## 1. What are mass expressions?




How can we identify a class of mass expressions as distinct from a
class of count expressions? In order to do so, one may try to employ
syntactic criteria or semantic criteria. We present them in turn. As we
shall see, only the syntactic criteria are satisfactory.


### 1.1 Syntactic criteria




At the syntactic level, there are two viable positions concerning
the mass / count distinction: it applies at the level of nouns; or it
only applies at the level of noun phrases.




The first position is the traditional, dominant view (e.g., Weinreich
1966; Krifka 1991; Gillon 1992). According to it, in a language
like English (or French, German, Greek, Italian, etc.), common nouns are
divided into two morphosyntactic subclasses, mass nouns and count
nouns. A defining characteristic of mass nouns, like *milk*,
*gold*, *furniture*, *wisdom* and *love*,
is that they are invariable in grammatical number, while count nouns,
like *rabbit*, *bottle*, *table*, *idea*
and *set*, can be used in the singular and in the plural.
Depending on the language, this basic morphosyntactic difference
between the two types of noun is supplemented by differences as to the
determiners they can combine with. Thus, in English, mass nouns can be
used with determiners like *much* and *a lot of,* but
neither with *one* nor *many*. On the contrary, count
nouns can be employed with numerals like *one* and determiners
like *many*, but not with *much.*




However, as is well known, mass nouns (like *milk*) can often
be used as count nouns: *You should take a hot milk with some
honey.* And vice versa: *You will find a lot of rabbit around
here*. For this reason, several researchers have denied that the
mass / count distinction applies at the level of nouns, and proposed
instead that it only applies at the level of noun phrases (e.g., Damourette
& Pichon 1927; Pelletier 1975, 2012; Ware 1975). Under this
view, at the level of nouns, there are only common nouns, but common
nouns can be used in a mass way or in a count way depending on the
morphosyntactic environment they are put into. Mass determiners, like
*much* or *a little*, lead to a mass use of the noun
(*much water*, *much table*), while count determiners,
like *a* or *two*, lead to a count use (*two
waters*, *two 
tables*).[1]




The two positions seem to be viable (cf. Pelletier &
Schubert 2003) and they have their respective proponents. For ease of
exposition, we adopt the dominant view in the rest of
the
 entry.[2]


### 1.2 Semantic criteria




At the semantic level, two properties have been proposed as being
characteristic of mass nouns: cumulative reference and distributive
reference. But as we shall see, cumulative reference is also a property
of plurals, while distributive reference doesn't apply to all mass
nouns.




Since Quine (1960), it is generally accepted that mass nouns refer
cumulatively. Consider a mass noun *M*. Suppose that we can
truly say of something *x* that *This is M* (with
*this* referring to *x*) and of something distinct,
*y*, that *This is M* (with *this* now referring to
*y*). Then in the same circumstance, we can also refer to
*x* and *y* together, and say of *x* and *y* that
*This is M*. This property of mass nouns is called cumulative
reference. Plural count nouns also have the same property. Let
*Ns* be a plural count noun. If these are Ns and those are Ns,
then we can refer to these and those together, and say of all of them
that they are Ns.




Various authors have proposed that mass nouns also refer
distributively (e.g., Cheng 1973; ter Meulen 1981). Let *M* be a
mass noun. Suppose that we can truly say of something *x* that *This
is M* (with *this* referring to *x*). Then in the same
circumstance, for anything *y* that is part of *x*, we can also say of y
that *This is M* (with *this* now referring to *y*).
However, with many mass nouns, the property doesn't apply when one
considers parts that are small enough. Water is made of oxygen and
hydrogen, but oxygen isn't water. And with mass nouns like
*furniture* or *silverware*, the problem appears even
more clearly, at the macroscopic level: a table is furniture, a leg of
the table is part of the table, but the leg isn't furniture.
Thus, the thesis that mass nouns refer distributively is mistaken
(e.g., Gillon 1992; Nicolas 2002).




A first attempt to defend the view would be as follows. Bunt (1985)
and others have proposed that, although modern science is in conflict
with the claim that the mass noun *water* refers distributively,
English speakers use the noun as if it did. The problem with this
suggestion is that it cannot be falsified, since it puts aside the very
cases that would do so. So it doesn't seem to be an empirical
hypothesis and it doesn't appear to make any prediction. What benefit
would there be to add this claim to any theory?




A better attempt would consist, in effect, in attributing a weaker
property to mass nouns. As we will see in section
2, an approach based on mereology may well, contrary to what is
usually assumed, want to use a non-extensional mereology, where the
notion of part is defined in terms of the notion of sum. If
distributive reference were understood using such a notion of part, it
could avoid the problems mentioned here. There is nothing wrong with
that, but then, the property attributed to mass nouns would be much
weaker than that originally advocated by its partisans (see Nicolas
2002, chapter 3, for a proposal of this kind). The claim made here is
simply that distributive reference, as usually conceived, is not a
property of mass nouns.




But more generally, it seems that there are no necessary and
sufficient semantic conditions that would specify what mass nouns are
and what count nouns are (see Gillon 1992, Koslicki 1999, and Nicolas
2002 for detailed arguments to this effect[3]; *a contrario,* see for
instance Landman 2011). A common noun that is mass in a language (like
*luggage* in English) may be count in another (like
*bagage* in French). The distinction between mass nouns and
count nouns should be drawn syntactically. A key feature of count nouns
is that they admit the singular / plural contrast while mass nouns do
not. So languages that do not mark the singular / plural contrast could
lack count nouns and all their common nouns function in a similar way
as mass nouns do in English. This might the case of certain classifier
languages, like Mandarin (Chierchia 1998,
 2010).[4]
Alternatively, the mass / count distinction could
just fail to apply to such languages. The proper way to draw the mass /
count distinction across languages is still debated (see Doetjes 2012, Rothstein 2017, and Bale & Barner 2018 for cross-linguistic overviews).




In the rest of this entry, we examine how sentences containing mass
nouns are interpreted, i.e., how their truth-conditions can be
specified. We consider several approaches to the semantics of mass
nouns. Even though a given approach may turn out to be unsatisfactory,
it is important to know in what precise respects it fails. For some of
its key proposals may be retained in (or transposed to) a better,
overall framework.


## 2. The purely mereological approach




We first consider the purely mereological approach (Moravcsik 1973),
which uses mereological sums as the denotata of mass nouns, and
interprets mass predication (e.g., *This is water*) in terms of
parthood.




Take the noun *water*. The view is that this noun denotes the
sum of all the water that there is. The notion of sum involved here
can be characterized formally, as done in mereology (cf. Varzi 2016, Cotnoir & Varzi forthcoming).
Intuitively, suppose there is some water in a bottle, *a*, and
some water in a cup, *b*. Then we can also refer to the water in
the bottle and the cup. This is a bigger portion of water, the sum of
*a* and *b*, noted *a*[?]*b* (or
*a*+*b*). *a* is part of *a*[?]*b*, and so
is *b*. More generally, we can sum all the portions of water
together. This is a very big portion of water, which is the denotation
of the noun *water*. Mass predication is then interpreted in
terms of these associated notions of sum and parthood:



*This is M* is true iff [*this*] [?] [*M*],




where [ * ] is the denotation function,
[*this*] is the sum of what is demonstrated and [*M*] is
the sum of all *M*. For instance:




*This is water* is true
iff [*this*] [?] [*water*], the sum of all water.





However, this quickly runs into a problem, since there are parts of
water (e.g., oxygen) that are too small to count as water. Water seems
to have "minimal parts", smallest parts that count as water. (As said
in section 1.2, this is even clearer with mass nouns like
*furniture*.)




Parsons (1970, reprinted in 1979: 150) points out a related
difficulty, the 'WOOD = FURNITURE' problem. Suppose that
all wood is used to make up furniture and all furniture is made of
wood. Then it would seem that the sum of the wood is identical with the
sum of the furniture. Therefore, all sentences of the form *The wood
P* and *The furniture P* are predicted to have the same
truth-value, for any predicate *P*. Yet, it might well be that
*The furniture is heterogeneous* is true, intuitively, while
*The wood is heterogeneous* is false.


**Remark 1:** The purely mereological approach is usually understood in
terms of classical extensional mereology. However, such an approach
doesn't seem to require the full power of this theory. The notion of
sum used could be the join operation [?] in a join semi-lattice.
And the notion of parthood employed could be the order relation [?]
defined in this way: *x* [?] *y* =def *x* [?] *y* =
*y*. The notion of join semi-lattice is more general and far less
constrained than classical extensional
 mereology (see Moltmann 1997, ch. 1, for other criticisms of the mereological
extensional view).


**Remark 2:** Thus, against Parsons, one may well want to deny that the
(sum or join of the) wood is identical to the (sum or join of the)
furniture. Indeed, if the furniture is broken, it ceases to exist,
while the wood does not. This line of argumentation is adopted more
generally by those who deny the identity of a ship and the wood it is
made of, of a man and its molecules, etc. (cf. Wasserman 2012).
Parsons' argument is based on a controversial metaphysical
assumption, one that a semantics of mass nouns need not
 make.[5]
(See Steen 2012, section 2.2 and
supplement 1, for other metaphysical considerations concerning the
purely mereological approach.)


**Remark 3:** Indeed, Moravcsik (1973) proposes to do something of this
kind in order to avoid the problem of minimal parts. The idea is to
associate to any mass noun its own part-whole relation. Let *M*
be a mass noun, [*M*] the sum it denotes, and
[?]M the associated part-whole relation. The sentence
*This is M* is represented as: [*this*]
[?]M [*M*]. Then, the leg of a table isn't a *furniture*-part of the furniture, so the problem of minimal
parts is avoided.


**Remark 4:** However, this doesn't account for the validity of
syllogisms like the following (Burge 1972: 266-267):



*This is gold. Gold is metal. Therefore, this is metal.*



Presumably, it represents it as follows:



[*this*] [?]GOLD [*gold*] &
[*gold*] [?]METAL [*metal*] - 
[*this*] [?]METAL [*metal*]



This is invalid, since only a uniform part-whole relation could
guarantee
 transitivity.[6]



**Remark 5:** The purely mereological approach faces yet another, very
general problem. One still needs a uniform framework for doing
semantics: for proper names, singular count nouns, plurals, mass nouns,
adjectives, verbs, etc. This has to be set theory, or something as
powerful like "non-singular" or "plural" logic
(see section 9).


## 3. The purely set-theoretic approach




By contrast, the purely set-theoretic approach (Burge 1972; Grandy
1973; Montague 1973[7])
treats
mass nouns as ordinary predicates that denote
 sets[8].
Mass predication is interpreted as set
membership. For any mass noun *M* and predicate *P*:



*This is M* is true iff [*this*] [?] [*M*]  
 *Some *M* P* is true iff [*M*] [?]
[*P*] [?] [?],



where [*this*] is the set whose elements are what is
demonstrated, [*M*] is the set having for elements everything
that is *M*, [*P*] is the set having for elements
everything that *P*.




A difficulty for this approach is to
specify, for any mass noun *M*, what the "portions of
M" are. This difficulty appears especially clearly with definite
descriptions, as in *The gold on the table weighs fifty grams.*
If the description *the gold on the table* denotes the set
having for elements everything that is gold on the table, then how can
we evaluate the truth of the sentence? It would not do to give the sum
of all weights (cf. Bunt 1985: 41). So it seems we must impose
restrictions on the elements of the set [*the gold on the
table*].[9]



Now comes a second and crucial difficulty concerning identity over
time. Consider:



*The clay that was on the desk on July 1st is identical
with the clay that was on the table on July 2nd.*  

(Context of utterance: three solid bits of clay were on the desk on
July 1st, and two solid bits of clay were on the table on
July 2nd. Example inspired by Cartwright 1965.)




Which set could make [*the clay that was on the desk on July
1st*] = [*the clay that was on the table on July
2nd*] true? What about the set of all minimal parts of
clay, i.e., the set of all the instances of clay that have no other
instance of clay as part? However, with mass nouns like
*garbage*, it is not clear what the minimal parts would be
(cf. Pelletier & Schubert 2003). And more crucially, one cannot
exclude *a priori* the possibility that what a given mass noun
applies to may be indefinitely divisible. So the semantics should not
require mass nouns to have minimal parts (cf. Bunt 1985; Gillon 1992).
(See section 9 for a solution to this problem
within the framework of non-singular logic; see Steen 2012, section
2.3, for various metaphysical considerations concerning the purely
set-theoretic approach.)


## 4. The mixed set-theoretic and mereological approach




From what precedes, one may be tempted to draw the following
lessons:


* Mass predication (as in *This is water*) should not be
understood in terms of parthood, but in terms of set membership.
* The denotation of a mass noun *M* (the set whose elements
are everything that is *M*) should be the join semi-lattice generated by
the sum or join operation on portions of
 M.[10]




This solves the problems encountered above by the purely
mereological approach and the purely set-theoretic approach. Indeed,
what precedes suggests that mass predication (*to be M*), like
count predication or adjectival predication, is well rendered in terms
of set membership. The purely set-theoretic approach has problems with
definite descriptions because it just uses sets, avoiding sums. But as
we saw earlier, mass nouns have the property of cumulative reference.
If there is some clay in two cups, then one can refer to all of the
clay as the clay in the two cups. This suggests that portions of clay
can be summed, and that the set of portions of clay should have the
structure of a join semi-lattice. When this is guaranteed for any mass
noun *M*, the semantic value of definite descriptions can easily
be specified. The description *the *M* that Q* denotes the sum of
everything that is some M that Q. It is such a sum that is weighted in
*The gold on the table weighs fifty grams*. And it is such a sum
whose identity over time is asserted in *The clay that was on the
desk on July 1st is identical with the clay that was on the
 table on July 2nd.* (Another treatment of identity over time is presented in 
section 9.)





Accordingly, we arrive at the mixed set-theoretic and mereological
approach:



*This is M* is true iff [*this*] [?]
[*M*]  

*Some *M* P* is true iff [*M*] [?] [*P*]
[?] [?]  

*The *M* (that *Q*) P* is true iff [*the *M* (that *Q*)*]
[?] [*P*],



where [*this*] is the set having for sole member the sum of
what is demonstrated, [*M*] is the set of everything that is M
(a join semi-lattice), [*the *M* (that *Q*)*] is the
set having for sole member the sum of everything that is some M (that
Q), [*P*] is the set having for members everything that P.




This
offers a simple way to accommodate the hindsights of the set-theoretic
and mereological approaches, while avoiding the previous pitfalls. A
decisive advantage of this view over the purely mereological one is
that the overall framework for doing semantics remains the usual one,
set theory. Gillon (1992) and Nicolas (2010) can be seen as instances
of such a mixed view,[11] with an additional component, namely
"aggregations" or "coverings"
(see section 8 below).


