Ideas of ěystein Linnebo, by Theme

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

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2. Reason / D. Definition / 12. Paraphrase
'Some critics admire only one another' cannot be paraphrased in singular first-order
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
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Na´ve logical sets
Na´ve set theory says any formula defines a set, and coextensive sets are identical
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
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
Plural plurals are unnatural and need a first-level ontology
Traditionally we eliminate plurals by quantifying over sets
Instead of complex objects like tables, plurally quantify over mereological atoms tablewise
Plural quantification may allow a monadic second-order theory with first-order ontology
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
In classical semantics singular terms refer, and quantifiers range over domains
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
The axioms of group theory are not assertions, but a definition of a structure
To investigate axiomatic theories, mathematics needs its own foundational axioms
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
You can't prove consistency using a weaker theory, but you can use a consistent theory
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Mathematics is the study of all possible patterns, and is thus bound to describe the world
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
'Deductivist' structuralism is just theories, with no commitment to objects, or modality
Non-eliminative structuralism treats mathematical objects as positions in real abstract structures
'Modal' structuralism studies all possible concrete models for various mathematical theories
'Set-theoretic' structuralism treats mathematics as various structures realised among the sets
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure
6. Mathematics / B. Foundations for Mathematics / 7. 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
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Logical truth is true in all models, so mathematical objects can't be purely logical
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Game Formalism has no semantics, and Term Formalism reduces the semantics
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 / C. Assigning Meanings / 3. Predicates
Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does?