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This introductory chapter surveys plural phenomena and argues for their logical significance. Terms may be classified as singular or plural, according to the number of things they are capable of denoting. Predicates and function signs may take plural terms as arguments, and function signs may express multivalued functions and produce plural functional terms (the wives of Henry VIII). Received philosophical and formal logic, however, makes no place for these plural phenomena. Strategies for a logic of plurals can therefore be broadly classified as singularist or pluralist. The singularist forces plurals into the old singular mould, whereas the pluralist develops a new plural logic that directly accommodates plural terms, plural predicates, multivalued functions, and plural quantification. In Chapters 3 and 4, general singularist strategies are eliminated. In this chapter, Michael Dummett’s more piecemeal singularist attempts to deal with plural phenomena are criticized.

Having previously dispatched singularism, this chapter turns to plural logic. First comes philosophical logic, beginning with the notion of a term. Singular and plural terms are different species of a common genus. As against a narrower Russellian conception, terms include definite descriptions and functional terms alongside proper names and demonstratives. Terms of any of these kinds may denote some thing(s) or may be empty. One aim of this book is to counterbalance the recent preoccupation with proper names and descriptions by placing functions and functional terms centre stage. Russell and Frege are criticized for failing to do justice to functions, despite their signal interest in mathematics. Of special interest are partial functions which map something to nothing, co-partial functions which map nothing to something, as well as functions which take several arguments at a given place, and multivalued functions which produce several values for a given choice of arguments.

This chapter argues against predicative analyses of plurality, which force plurals into the familiar mold of singular logic by turning an apparently plural term standing for several objects into a singular predicate standing for a concept or property. Michael Dummett enlists support from Fregean semantics in favor of a predicative analysis, but his arguments do not stand up, either as exegesis of Frege or on their own merits. As well as facing difficulties in eliminating plural content, predicative analyses are sunk by the equivocity objection: they misrepresent single English predicates as equivocal, by treating the predicate differently according as it combines with singular or plural arguments. Although George Boolos does not offer a predicative analysis, it is argued that his second-order treatment of plurals is also sunk by the equivocity objection. And Ian Rumfitt fails in his attempt to avoid the objection by modifying Boolos’s scheme.

This introductory chapter surveys plural phenomena and argues for their logical significance. Terms may be classified as singular or plural, according to the number of things they are capable of denoting. Predicates and function signs may take plural terms as arguments, and function signs may express multivalued functions and produce plural functional terms (the wives of Henry VIII). Received philosophical and formal logic, however, makes no place for these plural phenomena. Strategies for a logic of plurals can therefore be broadly classified as singularist or pluralist. The singularist forces plurals into the old singular mould, whereas the pluralist develops a new plural logic that directly accommodates plural terms, plural predicates, multivalued functions and plural quantification. In Chapters 3 and 4, general singularist strategies are eliminated. In this chapter, Michael Dummett's more piecemeal singularist attempts to deal with plural phenomena are criticised.

A list such as ‘Whitehead and Russell’ may be construed as a plural term denoting the two logicians. So regarded, a list is a kind of functional term, with the list-forming ‘and’ a function sign expressing a (multivalued) function. This chapter begins by putting the case for regarding lists as terms. A different account treats them as mere strings of separate items, and construes list-forming ‘and’ as a piece of punctuation rather than a function sign. There are no grounds to decide between the two accounts of lists: they are both tenable analyses. In particular, both can deal with nesting, order, repetition, and empty terms among the items in a list. Since both accounts need multigrade predicates, the appendix to the chapter rebuts arguments against them coming from Peirce, Dummett’s Frege, and P. F. Strawson.

Having previously dispatched singularism, this chapter turns to plural logic. First comes philosophical logic, beginning with the notion of a term. Singular and plural terms are different species of a common genus. As against a narrower Russellian conception, terms include definite descriptions and functional terms alongside proper names and demonstratives. Terms of any of these kinds may denote some thing(s) or may be empty. One aim of this book is to counterbalance the recent preoccupation with proper names and descriptions by placing functions and functional terms centre stage. Russell and Frege are criticized for failing to do justice to functions, despite their signal interest in mathematics. Of special interest are partial functions which map something to nothing, co-partial functions which map nothing to something, as well as functions which take several arguments at a given place, and multivalued functions which produce several values for a given choice of arguments.

This chapter is about multivalued functions and the corresponding functional terms, which are typically plural terms (the wives of Henry VIII, the square roots of 4). They are an everyday phenomenon, in ordinary life and mathematics, despite being ignored or even ruled out by contemporary logic. They cannot be identified with (or replaced by) relations, or otherwise eliminated. And, pace Frege, Russell, Carnap, and Church, their presence does not introduce an objectionable ambiguity.

