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E-grāmata: Many and the One: A Philosophical Study of Plural Logic

(University of Oslo), (University of Birmingham)
  • Formāts: 256 pages
  • Izdošanas datums: 22-Jul-2021
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780192509161
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Many and the One: A Philosophical Study of Plural LogicPalielināt
  • Formāts: 256 pages
  • Izdošanas datums: 22-Jul-2021
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780192509161
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Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic -- a logical system that takes plurals at face value -- has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic's applications than has been given thus far. Questions about the correct logic of plurals play a central role in the final chapters, where traditional plural logic is rejected in favor of a "critical" alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes.

Recenzijas

The volume by Florio and Linnebo is a most welcome contribution, and I believe it will be valuable not only for philosophers of logic and philosophical logicians, but also for philosophers of mathematics who are interested in the possible applications of plural logic to logico-mathematical theories and the broadly philosophical issues that these applications give rise to. * Francesca Boccuni, Philosophia Mathematica *

Preface xi
Acknowledgments xiii
1 Introduction
1(8)
I PRIMITIVE PLURALS
2 Taking Plurals at Face Value
9(22)
2.1 Some prominent views of plural sentences
9(5)
2.2 Taking plurals at face value
14(1)
2.3 The language of plural logic
15(4)
2.4 The traditional theory of plural logic
19(1)
2.5 The philosophical significance of plural logic
20(4)
2.6 Applications of plural logic
24(5)
2.7 Our methodology
29(2)
3 The Refutation of Singularism?
31(24)
3.1 Regimentation and singularism
31(3)
3.2 Substitution argument
34(2)
3.3 Incorrect existential consequences
36(2)
3.4 The paradox of plurality
38(3)
3.5 Plural Cantor: its significance
41(1)
3.6 Plural Cantor: its statement and proof
42(3)
3.7 Conclusion
45(2)
3.A Alternative formulations of Plural Cantor
47(8)
II COMPARISONS
4 Plurals and Set Theory
55(21)
4.1 A simple two-sorted set theory
55(2)
4.2 Plural logic and the simple set theory compared
57(2)
4.3 Plural logic vs. set theory: classifying the options
59(1)
4.4 Against the elimination of pluralities in favor of sets
60(4)
4.5 Against the elimination of sets in favor of pluralities
64(2)
4.6 The iterative conception of set
66(2)
4.7 Zermelo-Fraenkel set theory
68(2)
4.8 Proper classes as pluralities
70(2)
4.9 Are two applications of plural logic compatible?
72(1)
4.A Defining the translations
73(1)
4.B Defining the interpretation
74(2)
5 Plurals and Mereology
76(28)
5.1 Mereology
76(1)
5.2 Can mereology represent the plural?
77(4)
5.3 One-sorted plural logic
81(2)
5.4 Classifying some ways to talk about the many
83(2)
5.5 Mereological singularism in linguistic semantics
85(3)
5.6 Assessment of singularism in linguistic semantics
88(2)
5.7 The elimination of mereology in favor of plural logic
90(2)
5.8 Keeping both plural logic and mereology
92(4)
5.A Partial orders and principles of decomposition
96(2)
5.B Some notions of sum
98(3)
5.C Atomicity
101(1)
5.D One- and two-sorted plural logic compared
102(2)
6 Plurals and Second-Order Logic
104(19)
6.1 Second-order logic
104(3)
6.2 Plural logic and second-order logic compared
107(2)
6.3 The elimination of pluralities in favor of concepts
109(7)
6.4 The elimination of concepts in favor of pluralities
116(2)
6.5 Conclusion
118(5)
III PLURALS AND SEMANTICS
7 The Semantics of Plurals
123(28)
7.1 Regimentation vs. semantics
124(2)
7.2 Set-based model theory