## 5. Negation




However, a difficulty appears with negation (Roeper 1983;
Lonning 1987; Higginbotham 1994). Consider a positive sentence
of the form *The *M* P* and its negation *The *M*
not P*, where *M* is a mass noun and *P* a
predicate. For instance: *The gold is in the safe* and *The
gold is not in the safe.* Imagine that the universe of discourse
contains only two bits of gold, *a* and *b*, and their
sum *a*[?]*b*. Then under the mixed view, [*gold*] =
{*a*, *b*, *a*[?]*b*}, [*the gold*] =
{*a*[?]*b*}. Suppose further that only *a* is in the
safe: [*in the safe*] = {*a*}. Given what we said
in section 4:



*The gold is in the safe* is true iff [*the gold*]
[?] [*in the safe*]



So the sentence is predicted to be false.




Now, it seems plausible that the semantics should validate the
following equivalence: *The *M* not P* is true iff *The M
P* is false. Then, the sentence *The gold is not in the
safe* is predicted to be true. This is a problem for the mixed
approach developed so far, since one would like to ascribe the same
status to the positive sentence and its negation. Either because both
sentences are taken to be false. Or because both are judged
inapplicable in the circumstance, being as it were partly true and
partly false.




Consider also noun phrases of the form *the *M* that P* and
*the *M* that not P*. For instance: *the gold that is in the
safe* and *the gold that is not in the safe*. Here,
intuitions are very clear: the first noun phrase designates the solid
bit of gold *a*, while the second designates *b*. However, under the mixed
approach, *The gold is not in the safe* is true. This means that
*a*+*b* (the gold) is in the denotation of *is not in the safe*. So
it may seem that *a*+*b* is also in the denotation of *gold that is not
in the safe* and in that of *the gold that is not in the
safe*, contradicting speakers' intuitions.




How can we avoid these difficulties? Roeper (1983), Lonning
(1987) and Higginbotham (1994) propose that the solution lies in
defining predication and negation in a certain way within a
Boolean
 algebra.[12]
They consider only
cases where predicates (mass nouns included) are
"homogeneous", like above. Following ter Meulen (1981), a
predicate is said to be homogeneous if it applies both cumulatively and
distributively. Predicates like *gold* and *in the safe*
may indeed seem to apply distributively and cumulatively, if we stay at
the macroscopic level for *gold*. In this approach, mass nouns
and predicates denote elements in a certain Boolean algebra,
(*B*,[?],[?],[?]0,1). [?] is the order (or parthood)
relation. [?] is the join (or sum) operation. [?] is the meet
(or intersection) operation. 0 is the smallest element. 1 is the
largest element. As in any Boolean algebra, every element *x* has a
Boolean complement, noted [?]*x* (cf. Monk 2018).




Predication is understood in terms of Boolean intersection:



*This is M* is true iff [*this*] [?] [*M*]
= [*this*] iff [*this*] [?] [*M*]  

*The *M* P* is true iff [*M*] [?] [*P*] =
[*M*] iff [*M*] [?] [*P*]  

*Some *M* P* is true iff [*M*] [?] [*P*]
[?] 0,



where [*this*] is the join of what is demonstrated,
[*M*] is join of everything that is M, and [*P*] is the
join of everything that P.




And negation is defined in terms of
Boolean complement: [*not P*] =
[?][*P*]. So *The *M* not P* is true iff
[*M*] [?] [*not P*] = [?][*P*].


Applying this, both *The gold is in the safe* and *The
gold is not in the safe* are predicted to be false*.*
Indeed, in the situation imagined, the universe of discourse contains
only two bits of gold, *a* and *b*, and their join *a*[?]*b*. So
[*gold*] = *a*[?]*b* = 1, [*is in the safe*] = *a*,
[*is not in the safe*] = [?]*a* =
 *b*.[13]




The denotation of complex noun phrases is also built through Boolean
intersection: [*M that P*] = [*M*] [?] [*P*]. So [*gold that is in the safe*] = [*gold*][?][*is
in the safe*] = (*a*[?]*b*)[?]*a*
= *a*. And [*gold that is not in the safe*] = (*a*[?]*b*)[?]*b*
= *b*.


**Remark 1:** Under this approach, the whole universe of discourse (for
mass nouns and their predicates) is specified by a single Boolean
algebra, with uniformly defined join (sum), meet (intersection), and
order (parthood). Predication is defined in terms of Boolean
intersection (or equivalently, order or parthood in the case of
definite subjects). This works with mass nouns and predicates that are
homogeneous (i.e., refer distributively and cumulatively). But mass
nouns like *furniture* are clearly not homogeneous. And a
predicate like *made by John* isn't homogeneous either.
When something is made by John (e.g., a piece of furniture), it
doesn't imply that any part of it (e.g., some wood used to make it
up) is also made by John. For the reasons given in section 2, the
Boolean approach may ascribe incorrect truth-conditions to sentences
like *This is furniture*, *Some furniture is made by
John*, and *The furniture is made by John*. For instance,
[*this*] [?] [*furniture*] does not guarantee that
what is demonstrated is furniture, since a piece of wood may be a part
of a piece of furniture without being furniture. So there are mass
nouns and predicates to which the approach doesn't seem to apply,
even though the same difficulties with negation appear with them. If in
the examples above we replace *gold* by *furniture* and
*in the safe* by *made by John*, we meet the same
problems with negation. (If there are two pieces of furniture, only one
of which is made by John, is it true or false that *The furniture is
made by John*?) So an appropriate solution had better not be tied
to the assumption of
 homogeneity.[14]


