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Number word constructions, degree semantics and the metaphysics of degrees

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Abstract

A central question for ontology is the question of whether numbers really exist. But it seems easy to answer this question in the affirmative. The truth of a sentence like (1) ‘Seven students came to the party’ can be established simply by looking around at the party and counting students. A trivial paraphrase of (1) is (2) ‘The number of students who came to the party is seven’. But (2) appears to entail the existence of a number, and so it seems that we must conclude that numbers exist. This is sometimes called the puzzle of how we can get something from nothing. Most extant attempts to solve the puzzle take it for granted that (1) is ontologically innocent, and focus their attention either on (2) or on the transition from (1) to (2). We argue that both attempts go wrong at the first step: the assumption that (1) is ontologically innocent is undermined by a highly attractive and independently well-motivated degree-based account of number word constructions. Thus the degree-based account provides a straightforward linguistic resolution of the puzzle of how we can get something from nothing. But the paper also has a second aim. The semantics we present treats ‘seven’ as a referring expression that refers to a degree of a certain sort. But what are degrees? We consider various anti-platonist proposals that seek to account for degrees in terms of relations between concrete entities, and argue that they are incompatible with the Universal Density of Measurement hypothesis (UDM) of Fox and Hackl (Linguist Philos 29:537–586, 2006). While the UDM cannot yet claim to be the consensus view among degree-based semanticists, our aim is to use it to illustrate how views about the nature of degrees can be evaluated by considering the properties degrees must have if they are to play the explanatory roles they are called upon to play in linguistics. In the present state of development of degree-based semantics there are difficult open questions about what these properties are. These questions need to be addressed if we are to develop a clear picture of what natural language semantics has to contribute to ontology and metaphysics.

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Acknowledgements

The research for this essay was carried out at the Zukunftskolleg at the University of Konstanz; we are grateful to the Zukunftskolleg for providing such a supportive environment for interdisciplinary collaboration. We want to thank two anonymous referees as well as the editors Craige Roberts and Stewart Shapiro for their valuable comments. We would also like to express our gratitude to Magdalena Balcerak Jackson, David Liebesman and Maribel Romero, as well as to the members of the Konstanz Working Group on the Foundations of Semantics, for their feedback and encouragement. Funding was provided by Deutsche Forschungsgemeinschaft.

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Balcerak Jackson, B., Penka, D. Number word constructions, degree semantics and the metaphysics of degrees. Linguist and Philos 40, 347–372 (2017). https://doi.org/10.1007/s10988-017-9213-z

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