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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.]
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2007
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Philosophy of Mathematics
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§2.3
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p.7
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10881
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The concept of 'ordinal number' is set-theoretic, not arithmetical
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§2.4
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p.8
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10882
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Predicative definitions only refer to entities outside the defined collection
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§5.2
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p.23
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10884
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A theory is 'categorical' if it has just one model up to isomorphism
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§5.3
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p.26
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10885
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Computer proofs don't provide explanations
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01.1
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p.2
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15323
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Truth is a property, because the truth predicate has an extension
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01.1
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p.3
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15324
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Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles
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01.1
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p.4
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15325
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Inferential deflationism says truth has no essence because no unrestricted logic governs the concept
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01.2
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p.5
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15329
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Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well
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01.2
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p.5
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15326
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Doubt is thrown on classical logic by the way it so easily produces the liar paradox
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01.4
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p.7
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15328
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A theory is 'non-conservative' if it facilitates new mathematical proofs
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01.4
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p.8
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15331
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Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength
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01.4
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p.8
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15332
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'Reflexive' truth theories allow iterations (it is T that it is T that p)
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01.4
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p.8
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15330
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Friedman-Sheard theory keeps classical logic and aims for maximum strength
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02.1
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p.12
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15333
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Modern correspondence is said to be with the facts, not with true propositions
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02.1
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p.13
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15337
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The correspondence 'theory' is too vague - about both 'correspondence' and 'facts'
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02.1
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p.13
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15334
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The coherence theory allows multiple coherent wholes, which could contradict one another
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02.1
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p.13
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15336
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The pragmatic theory of truth is relative; useful for group A can be useless for group B
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02.1
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p.13
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15338
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We may believe in atomic facts, but surely not complex disjunctive ones?
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02.2
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p.17
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15340
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Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa
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02.2
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p.18
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15341
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Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms
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02.3
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p.20
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15344
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Deflationism skips definitions and models, and offers just accounts of basic laws of truth
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02.3
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p.21
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15345
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Semantic theories have a regress problem in describing truth in the languages for the models
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02.3
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p.21
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15346
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Axiomatic approaches to truth avoid the regress problem of semantic theories
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02.4
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p.23
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15348
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Propositions have sentence-like structures, so it matters little which bears the truth
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02.4
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p.23
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15347
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A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding
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02.5
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p.25
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15349
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It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F)
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03.5.2
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p.38
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15350
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The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar
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04.2
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p.49
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15351
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Axiomatic theories take truth as primitive, and propose some laws of truth as axioms
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04.2
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p.51
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15352
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A definition should allow the defined term to be eliminated
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04.3
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p.52
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15353
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The first incompleteness theorem means that consistency does not entail soundness
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04.5
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p.55
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15354
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Tarski's hierarchy lacks uniform truth, and depends on contingent factors
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04.6
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p.58
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15355
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Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated
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05 Intro
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p.59
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15356
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Deflationism concerns the nature and role of truth, but not its laws
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05.1
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p.60
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15358
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Deflationism says truth isn't a topic on its own - it just concerns what is true
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05.1
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p.60
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15357
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Philosophy is the most general intellectual discipline
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05.2.2
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p.63
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15359
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Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property
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05.2.3
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p.65
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15360
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ZFC showed that the concept of set is mathematical, not logical, because of its existence claims
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06.1
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p.70
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15361
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A good theory of truth must be compositional (as well as deriving biconditionals)
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06.2
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p.72
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15362
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If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't
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06.2
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p.73
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15363
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In the supervaluationist account, disjunctions are not determined by their disjuncts
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06.3
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p.73
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15364
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English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable
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06.3
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p.74
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15366
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Satisfaction is a primitive notion, and very liable to semantical paradoxes
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06.4
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p.77
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15367
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By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!
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07.5
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p.92
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15368
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This deflationary account says truth has a role in generality, and in inference
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07.5
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p.93
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15369
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Set theory is substantial over first-order arithmetic, because it enables new proofs
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07.7
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p.100
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15370
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Predicativism says mathematical definitions must not include the thing being defined
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07.7
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p.101
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15371
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An axiomatic theory needs to be of maximal strength, while being natural and sound
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09.3
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p.128
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15372
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Some claim that indicative conditionals are believed by people, even though they are not actually held true
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10.1
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p.141
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15373
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Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models
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10.2.3
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p.146
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15374
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Truth has no 'nature', but we should try to describe its behaviour in inferences
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