**Remark 2:** Indeed, the proposed treatment of negation can be adapted
within the mixed approach. The basic idea is that if something *x* P and
something *y* not P, then *x* and *y* do not overlap (have no part in common,
have 0 as intersection). So, within the mixed view, one may define
[*not P*] as the set comprising anything that does not overlap
[?][*P*], the sum of everything that P. This solves the
problems above, without requiring homogeneity.


**Remark 3:** However, defining negation in terms of Boolean complement
or non-overlap doesn't work with all predicates. Consider the
adjective *cheap*. Fix the context of speech so that what counts
as cheap is specified and what counts as not cheap is also specified.
The pieces of furniture *a* and *b* may each count as cheap, while together
they (*a*[?]*b*) count as not cheap. So non-overlap isn't
satisfied here: something not cheap overlaps with something cheap.
*Cheap* is a vague predicate. But the same phenomenon is
observed with an exact predicate like *costs fifty euros*: *a* and
*b* may each cost fifty euros, while together they don't, costing
ninety euros, for instance. So non-overlap should not be required. In
general, [*not P*] cannot be defined in terms of [*P*].
Instead, it seems that [*P*] and [*not P*] should be
separately specified. (This is done in many approaches to
vagueness.)


**Remark 4:** The same difficulties also appear with plurals, as we can
see if we replace *gold* by *pieces of furniture* and
*in the safe* by *made by John* in the examples above.
There is no agreement on what the proper treatment of negated plural
sentences is. Still, a popular view is the following (Krifka 1996;
Lobner 2000; see Breheny 2005 *a contrario*). Sentences like *The
pieces of furniture are in the safe* and *The pieces of
furniture are not in the safe* make a presupposition of
"indivisibility": they can be used felicitously only if all
the pieces of furniture are in safe or if none
 is.[15]
The same could be proposed concerning mass
nouns. In any case, a unified treatment of negation would be welcome,
given that negation seems to create the same basic problem with mass
nouns and with plurals.


**Remark 5:** But the problem is even more general, since it also
appears with subjects that are count and singular, like *table*.
Is the table in the living-room, when one half of it is, while the
other half is in the bedroom? The application of a predicate to an
entity (or the application of the negated predicate) seems to be often
sensitive to the part structure of the entity (Lobner 2000;
Corblin 2008). More work on this is needed to understand how
predication and negation work in connection to part-structure.


## 6. Quantifiers




What is the semantics of quantifiers combining with mass nouns:
*some, all, no, only, little, much, most, two liters
of...*? Higginbotham and May (1981) propose that the semantics
of quantifiers combining with count nouns (*some, all, no, only,
few, many, most, two*...) can be captured within the framework
of generalized quantification. Inspired by Roeper (1983) and
Lonning (1987), Higginbotham (1994) applies similar ideas to the
case of mass nouns. His proposals are made in the Boolean approach
criticized in the last section. So we transpose them directly into the
mixed set-theoretic and mereological framework. This also has the
advantage that the same framework is used for count quantifiers and
mass quantifiers.




We consider sentences of the form *Q *M* P*, where *Q*
is a quantifier, *M* a mass noun and *P* a predicate.
[*M*] is the denotation of the mass noun, i.e., the set that has
for members everything that is M (a join semi-lattice). [*P*] is
the set that has for members everything that P. Using set-theoretic intersection [?], one may propose:



*Some M P* is true iff [*M*] [?] [*P*] [?] [?]  

*All M P* is true iff [*M*] [?] [*P*] =
[*M*][16]  

*No M P* is true iff [*M*] [?] [*P*] =
[?]  

*Only M P* is true iff [*M*] [?] [*P*] =
[*P*]



This applies to sentences like *Some / All / No / Only gold was
stolen*.




With the other quantifiers (*little*, *much*,
*most*, *two liters of*...), one seems to say
something about the quantity of M (*little gold*) or intensity
of M (*little wisdom*). Let us suppose that a mass noun
*M* has an associated function, m, that measures quantity or
intensity. We focus here on mass nouns that apply to concrete entities,
like *water* or *furniture* (see section 10 for
"abstract" mass nouns). In that case, it is handy (though
perhaps not necessary) to suppose that m is monotonic:



*x* [?] *y* - m(*x*) [?] m(*y*)



and additive (if *x* and *y* don't overlap, the measure of their sum is
the sum of their measures):



![?]*z* (*z* [?] *x* & *z* [?] *y*) -
 m(*x*[?]*y*) =
m(*x*) + m(*y*)



(The measure function m is associated to a certain mass noun
*M*. But of course, some mass nouns may share the same measure
function. And with a single mass noun *M*, different measure
functions may perhaps be used, in different contexts, to measure the
"quantity of M" that is relevant in the context.)




