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Ideas of Øystein Linnebo, by Text
[Norwegian, fl. 2006, Lecturer at Bristol University, then Birkbeck, London.]
2003
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Plural Quantification Exposed
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§0
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p.71
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10778
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Can second-order logic be ontologically first-order, with all the benefits of second-order?
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§1
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p.73
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10779
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A comprehension axiom is 'predicative' if the formula has no bound second-order variables
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§1
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p.75
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10781
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A 'pure logic' must be ontologically innocent, universal, and without presuppositions
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§2
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p.78
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10782
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The modern concept of an object is rooted in quantificational logic
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§4
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p.88
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10783
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Plural quantification depends too heavily on combinatorial and set-theoretic considerations
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2008
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Plural Quantification
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1
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p.2
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10633
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'Some critics admire only one another' cannot be paraphrased in singular first-order
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1.1
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p.4
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10634
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Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does?
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2
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p.5
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10635
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Second-order quantification and plural quantification are different
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2.4
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p.8
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10636
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Plural plurals are unnatural and need a first-level ontology
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2.4
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p.9
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10637
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Ordinary speakers posit objects without concern for ontology
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3
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p.10
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10638
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A pure logic is wholly general, purely formal, and directly known
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4.4
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p.14
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10639
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Plural quantification may allow a monadic second-order theory with first-order ontology
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4.5
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p.4
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10640
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Instead of complex objects like tables, plurally quantify over mereological atoms tablewise
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5
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p.15
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10641
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Traditionally we eliminate plurals by quantifying over sets
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5.4
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p.20
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10643
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We speak of a theory's 'ideological commitments' as well as its 'ontological commitments'
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2008
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Structuralism and the Notion of Dependence
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Intro
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p.59
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14083
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Structuralism is right about algebra, but wrong about sets
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1
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p.60
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14085
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'Deductivist' structuralism is just theories, with no commitment to objects, or modality
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I
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p.60
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14084
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Non-eliminative structuralism treats mathematical objects as positions in real abstract structures
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I
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p.60
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14086
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'Modal' structuralism studies all possible concrete models for various mathematical theories
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I
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p.61
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14087
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'Set-theoretic' structuralism treats mathematics as various structures realised among the sets
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II
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p.65
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14088
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An 'intrinsic' property is either found in every duplicate, or exists independent of all externals
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III
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p.66
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14089
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Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure
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III
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p.68
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14090
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In mathematical structuralism the small depends on the large, which is the opposite of physical structures
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V
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p.73
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14091
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There may be a one-way direction of dependence among sets, and among natural numbers
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2017
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Philosophy of Mathematics
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11.1
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p.155
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23448
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Mathematics is the study of all possible patterns, and is thus bound to describe the world
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2
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p.21
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23441
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Logical truth is true in all models, so mathematical objects can't be purely logical
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3.3
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p.44
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23442
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Game Formalism has no semantics, and Term Formalism reduces the semantics
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3.5
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p.52
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23443
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The axioms of group theory are not assertions, but a definition of a structure
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4.1
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p.56
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23444
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To investigate axiomatic theories, mathematics needs its own foundational axioms
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4.2
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p.62
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23445
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Naïve set theory says any formula defines a set, and coextensive sets are identical
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4.6
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p.71
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23446
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You can't prove consistency using a weaker theory, but you can use a consistent theory
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7.1
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p.101
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23447
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In classical semantics singular terms refer, and quantifiers range over domains
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