126(4)
7.3 Plurality-based model theory
130(5)
7.4 Criticisms of the set-based model theory
135(4)
7.5 The semantics of plural predication
139(5)
7.6 The problem of choice
144(2)
7.7 Absolute generality as a constraint
146(1)
7.8 Parity constraints
147(3)
7.9 Conclusion
150(1)
8 On the Innocence and Determinacy of Plural Quantification
151(23)
8.1 Introduction
151(3)
8.2 A plurality-based Henkin semantics
154(1)
8.3 The legitimacy of ascending one order
155(2)
8.4 Does ontological innocence ensure determinacy?
157(3)
8.5 The semantic determinacy of plural quantification
160(2)
8.6 The metaphysical determinacy of plural quantification
162(1)
8.7 A generalized notion of ontological commitment
163(5)
8.8 Applications reconsidered
168(2)
8.A Henkin semantics
170(2)
8.B Completeness of the Henkin semantics
172(2)
9 Superplurals
174(31)
9.1 Introduction
174(1)
9.2 What superplural reference would be
174(4)
9.3 Grades of superplural involvement
178(1)
9.4 Possible examples from natural language
179(3)
9.5 The possible examples scrutinized
182(3)
9.6 The multigrade analysis
185(3)
9.7 Covers
188(5)
9.8 Mixed-level predications
193(2)
9.9 Mixed-level terms, order, and repetition
195(3)
9.10 Conclusion
198(2)
9.A The notion of upwards closure
200(5)
IV THE LOGIC AND METAPHYSICS OF PLURALS
10 Plurals and Modals
205(35)
10.1 Introduction
205(2)
10.2 Why plural rigidity matters
207(1)
10.3 Challenges to plural rigidity
208(2)
10.4 An argument for the rigidity of sets
210(6)
10.5 An argument for plural rigidity
216(3)
10.6 Towards formal arguments for plural rigidity
219(2)
10.7 The argument from uniform adjunction
221(2)
10.8 The argument from partial rigidification
223(1)
10.9 The argument from uniform traversability
224(2)
10.10 Pluralities as extensionally definite
226(3)
10.11 The status of plural comprehension
229(2)
10.A Traversability and quasi-combinatorial reasoning
231(1)
10.B Proofs
232(8)
11 Absolute Generality and Singularization
240(22)
11.1 Absolute generality
240(1)
11.2 A challenge to absolute generality
241(3)
11.3 Atrilemma
244(2)
11.4 Relativism and inexpressibility
246(3)
11.5 Traditional absolutism and ascent
249(4)
11.6 Ascent and inexpressibility
253(3)
11.7 Lifting the veil of type distinctions
256(6)
11 A The Ascent Theorem
262(6)
12 Critical Plural Logic
268(29)
12.1 Introduction
268(1)
12.2 The extendability argument
269(3)
12.3 Our liberal view of definitions
272(4)
12.4 Why plural comprehension has to be restricted
276(2)
12.5 The principles of critical plural logic
278(6)
12.6 Extensions of critical plural logic
284(2)
12.7 Critical plural logic and set theory
286(4)
12.8 Generalized semantics without a universal plurality?
290(3)
12.9 What we have learnt
293(4)
References 297(10)
Index 307
Salvatore Florio is a Senior Lecturer in Philosophy at the University of Birmingham. He specializes in philosophy of language, philosophical logic, and philosophy of mathematics. His published work, which focuses on questions about the nature of logic and the foundations of semantics, has appeared in journals such as Australasian Journal of Philosophy, Mind, Nous, Philosophers' Imprint, and Philosophy and Phenomenological Research. Since 2019 he has been an editor of The Review of Symbolic Logic.



Ųystein Linnebo is Professor of Philosophy at the University of Oslo. His main research interests lie in the philosophies of logic and mathematics, metaphysics, and early analytic philosophy (especially Frege). A recipient of an ERC Starting Grant, he has published more than sixty scientific articles and is the author of Philosophy of Mathematics (Princeton University Press, 2017) and Thin Objects: An Abstractionist Account (Oxford University Press, 2018).
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