One can also define the measure of a set *E*:



m(*E*) =def m([?]*E*),



where [?]*E* is the sum (or join) of the elements of *E*.




With this in hand, the meanings of *little*, *much*
and *most* can be specified as follows, the numerical values
*p*, *q*, *r* and *s* being specified
contextually when a sentence is uttered:



*Little*1 *M P* is true iff m([*M*]
[?] [*P*]) [?] *p*  

*Little*2 *M P* is true iff m([*M*]
[?] [*P*]) [?] *r*\*m([*M*])  

*Much*1 *M P* is true iff m([*M*]
[?] [*P*]) [?] *q*  

*Much*2 *M P* is true iff m([*M*]
[?] [*P*]) [?] *s*\*m([*M*])  

*Most M P* is true iff m([*M*] [?] [*P*])
[?] m([*M*]) / 2  

*Two liters of M P* is true iff m([*M*] [?]
[*P*]) = 2  

with a function m measuring in liters



Above, *little* and *much* are given two meanings, an
"absolute" one and a "proportional" one. Thus,
a sentence like *Much gold was stolen* may mean that:


* The stolen gold was a large quantity of gold (absolute
interpretation):


m([*M*] [?] [*P*]) [?] *q*, where *q* is specified contextually.
* The stolen gold was a great proportion of the gold
(proportional
 interpretation):[17]


m([*M*] [?] [*P*]) [?] *s*\*m([*M*]), where *s* is specified contextually.


**Remark 1:** What precedes allows one to characterize
the meaning(s) of the various mass quantifiers. But of course, it
leaves room for improvement concerning the specific meanings
attributed. For instance, Solt (2009) argues in favor of a different
condition for the quantifier
*most*.


**Remark 2:** In the case of the count quantifiers *few* and
*many*, there is evidence that each quantifier is really
ambiguous between two interpretations, the absolute one and the
proportional one (Partee 1989). It remains to be seen whether there is
similar evidence in the case of *little* and *much*.


**Remark 3:** Adding negation to this picture creates the same
difficulties as we saw above in the case of definites. In certain
cases, [*not P*] can be defined in terms of [*P*] and
non-overlap. But in general, [*P*] and [*not P*] should
be separately specified.


## 7. Logical relations




In what precedes, we have looked at the semantics of various kinds
of sentences where mass nouns appear. But we haven't considered whether
there are logical relations between such sentences, i.e., whether the
semantics makes up for an adequate logic of mass nouns. This is the
topic of this section. An appropriate semantics for mass nouns should
guarantee things like what follows. (For a more detailed discussion,
see Pelletier & Schubert 2003: 63-74.)




Existential generalization: there are many sentences whose truth
entails the truth of an existential generalization. For instance:



*The wine is on the table. So some wine is on the table.*



Universal instantiation: as mentioned in section 2, this kind of
reasoning also seems to be valid:



*This is gold. All gold is metal. Therefore, this is
metal.*



Also, sentences like the following should come out as always true,
given the meanings of the words involved: *All gold is gold.*
And in any situation in which there is some gold in Zurich, this should
also be true: *The gold in Zurich is gold.*




Mass nouns can also be used in generic sentences, which express
generalizations: *Gold is metal*. So one would need a semantics
for generic sentences to check, for instance, if this reasoning is
validated: *This is gold. Gold is metal. Therefore, this is
metal.* However, the semantics of genericity is a vast topic, which
falls out of the scope of this entry (see also note 7).




Finally, mass nouns can also be used as count nouns: *Gold is a
metal*. So a full-blown semantics, covering both mass nouns and
count nouns, should be able to validate syllogisms like the following,
which involve mass and count uses of a mass noun: *This is gold.
Gold is a metal. Therefore, this is a metal.*




For illustration, let us see how the mixed set-theoretic and
mereological framework developed in sections 4 and 6 handles some of
these cases. According to section 4:



*The wine is on the table* is true iff [*the wine*]
[?] [*on the table*],



where [*the wine*] denotes the
sum of the wine, and [*on the table*] denotes the set containing
everything that is on the table.



*Some wine is on the table* is true iff [*wine*]
[?] [*on the table*] [?] [?],



where [*wine*] is the set containing everything that is wine (a
join semi-lattice).




Since [*wine*] is the join semi-lattice
containing everything that is wine, it contains in particular the sum
of the wine. So, given how the semantics is set-up, the truth
of *The wine is one the table* guarantees that of *Some wine
is on the table*.




For a different case, according to sections 4 and 6:



*This is gold* is true iff [*this*] [?]
[*gold*]  

*All gold is metal* is true iff [*gold*] [?]
[*metal*] = [*gold*]



So the truth of *This is gold* and *All gold is metal*
guarantees that [*this*] [?] [*metal*] and so that
*This is metal* is true.