The most common singularist strategy is changing the subject, which replaces a plural term apparently denoting several things by a singular term standing for a single thing, a set or aggregate or group. This chapter argues that no version of changing the subject works. Naive versions can be quickly dismissed. More sophisticated variants, in which the predicate as well as the subject is changed, fall foul of an analogue of Russell’s paradox. The appendix to this chapter criticizes extensions of Donald Davidson’s event-analysis of singular verbs of action to plural predication in general.

The notion of a plural term is spelled out using the relation of plural denotation. This chapter investigates whether plural denotation is distributive or collective. Different authors—Peter Simons, Keith Hossack, Mark Sainsbury, Byeong-uk Yi—have argued for one or other answer, but their arguments are unsound. The conclusion is that plural denotation is indeterminate in this respect, since both kinds of denotation—distributive and collective—produce the same correct truth conditions for plural predications. Five ways of formulating truth conditions are investigated, including those featuring so-called free relatives, such as the multiply ambiguous what-phrase in ‘F(a) is true iff F is true of what a denotes’.

This chapter argues against predicative analyses of plurality, which force plurals into the familiar mould of singular logic by turning an apparently plural term standing for several objects into a singular predicate standing for a concept or property. Michael Dummett enlists support from Fregean semantics in favour of a predicative analysis, but his arguments do not stand up, either as exegesis of Frege or on their own merits. As well as facing difficulties in eliminating plural content, predicative analyses are sunk by the equivocity objection: they misrepresent single English predicates as equivocal, by treating the predicate differently according as it combines with singular or plural arguments. Although George Boolos does not offer a predicative analysis, it is argued that his second-order treatment of plurals is also sunk by the equivocity objection. And Ian Rumfitt fails in his attempt to avoid the objection by modifying Boolos's scheme.

A list such as ‘Whitehead and Russell’ may be construed as a plural term denoting the two logicians. So regarded, a list is a kind of functional term, with the list-forming ‘and’ a function sign expressing a (multivalued) function. This chapter begins by putting the case for regarding lists as terms. A different account treats them as mere strings of separate items, and construes list-forming ‘and’ as a piece of punctuation rather than a function sign. There are no grounds to decide between the two accounts of lists: they are both tenable analyses. In particular, both can deal with nesting, order, repetition, and empty terms among the items in a list. Since both accounts need multigrade predicates, the appendix to the chapter rebuts arguments against them coming from Peirce, Dummett's Frege, and P. F. Strawson.

The most common singularist strategy is changing the subject, which replaces a plural term apparently denoting several things by a singular term standing for a single thing, a set or aggregate or group. This chapter argues that no version of changing the subject works. Naive versions can be quickly dismissed. More sophisticated variants, in which the predicate as well as the subject is changed, fall foul of an analogue of Russell's paradox. The appendix to this chapter criticises extensions of Donald Davidson's event-analysis of singular verbs of action to plural predication in general.

This chapter is about multivalued functions and the corresponding functional terms, which are typically plural terms (the wives of Henry VIII, the square roots of 4). They are an everyday phenomenon, in ordinary life and mathematics, despite being ignored or even ruled out by contemporary logic. They cannot be identified with (or replaced by) relations, or otherwise eliminated. And, pace Frege, Russell, Carnap, and Church, their presence does not introduce an objectionable ambiguity.

The notion of a plural term is spelled out in this chapter using the relation of plural denotation. The chapter investigates whether plural denotation is distributive or collective. Different authors—Peter Simons, Keith Hossack, Mark Sainsbury, Byeong-uk Yi—have argued for one or other answer, but their arguments are unsound. The conclusion is that plural denotation is indeterminate in this respect, since both kinds of denotation—distributive and collective—produce the same correct truth conditions for plural predications. Five ways of formulating truth conditions are investigated, including those featuring so-called free relatives, such as the multiply ambiguous what-phrase in ‘F(a) is true iff F is true of what a denotes’.

Chapter 3 identifies features of plural group agents (picked out using plural referring terms) to contrast them with singular group agents (picked out with grammatically singular referring terms). On the basis of the contrasts, it develops the prima facie case against a reductive account of singular group action sentences. The main contrasts developed are that (i) many singular group action sentences appear not to admit of a distributive reading, (ii) membership in a singular group agent requires a special social status, (iii) singular group agents persist through changes in membership, (iv) could have had different members, (v) can act through periods during which their membership changes entirely, and (vi) appear to be able to act though not all their members contribute, in contrast to plural group agents.