As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc.
for x, we may substitute both Alice and Bob, or all the numbers between 0 and 10, or all the buildings in London over 20 stories.
The view is commonly associated with George Boolos, though it is older (see notably Simons 1982), and is related to the view of classes defended by John Stuart Mill and other nominalist philosophers.
Mill argued that universals or "classes" are not a peculiar kind of thing, having an objective existence distinct from the individual objects that fall under them, but "is neither more nor less than the individual things in the class".
See also Gottlob Frege 1895 for a critique of an earlier view defended by Ernst Schroeder.
129–133) Interest revived in plurals with work in linguistics in the 1970s by Remko Scha, Godehard Link, Fred Landman, Friederike Moltmann, Roger Schwarzschild, Peter Lasersohn and others, who developed ideas for a semantics of plurals.
Sentences like are said to involve a multigrade (also known as variably polyadic, also anadic) predicate or relation ("cooperate" in this example), meaning that they stand for the same concept even though they don't have a fixed arity (cf.
The notion of multigrade relation/predicate has appeared as early as the 1940s and has been notably used by Quine (cf.
Plural quantification deals with formalizing the quantification over the variable-length arguments of such predicates, e.g. "xx cooperate" where xx is a plural variable.
Note that in this example it makes no sense, semantically, to instantiate xx with the name of a single person.
Broadly speaking, nominalism denies the existence of universals (abstract entities), like sets, classes, relations, properties, etc.
Thus the plural logics were developed as an attempt to formalize reasoning about plurals, such as those involved in multigrade predicates, apparently without resorting to notions that nominalists deny, e.g. sets.
Standard first-order logic has difficulties in representing some sentences with plurals.
Kaplan proved that it is nonfirstorderizable (the proof can be found in that article).
Hence its paraphrase into a formal language commits us to quantification over (i.e. the existence of) sets.
[1] Later, Oliver & Smiley (2001), Rayo (2002), Yi (2005) and McKay (2006) argued that sentences such as also cannot be interpreted in monadic second-order logic.
This is because predicates such as "are shipmates", "are meeting together", "are surrounding a building" are not distributive.
Yet such sentences also seem innocent of any existential assumptions, and do not involve quantification.
have suggested that plural logic opens the prospect of simplifying the foundations of mathematics, avoiding the paradoxes of set theory, and simplifying the complex and unintuitive axiom sets needed in order to avoid them.
[clarification needed] Recently, Linnebo & Nicolas (2008) have suggested that natural languages often contain superplural variables (and associated quantifiers) such as "these people, those people, and these other people compete against each other" (e.g. as teams in an online game), while Nicolas (2008) has argued that plural logic should be used to account for the semantics of mass nouns, like "wine" and "furniture".
Other logical symbols definable in terms of these can be used freely as notational shorthands.
, the domain does not have to include sets, and therefore plural logic achieves ontological innocence while still retaining the ability to talk about the extensions of a predicate.
does not yield Russell's paradox because the quantification of plural variables does not quantify over the domain.
Another aspect of the logic as Boolos defines it, crucial to this bypassing of Russell's paradox, is the fact that sentences of the form
This can be taken as the simplest, and most obvious argument that plural logic as Boolos defined it is ontologically innocent.