## 8. Collective and non-collective construals, coverings




According to Gillon (1992), a sentence containing a mass noun may
receive so-called "collective" and
"distributive" construals, modulo the meaning of the
particular lexical items composing the sentence, context of speech and
knowledge of the world. (Sentences containing plurals also receive such
construals. This is well documented by Gillon (1992, 1996) and
Schwarzschild (1996). This may be confirmed by replacing mass nouns
with plurals in the examples we give below.)




Take the following sentence: *This silverware costs a hundred
euros.* The sentence may be true if the silverware costs,
altogether, a hundred euros: this is the collective construal of the
sentence. It may be true if each piece of silverware, by itself, costs
a hundred euros: this is the distributive construal. It may also be
true if the silverware demonstrated consists in two sets of silverware,
each set of silverware costing, by itself, a hundred euros: this may be
called an "intermediate" construal.




A partly similar range of construals is observed with a mass noun
like *wine*: *This wine costs a hundred euros.* A
collective construal would assert that the wine, altogether, costs a
hundred euros. A non-collective construal may be obtained, for
instance, when the wine demonstrated consists of two cases of wine. The
speaker could then assert that each case of wine costs a hundred euros.
What about a distributive construal of the sentence? In fact, this
notion does not apply in this case, since a mass noun like
*wine* has no linguistically specified minimal part.




Therefore, the distinction that concerns all mass nouns is not that
between collective and distributive construals (nor that between
collective, distributive and intermediate readings), but rather, that
between collective and non-collective construals. What happens is that,
in the specific case of mass nouns like *silverware*, one can
identify, among the non-collective construals, one reading that may be
called distributive and other readings that may be called
intermediate.




The specific meanings of the verbal expression and its arguments,
combined with knowledge of the world and context of speech, may render
a type of construal more, or less, plausible. In particular,
non-collective, "intermediate" construals may be harder to
get than the collective reading, or than the "distributive"
reading if there is one. Such construals require specific information
about the context in order to become available. They are often easier
to get when the verb has several arguments, as in the following example
due to Gillon (1992): *This fruit was wrapped in that paper.* A
non-collective, "intermediate" construal with respect to
its first argument (*this fruit*) would be one where there are
several pieces of paper, each enclosing several pieces of fruit.




In the examples we have given so far (as well as in those considered
by Gillon), non-collective construals always correspond to partitions
of the denotation of the mass noun phrase. However, some
interpretations correspond to a more general notion, that of a
"covering": a set *X* is a covering of a set *Y* just in case
the sum of the elements of *X* is identical with the sum of the elements
of
 *Y*.[18]
Thus, with *This
livestock carried that furniture,* it may be that some pieces of
furniture were repeatedly part of the furniture carried by some of the
livestock. The relation of carrying thus applies between elements of a
covering of [*this livestock*] and elements of a covering of
[*that furniture*]. So it seems that the semantics of mass nouns
should leave room, not only for partitions, but also for all kinds
of
 coverings.[19]




Let us now see how Gillon (1992, 1996, 2012) accounts for these
data. We follow him quite closely, but introduce a few technical
modifications to ensure that everything works.




The denotation of a mass noun *M* is the set [*M*]
that has for elements everything that is M (a join semi-lattice). This
is needed in order to specify correctly the truth-conditions of
 *This is M*:[20]



*This is M* is true iff [*this*] [?]
[*M*],



where [*this*] is the set having for sole member the sum of
what is demonstrated.




A set *Y* is an *M*-covering of a
set *Z* just in case these two conditions are satisfied:


* *Y* is a subset of [*M*]: *Y* [?]
[*M*].
* The sum of the elements of *Y* is identical to the sum of the
elements of *Z*.




The interpretation of sentences like the following depends on the
choice of an *M*-covering of the
noun's
 denotation.[21]
Relative to this
choice of covering *C*:



*The M P* is true iff *C* [?] [*P*]  

*Some M P* is true iff *C* [?] [*P*] [?]
[?]  

*All M P* is true iff *C* [?] [*P*] = *C*



Gillon doesn't extend his account to other quantified
statements. However, this is easily done, following section 6. Thus,
one may define the measure of a set *E*:



m(*E*) =def m([?]*E*),



where [?]*E* is the sum of the
elements of *E*.




And propose that:



*Most M P* is true iff m(*C* [?] [*P*]) [?]
m(*C*) / 2



And similarly for the other quantifiers whose interpretation involve
a measure.


**Remark:** For Gillon, each choice
of covering determines a specific interpretation of the sentence. The
sentence is thus many-ways ambiguous. Schwarzschild (1996) offers a
detailed defense for a similar position in the case of plurals (see also Champollion 2017). But
the view also has opponents, like Lasersohn (1995). Among the
alternatives, one could propose that a sentence of the form *The M
P* is true just in case there exists a covering *C* of
[*M*] such that *C* [?] [*P*]. The sentence
wouldn't be ambiguous but indeterminate with respect to coverings. A
problem is that this wouldn't predict the distributive / collective
ambiguity, which does seem real (see Gillon 1992 for evidence).


