In linguistic semantics

Semantics

Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....

, an expression X is said to have cumulative reference if and only if the following holds: If X is true of both of a and b, then it is also true of the combination of a and b. Example: If two separate entities can be said to be "water", then combining them into one entity will yield more "water". If two separate entities can be said to be "a house", their combination cannot be said to be "a house". Hence, "water" has cumulative reference, while the expression "a house" does not. The plural

English plural

In the English language, nouns are inflected for grammatical number —that is, singular or plural. This article discusses the variety of ways in which English plurals are formed for nouns...

 form "houses", however, does have cumulative reference. If two (groups of) entities are both "houses", then their combination will still be "houses".

Cumulativity has proven relevant to the linguistic treatment of the mass/count

Mass noun

In linguistics, a mass noun is a noun that refers to some entity as an undifferentiated unit rather than as something with discrete subsets. Non-count nouns are best identified by their syntactic properties, and especially in contrast with count nouns. The semantics of mass nouns are highly...

 distinction and for the characterization of grammatical telicity

Telicity

In linguistics, telicity is the property of a verb or verb phrase that presents an action or event as being complete in some sense...

.

Formally, a cumulativity predicate CUM can be defined as follows, where capital X is a variable

Variable (mathematics)

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 over sets, U is the universe of discourse, p is a mereological

Mereology

In philosophy and mathematical logic, mereology treats parts and the wholes they form...

 part structure

Mathematical structure

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....

 on U, and is the mereological

Mereology

In philosophy and mathematical logic, mereology treats parts and the wholes they form...

sum operation.



In later work, Krifka has generalized the notion to n-ary predicates, based on the phenomenon of cumulative quantification. For example, the two following sentences appear to be equivalent:

John ate an apple and Mary ate a pear.

John and Mary ate an apple and a pear.

This shows that the relation "eat" is cumulative. In general, an n-ary predicate R is cumulative if and only if the following holds: