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Multivalued analytic functions are considered to be functions on a Riemann surface, which can be defined either by cutting and pasting or by a graph construction. Multivalued maps in general are considered, and it is shown how the concept of multivalued analytic functions generalizes to harmonic morphisms using the graph construction. An alternative treatment for space forms is presented, and some specific examples are given. The chapter concludes with a discussion of the behaviour on the branching set of the projection map for a multivalued harmonic morphism on a three-dimensional space form.

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.

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.

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 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.

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.