## 9. Non-singular terms




There are many similarities between the semantics of mass nouns and
plurals, cf. sections 5, 6, and 8. Also, at a very intuitive level, if
there are eight pieces of silverware on the table, then the speaker
seems to refer to eight things at once when he says: *The silverware
that is on the table comes from Italy*. If this intuition is taken
seriously, then a mass noun isn't a singular term. Rather, it is a
non-singular term that may refer to one or several things at once.




Nicolas (2008) puts forwards a semantics of mass nouns that does
justice to this
 intuition (see Laycock (2006), Cocchiarella (2009) and McKay (2016) for related proposals).
It is cashed out in "non-singular" or "plural"
logic. In usual logic frameworks, like predicate logic, constants and
variables are singular in the following sense. Under any
interpretation, a constant is interpreted as one individual, and under
any assignment, a variable is interpreted as one individual. By
contrast, non-singular or plural logic possesses singular and
non-singular constants and variables. Under any interpretation and
variable assignment, a non-singular term (a constant or a variable) may
be interpreted as one or more individuals in the domain of
interpretation. In particular, a formula consisting of a predicate
whose argument is a non-singular constant is true just in case the
constant is interpreted as one or more individuals that jointly satisfy
the predicate (cf. Linnebo
 2017[22]).


**Remark:** The claim is not that mass nouns are plurals. It is that
mass nouns and plurals share a common property, namely the ability to
refer non-singularly, to one or several things at once.



The resulting semantics has the following features:


* Axioms guaranteeing the existence of mereological sums are
replaced by non-singular or plural reference. (Cf. also Nicolas
2009, 2017.)
* Combined with a generalized notion of covering, this allows a
treatment of identity statements different from that offered by the
mixed set-theoretic and mereological approach (where what remains
identical over time is a certain mereological sum).



Imagine that three solid bits of clay, the *as*, were on the
desk on July 1st, and two solid bits of clay, the *bs*, were
on the table on July 2nd. Now consider the statement:



*The clay that was on the desk on July 1st is identical
with the clay that was on the table on July
2nd.*



According to Nicolas (2008), the sentence is true when a common
non-singular covering can be chosen for the *as* and the
*bs*.[23]
This means that there are some
small bits of clay, each of which has retained its identity over time.
On July 1st, these bits of clay were so arranged that they
made up the *as* (i.e., they were a covering of the *as*). On July
2nd, they were arranged differently, in such a way that they
made up the *bs*. This doesn't require the existence of minimal parts of
clay. It only requires the existence of a common division of the *as* and
the *bs* into certain bits of clay. (See Steen 2012, section 2.4, for
metaphysical considerations concerning the non-singularist or pluralist
approach.)


## 10. Abstract mass nouns




"Abstract" nouns like *sadness* and
*wisdom* and "concrete" nouns like *water*
and *furniture* all belong to the morphosyntactic class of mass
nouns. However, semantics for mass nouns have generally focused on
concrete terms, i.e., terms that apply to concrete entities. This raises
an important question: are abstract mass nouns a separate species of
mass nouns, with their own semantic properties? Or can a general
account be proposed, which would work both for concrete mass nouns and
abstract ones?




Nicolas (2004, 2010) shows that a general account of the semantics
of mass nouns can indeed be proposed, provided one takes a more general
stance than when focusing on concrete mass nouns alone (see also Grimm 2014). Several issues
arise.




Reference: concrete common nouns can be used in definite
descriptions, where they seem to refer to entities of various types. Do
abstract mass nouns refer to something when they are used in definite
descriptions? And if so, what do they refer to? Consider sentences like
these: *Julie's wisdom attracted Tom. Julie's love for Tom lasted
several years.* Nicolas proposes that their subjects, which are
headed by abstract mass nouns, do refer (or make as if to refer) to
instances of properties or relations, thereby introducing these as
referents in the
 discourse (Moltmann 2007 proposes something similar).
He argues that this provides the most unified explanation of the
various uses of abstract mass nouns.




Nominalization: many abstract mass nouns are derived from an
adjective or verb. What is the semantic effect of nominalization?
Nicolas proposes that its intuitive effect, namely reification, a
"something-from-nothing transformation", is adequately captured by
meaning postulates. Thus, a meaning postulate relates the meanings of
the mass noun *love* and the verb *to love*. It ensures
that an instance of love from John towards Mary exists if and only if
John loves Mary.




Distributive, collective, and intermediate construals: sentences
with concrete mass nouns or plurals may receive so-called distributive,
collective, and intermediate construals (cf. section 8 above). Is it
also the case with abstract mass nouns? Nicolas (2010) suggests that
this is so. Take the sentence *The strength of these men is
impressive*, uttered in a context where two strong teams are
competing. The sentence can assert that the strength of each team is
impressive. This corresponds to a construal which is neither
distributive (the strength of each man is impressive), nor collective
(the strength of the men all together is impressive), but intermediate
between distributive and collective. Nicolas shows that these
construals can be accounted for by Gillon's (1996) rule for the
interpretation of complex noun phrases containing a prepositional
phrase.




Overall, then, it seems that a unified semantics can be specified
for all mass nouns.