a non-empty subset of M,then M contains an element x such that (y)((z)(zEA - > zRy) H xRy); and (ii) if x and y are elements of M such that — xRy, then M contains an element z such that Vu(uRZHURx — (Ev) (vRu vRy)). Thus, where R is q,(i) requires that fusions of non-empty subsets of Q are quantities of water, (ii) rules out the fusion of the empty set, and (i) and (ii) require that for every pair of distinct quantities of water, x and y, either x—y or y—x is a quantity of water. This definition, though perhaps less elegant, is equivalent to Tarski’s; and it has the virtue of making a mereology look as much like an algebra of sets (Boolean algebra) as possible. Where B is a set of sets, (B, c) is a mereology just in case it is closed under fusions (unions) and non-empty set-theoretic differences. The only mereology of sets which is an algebra of sets is ({.} ç), and if {y$} and (B, g_) is a mereology of sets, it would be an algebra of sets but for the empty set. There is no ‘empty’ individual in a (non-trivial) mereology.
A World of Individuals’ (1956), reprinted in Problems and Projects ( 1972 ). Goodman puts the suggestion in terms of a ‘generating relation’ which, unlike quantity inclusion, is asymmetric; but this is a minor difference. I do not mean to suggest that Goodman holds any version of the atomism discussed below, but generating relations are defined in this article only for systems based on sets of atoms.
A variation on this view of the matter would be to suppose m assigns one only to the largest (or smallest) atoms of the same size; but since atoms have no sub-quantities in common m(x—y)=m(x) for every pair of atoms, x and y, and it is hard to see how non-integral values are to be assigned to them without invoking a spatio-temporal metric in something like the manner discussed below. To the extent that I understand him, Henry Laycock has recently expressed views which would seem to commit him to some version of atomism in ‘Some Questions of Ontology’, Philosophical Review 81 (1972).
And in fact this seems not even to be sound science in view of such phenomena as ionization.
This is pretty clearly the view of W. V. Quine (in Word and Object and elsewhere) and, less clearly, that of others. It is also Parsons’ view in the article cited above; see pp. 366, 368. R. E. Grandy expresses justified discomfort at the introduction of a ‘new primitive’ (our quantity inclusion) in ‘Comments on Moravcsik’, Approaches to Natural Language, Hintikka, Moravcsik, and Suppes, eds. (1973). And he has recently expressed a view which suggests the scepticism discussed below.
Here I am relying on Halmos, op. cit.
It follows that if x and y are distinct elements of the domain of m, and yqx, there is an element z of the domain of m such that x=Fu{x, y}; z is x—y,and m(x)=m(y)+ +m(x—y),It does not follow that for every pair of distinct elements of the domain of m, there is an element z such that if x is as much water as y, x is quantitatively equal to Fu{y, z} unless y is a sub-quantity of x; and the difficulty for the variants of atomism discussed in note (12) remains. Where x and y are distinct atoms, neither is a sub-quantity of the other, and there are elements u and y in the domain of m such that m(x)=m(y)+(m(u)—m(v)) only if x and y are sub-quantities of unit quantities; but there need be nothing in the domain of m such that its fusion with y is quantitatively equal to x (unless, again, quantity inclusion is defined in terms of spatio-temporal occupancy).
Given the added condition that if m’ is a function that satisfies (i) and (ii), there is a number n such that m’(x)=nm(x),for every x in the domain of in (see below), I take it that according to Krantz, et al. cited above, in represents a ‘positive closed extensive structure’, the structure (D,> o), where >‘means ‘as much water as’ and o is an operation such that a o b=Fu {a, b},if a and b have no common sub-quantities, and, if they do, a o b=Fu{a, b} — Fu{u:uqa • uqb}.
Philosophical Foundations of Physics,Martin Gardner, ed., p. 71, my emphasis. Compare ‘volume of business’ or ‘… work’; surely ‘volume’ here just means ‘amount’.
This is roughly Parsons’ rendering of the analogue of (8), given that ‘… we have the formula ‘3Gy’ which is true of the amount three gallons and nothing else’ (p. 379 of the article cited above). Thus, ‘3Gy’ represents a function, and we may put ‘y=: g(3)’ instead.
Under ‘measurement’.
Principles of Mathematics,Section 156. I think my use of ‘kind’ is in fact his (in more favorable cases), and my explanation of its use may be regarded as an attempt to interpret what he says in sections 155-57 about “the absolute theory”.
On Generation and Corruption,Bk. I, Ch. 10. This Passage and that problems here involved were most recently called to my attention by Richard Sharvy and Lucy Carol. As a matter of chemistry there may be more water in the case in question; but surely not as a matter of semantics. Compare adding color or clay to some water or adding water to whiskey.
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Cartwright, H.M. (1975). Amounts and Measures of Amount. In: Pelletier, F.J. (eds) Mass Terms: Some Philosophical Problems. Synthese Language Library, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-4110-5_13
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