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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 The concept of 'ordinal number' is set-theoretic, not arithmetical
§2.4 p.8 Predicative definitions only refer to entities outside the defined collection
§5.1 p.22 Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers
§5.2 p.23 A theory is 'categorical' if it has just one model up to isomorphism
§5.3 p.26 Computer proofs don't provide explanations
2011 The Tarskian Turn
01.1 p.2 Truth is a property, because the truth predicate has an extension
01.1 p.3 Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles
01.1 p.4 Inferential deflationism says truth has no essence because no unrestricted logic governs the concept
01.2 p.5 Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well
01.2 p.5 Doubt is thrown on classical logic by the way it so easily produces the liar paradox
01.4 p.7 A theory is 'non-conservative' if it facilitates new mathematical proofs
01.4 p.8 Friedman-Sheard theory keeps classical logic and aims for maximum strength
01.4 p.8 'Reflexive' truth theories allow iterations (it is T that it is T that p)
01.4 p.8 Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength
02.1 p.12 Modern correspondence is said to be with the facts, not with true propositions
02.1 p.13 The pragmatic theory of truth is relative; useful for group A can be useless for group B
02.1 p.13 The correspondence 'theory' is too vague - about both 'correspondence' and 'facts'
02.1 p.13 We may believe in atomic facts, but surely not complex disjunctive ones?
02.1 p.13 The coherence theory allows multiple coherent wholes, which could contradict one another
02.2 p.17 Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa
02.2 p.18 Tarski proved that any reasonably expressive language suffers from the liar paradox
02.2 p.18 Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms
02.3 p.20 Deflationism skips definitions and models, and offers just accounts of basic laws of truth
02.3 p.21 Semantic theories have a regress problem in describing truth in the languages for the models
02.3 p.21 Axiomatic approaches to truth avoid the regress problem of semantic theories
02.4 p.23 Propositions have sentence-like structures, so it matters little which bears the truth
02.4 p.23 A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding
02.5 p.25 It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F)
03.5.2 p.38 The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar
04.2 p.49 Axiomatic theories take truth as primitive, and propose some laws of truth as axioms
04.2 p.51 A definition should allow the defined term to be eliminated
04.3 p.52 The first incompleteness theorem means that consistency does not entail soundness
04.5 p.55 Tarski's hierarchy lacks uniform truth, and depends on contingent factors
04.6 p.58 Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated
05 Intro p.59 Deflationism concerns the nature and role of truth, but not its laws
05.1 p.60 Deflationism says truth isn't a topic on its own - it just concerns what is true
05.1 p.60 Philosophy is the most general intellectual discipline
05.2.2 p.63 Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property
05.2.3 p.65 ZFC showed that the concept of set is mathematical, not logical, because of its existence claims
06.1 p.70 A good theory of truth must be compositional (as well as deriving biconditionals)
06.2 p.72 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 In the supervaluationist account, disjunctions are not determined by their disjuncts
06.3 p.73 English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable
06.3 p.74 Satisfaction is a primitive notion, and very liable to semantical paradoxes
06.4 p.77 By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!
07.5 p.92 This deflationary account says truth has a role in generality, and in inference
07.5 p.93 Set theory is substantial over first-order arithmetic, because it enables new proofs
07.7 p.100 Predicativism says mathematical definitions must not include the thing being defined
07.7 p.101 An axiomatic theory needs to be of maximal strength, while being natural and sound
09.3 p.128 Some claim that indicative conditionals are believed by people, even though they are not actually held true
10.1 p.141 Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models
10.2.3 p.146 Truth has no 'nature', but we should try to describe its behaviour in inferences