2007 | Philosophy of Mathematics |
§2.3 | p.7 | 10881 | The concept of 'ordinal number' is set-theoretic, not arithmetical |
§2.4 | p.8 | 10882 | Predicative definitions only refer to entities outside the defined collection |
§5.1 | p.22 | 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers |
§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 |
2011 | The Tarskian Turn |
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 'non-conservative' if it facilitates new mathematical proofs |
01.4 | p.8 | 15330 | Friedman-Sheard theory keeps classical logic and aims for maximum strength |
01.4 | p.8 | 15331 | Kripke-Feferman 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 | 15334 | The coherence theory allows multiple coherent wholes, which could contradict one another |
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 | 15338 | We may believe in atomic facts, but surely not complex disjunctive ones? |
02.2 | p.17 | 15340 | Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa |
02.2 | p.18 | 15341 | Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms |
02.2 | p.18 | 15342 | Tarski proved that any reasonably expressive language suffers from the liar paradox |
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 sentence-like 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 truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F) |
03.5.2 | p.38 | 15350 | The Naďve Theory takes the bi-conditionals 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 truth-property |
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 first-order 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 |