Ideas of Řystein Linnebo, by Theme

[Norwegian, fl. 2006, Lecturer at Bristol University, then Birkbeck, London.]

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
A comprehension axiom is 'predicative' if the formula has no bound second-order variables
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
A 'pure logic' must be ontologically innocent, universal, and without presuppositions
A pure logic is wholly general, purely formal, and directly known
5. Theory of Logic / G. Quantification / 6. Plural Quantification
'Some critics admire only one another' cannot be paraphrased in singular first-order
Can second-order logic be ontologically first-order, with all the benefits of second-order?
Plural quantification depends too heavily on combinatorial and set-theoretic considerations
Second-order quantification and plural quantification are different
Traditionally we eliminate plurals by quantifying over sets
Instead of complex objects like tables, plurally quantify over mereological atoms tablewise
Plural plurals are unnatural and need a first-level ontology
Plural quantification may allow a monadic second-order theory with first-order ontology
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / b. Varieties of structuralism
'Deductivist' structuralism is just theories, with no commitment to objects, or modality
'Modal' structuralism studies all possible concrete models for various mathematical theories
Non-eliminative structuralism treats mathematical objects as positions in real abstract structures
'Set-theoretic' structuralism treats mathematics as various structures realised among the sets
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
Structuralism is right about algebra, but wrong about sets
In mathematical structuralism the small depends on the large, which is the opposite of physical structures
7. Existence / C. Structure of Existence / 4. Ontological Dependence
There may be a one-way direction of dependence among sets, and among natural numbers
7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
We speak of a theory's 'ideological commitments' as well as its 'ontological commitments'
7. Existence / D. Theories of Reality / 10. Ontological Commitment / e. Ontological commitment problems
Ordinary speakers posit objects without concern for ontology
8. Modes of Existence / B. Properties / 4. Intrinsic Properties
An 'intrinsic' property is either found in every duplicate, or exists independent of all externals
9. Objects / A. Existence of Objects / 1. Physical Objects
The modern concept of an object is rooted in quantificational logic
19. Language / A. Language / 6. Predicates
Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does?