Quantifiers and Mass Terms

Quantifiers and Mass Terms

The usual approach to quantification focuses on quantifier words combined with count nouns, as in “All men are mortal” and “Some sheep are black”. We are told that such sentences require a paraphrase by means of variables ranging over objects (the “domain of quantification”)—“for some object x etc.”. But there are quantified sentences that employ mass terms not count nouns, as in “All coal is black” or “There is some milk in the fridge”. No objectseems to be meant—no x such that… It makes sense to ask how many objects are thus and so in response to a quantification using a count noun, but we can’t ask how many coal are black or what the number of milk is. We can ask how much coal or milk there is in a certain location, but not how many coal or milk is there. Mass term quantifications don’t “range over” a class of countable objects that can be assigned to bound variables. That is not how their semantics works. Yet the logic of such sentences follows the logic of count noun quantifications: from “All coal is black” and “This is coal” we can infer “This is black”. So it looks as if the standard semantics doesn’t capture the logical implications that are involved. This means that predicate logic is inadequate to capture quantifier entailments. Not good!

It might be thought (hoped) that mass term quantification can be analyzed by means of count noun quantification; we just need to bring in an ontology of chunks, lumps, pieces, volumes, morsels, and smidgens. Thus we have “All chunks of coal are black” and “There is a volume of milk in the fridge”. This already sounds stilted, but it also trades on a correspondence that doesn’t amount to a paraphrase. It is true that we can manufacture a truth about objects by employing such dummy sortals, but that doesn’t mean that the two mean the same thing. Likewise we can contrive a mass term quantification for any count noun quantification, but it would be wrong to claim semantic equivalence. Consider “There are many cod fish in the North Sea” and “There is a large amount of codfish in the North Sea”, where “codfish” functions as a mass term like “salmon” or “halibut”. Where there are fishy objects there is fishy stuff, and where there is fishy stuff there are fishy objects. But does anyone think we can analyze “All men are mortal” as “All man stuff is mortal”? One type of sentence speaks of objects of a certain kind; the other type speaks of stuff of a certain kind. Objects and stuff go together, but talking of one is not talking of the other. Also, the paraphrase in terms of dummy sortals doesn’t always get the truth conditions right: it may be true that all coal is black but not that all pieces of coal are black, because some pieces might be too small to have color; and the milk in the fridge might be scattered about not collected into a discrete volume. And isn’t it consistent to reject an ontology of objects while accepting that the world contains stuffs? There is coal and milk and gold and blood but there are no objects corresponding to these stuffs (“stuff metaphysics”). Stuffs are manifested at locations, according to this view, but there are no real objects that fall under mass terms. You can be an eliminativist about objects but a realist about stuffs. There is certainly no obligation to accept the count noun paraphrase, clunky as it is. It looks like an ad hoc attempt to save a theory not a natural semantic analysis.

Note that this point also destroys Russell’s theory of descriptions: for we also have “The milk in the fridge is off”, and this can’t be paraphrased by quantifying over milky objects (“There is a unique volume v such that v is milk and v is in the fridge and v is off”). Mass term definite descriptions don’t mean the same as corresponding count noun definite descriptions. Russell’s theory works smoothly enough for descriptions that speak of objects but not for descriptions that speak of stuffs—kings of France but not amounts of butter (“The butter on the table is rancid”). True, we can still use quantifier expressions to paraphrase such descriptions, but these expressions can’t be analyzed by using the standard apparatus of variables and domains of quantification. This is no more plausible than supposing that object quantification can be analyzed using an ontology of stuffs—as with “cod fish” and “codfish”. Objects are made of a certain kind of stuff, but talking about objects is not talking about the stuff they are made of. So it turns out that Russell’s theory, as normally formulated, is too wedded to the standard analysis of quantification; we need to broaden that analysis so as to take in descriptions that employ mass terms.

So what is the correct analysis of such quantification? I don’t know: it seems to defy the usual type of semantic construction built out of words for objects and properties. What does “some milk” mean? Part of the problem is not knowing what “milk” means: does it refer to the aggregate of all volumes of milk (whatever exactly that means), or is it an abstract singular term denoting the Platonic form MILK, or is it not referential at all? And is some milk a part of the reference of “milk”, or an instance of it, or not a relation to it at all? The semantics of quantifier phrases like this (“most milk”, “a lot of milk”, “nearly all milk”, etc.) is obscure. So we really don’t have a decent semantics for quantifier words as they occur in natural language; and predicate logic (“quantification theory”) is a poor representation of the logic of quantifiers. At best it deals with a restricted class of quantificational inferences. People like to proclaim the great success of modern logical symbolism in capturing the logic of quantifiers, but it turns out that this is only half the story; a lot of quantifier logic is not captured by this symbolism. It works nicely for mathematical quantification because arithmetic isn’t about numerical stuff, but for stuff-related quantification it doesn’t even get off the ground. We already knew that mass terms make trouble for the semantic categories recognized in formal logic (are they predicates or individual constants?); well, it turns out that they also upset the usual theory of quantifiers, both standard and non-standard. Logicians might want to drink some warm milk.[1]

[1] I wouldn’t be surprised if linguists have noted the linguistic phenomena here recorded, but my knowledge of linguistics doesn’t extend far enough to be sure. There was a flurry of work about mass terms a couple of decades ago in philosophy, but I don’t recall any mention of the difficulties posed by mass term quantification.

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