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Ideas of Øystein Linnebo, by Text
[Norwegian, fl. 2006, Lecturer at Bristol University, then Birkbeck, London.]
2003

Plural Quantification Exposed

§0

p.71

10778

Can secondorder logic be ontologically firstorder, with all the benefits of secondorder?

§1

p.73

10779

A comprehension axiom is 'predicative' if the formula has no bound secondorder variables

§1

p.75

10781

A 'pure logic' must be ontologically innocent, universal, and without presuppositions

§2

p.78

10782

The modern concept of an object is rooted in quantificational logic

§4

p.88

10783

Plural quantification depends too heavily on combinatorial and settheoretic considerations

2008

Plural Quantification

1

p.2

10633

'Some critics admire only one another' cannot be paraphrased in singular firstorder

1.1

p.4

10634

Predicates are 'distributive' or 'nondistributive'; do individuals do what the group does?

2

p.5

10635

Secondorder quantification and plural quantification are different

2.4

p.8

10636

Plural plurals are unnatural and need a firstlevel ontology

2.4

p.9

10637

Ordinary speakers posit objects without concern for ontology

3

p.10

10638

A pure logic is wholly general, purely formal, and directly known

4.4

p.14

10639

Plural quantification may allow a monadic secondorder theory with firstorder ontology

4.5

p.4

10640

Instead of complex objects like tables, plurally quantify over mereological atoms tablewise

5

p.15

10641

Traditionally we eliminate plurals by quantifying over sets

5.4

p.20

10643

We speak of a theory's 'ideological commitments' as well as its 'ontological commitments'

2008

Structuralism and the Notion of Dependence

Intro

p.59

14083

Structuralism is right about algebra, but wrong about sets

1

p.60

14085

'Deductivist' structuralism is just theories, with no commitment to objects, or modality

I

p.60

14084

Noneliminative structuralism treats mathematical objects as positions in real abstract structures

I

p.60

14086

'Modal' structuralism studies all possible concrete models for various mathematical theories

I

p.61

14087

'Settheoretic' structuralism treats mathematics as various structures realised among the sets

II

p.65

14088

An 'intrinsic' property is either found in every duplicate, or exists independent of all externals

III

p.66

14089

Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure

III

p.68

14090

In mathematical structuralism the small depends on the large, which is the opposite of physical structures

V

p.73

14091

There may be a oneway direction of dependence among sets, and among natural numbers

2017

Philosophy of Mathematics

11.1

p.155

23448

Mathematics is the study of all possible patterns, and is thus bound to describe the world

2

p.21

23441

Logical truth is true in all models, so mathematical objects can't be purely logical

3.3

p.44

23442

Game Formalism has no semantics, and Term Formalism reduces the semantics

3.5

p.52

23443

The axioms of group theory are not assertions, but a definition of a structure

4.1

p.56

23444

To investigate axiomatic theories, mathematics needs its own foundational axioms

4.2

p.62

23445

Naïve set theory says any formula defines a set, and coextensive sets are identical

4.6

p.71

23446

You can't prove consistency using a weaker theory, but you can use a consistent theory

7.1

p.101

23447

In classical semantics singular terms refer, and quantifiers range over domains
