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Ideas of Leon Horsten, by Text
[Belgian, fl. 2007, Professor at the Catholic University of Leuven, then at University of Bristol.]
2007

Philosophy of Mathematics

§2.3

p.7

10881

The concept of 'ordinal number' is settheoretic, not arithmetical

§2.4

p.8

10882

Predicative definitions only refer to entities outside the defined collection

§5.2

p.23

10884

A theory is 'categorical' if it has just one model up to isomorphism

§5.3

p.26

10885

Computer proofs don't provide explanations

01.1

p.2

15323

Truth is a property, because the truth predicate has an extension

01.1

p.3

15324

Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles

01.1

p.4

15325

Inferential deflationism says truth has no essence because no unrestricted logic governs the concept

01.2

p.5

15329

Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well

01.2

p.5

15326

Doubt is thrown on classical logic by the way it so easily produces the liar paradox

01.4

p.7

15328

A theory is 'nonconservative' if it facilitates new mathematical proofs

01.4

p.8

15330

FriedmanSheard theory keeps classical logic and aims for maximum strength

01.4

p.8

15331

KripkeFeferman has truth gaps, instead of classical logic, and aims for maximum strength

01.4

p.8

15332

'Reflexive' truth theories allow iterations (it is T that it is T that p)

02.1

p.12

15333

Modern correspondence is said to be with the facts, not with true propositions

02.1

p.13

15337

The correspondence 'theory' is too vague  about both 'correspondence' and 'facts'

02.1

p.13

15336

The pragmatic theory of truth is relative; useful for group A can be useless for group B

02.1

p.13

15334

The coherence theory allows multiple coherent wholes, which could contradict one another

02.1

p.13

15338

We may believe in atomic facts, but surely not complex disjunctive ones?

02.2

p.17

15340

Tarski Biconditional: if you'll assert φ you'll assert φistrue  and also vice versa

02.2

p.18

15341

Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms

02.3

p.20

15344

Deflationism skips definitions and models, and offers just accounts of basic laws of truth

02.3

p.21

15345

Semantic theories have a regress problem in describing truth in the languages for the models

02.3

p.21

15346

Axiomatic approaches to truth avoid the regress problem of semantic theories

02.4

p.23

15348

Propositions have sentencelike structures, so it matters little which bears the truth

02.4

p.23

15347

A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding

02.5

p.25

15349

It is easier to imagine truthvalue gaps (for the Liar, say) than for truthvalue gluts (both T and F)

03.5.2

p.38

15350

The Naďve Theory takes the biconditionals as axioms, but it is inconsistent, and allows the Liar

04.2

p.49

15351

Axiomatic theories take truth as primitive, and propose some laws of truth as axioms

04.2

p.51

15352

A definition should allow the defined term to be eliminated

04.3

p.52

15353

The first incompleteness theorem means that consistency does not entail soundness

04.5

p.55

15354

Tarski's hierarchy lacks uniform truth, and depends on contingent factors

04.6

p.58

15355

Strengthened Liar: 'this sentence is not true in any context'  in no context can this be evaluated

05 Intro

p.59

15356

Deflationism concerns the nature and role of truth, but not its laws

05.1

p.60

15358

Deflationism says truth isn't a topic on its own  it just concerns what is true

05.1

p.60

15357

Philosophy is the most general intellectual discipline

05.2.2

p.63

15359

Deflation: instead of asserting a sentence, we can treat it as an object with the truthproperty

05.2.3

p.65

15360

ZFC showed that the concept of set is mathematical, not logical, because of its existence claims

06.1

p.70

15361

A good theory of truth must be compositional (as well as deriving biconditionals)

06.2

p.72

15362

If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't

06.2

p.73

15363

In the supervaluationist account, disjunctions are not determined by their disjuncts

06.3

p.73

15364

English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable

06.3

p.74

15366

Satisfaction is a primitive notion, and very liable to semantical paradoxes

06.4

p.77

15367

By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!

07.5

p.92

15368

This deflationary account says truth has a role in generality, and in inference

07.5

p.93

15369

Set theory is substantial over firstorder arithmetic, because it enables new proofs

07.7

p.100

15370

Predicativism says mathematical definitions must not include the thing being defined

07.7

p.101

15371

An axiomatic theory needs to be of maximal strength, while being natural and sound

09.3

p.128

15372

Some claim that indicative conditionals are believed by people, even though they are not actually held true

10.1

p.141

15373

Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models

10.2.3

p.146

15374

Truth has no 'nature', but we should try to describe its behaviour in inferences
