95 ideas
| 1798 | He studied philosophy by suspending his judgement on everything [Pyrrho, by Diog. Laertius] |
| 16325 | Analysis rests on natural language, but its ideal is a framework which revises language [Halbach] |
| 1800 | Sceptics say reason is only an instrument, because reason can only be attacked with reason [Pyrrho, by Diog. Laertius] |
| 16292 | An explicit definition enables the elimination of what is defined [Halbach] |
| 15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine] |
| 16307 | Don't trust analogies; they are no more than a guideline [Halbach] |
| 16339 | Truth axioms prove objects exist, so truth doesn't seem to be a logical notion [Halbach] |
| 16330 | Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth [Halbach] |
| 16324 | Any definition of truth requires a metalanguage [Halbach] |
| 15647 | Truth definitions don't produce a good theory, because they go beyond your current language [Halbach] |
| 16293 | Traditional definitions of truth often make it more obscure, rather than less [Halbach] |
| 16301 | If people have big doubts about truth, a definition might give it more credibility [Halbach] |
| 15649 | In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage [Halbach] |
| 16297 | Semantic theories avoid Tarski's Theorem by sticking to a sublanguage [Halbach] |
| 16337 | Disquotational truth theories are short of deductive power [Halbach] |
| 15648 | Instead of a truth definition, add a primitive truth predicate, and axioms for how it works [Halbach] |
| 15655 | Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms? [Halbach] |
| 15654 | If truth is defined it can be eliminated, whereas axiomatic truth has various commitments [Halbach] |
| 16294 | Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage [Halbach] |
| 16311 | To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction' [Halbach] |
| 16318 | Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach] |
| 16326 | The main semantic theories of truth are Kripke's theory, and revisions semantics [Halbach] |
| 16299 | Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory [Halbach] |
| 16340 | Truth axioms need a base theory, because that is where truth issues arise [Halbach] |
| 16322 | CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA [Halbach] |
| 16305 | We know a complete axiomatisation of truth is not feasible [Halbach] |
| 16313 | A theory is 'conservative' if it adds no new theorems to its base theory [Halbach, by PG] |
| 16315 | The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals [Halbach] |
| 16314 | Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free' [Halbach] |
| 15650 | Axiomatic theories of truth need a weak logical framework, and not a strong metatheory [Halbach] |
| 16327 | Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth [Halbach] |
| 16329 | Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts [Halbach] |
| 16331 | The KF is much stronger deductively than FS, which relies on classical truth [Halbach] |
| 16332 | The KF theory is useful, but it is not a theory containing its own truth predicate [Halbach] |
| 15656 | Deflationists say truth merely serves to express infinite conjunctions [Halbach] |
| 16338 | Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge [Halbach] |
| 16317 | The main problem for deflationists is they can express generalisations, but not prove them [Halbach] |
| 16316 | Deflationists say truth is just for expressing infinite conjunctions or generalisations [Halbach] |
| 16319 | Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism [Halbach] |
| 16320 | Some say deflationism is axioms which are conservative over the base theory [Halbach] |
| 16335 | In Strong Kleene logic a disjunction just needs one disjunct to be true [Halbach] |
| 16334 | In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value [Halbach] |
| 17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo] |
| 17607 | Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo] |
| 15657 | To prove the consistency of set theory, we must go beyond set theory [Halbach] |
| 16309 | Every attempt at formal rigour uses some set theory [Halbach] |
| 10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg] |
| 13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
| 17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
| 9565 | Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara] |
| 3339 | For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn] |
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| 13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy] |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| 13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
| 13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD] |
| 13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy] |
| 16333 | The underestimated costs of giving up classical logic are found in mathematical reasoning [Halbach] |
| 15652 | We can use truth instead of ontologically loaded second-order comprehension assumptions about properties [Halbach] |
| 15651 | Instead of saying x has a property, we can say a formula is true of x - as long as we have 'true' [Halbach] |
| 16310 | A theory is some formulae and all of their consequences [Halbach] |
| 16344 | Soundness must involve truth; the soundness of PA certainly needs it [Halbach] |
| 16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |
| 16341 | Normally we only endorse a theory if we believe it to be sound [Halbach] |
| 16347 | Many new paradoxes may await us when we study interactions between frameworks [Halbach] |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| 16336 | The liar paradox applies truth to a negated truth (but the conditional will serve equally) [Halbach] |
| 13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD] |
| 15897 | Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine] |
| 16321 | The compactness theorem can prove nonstandard models of PA [Halbach] |
| 16343 | The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach] |
| 18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy] |
| 13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy] |
| 16312 | To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach] |
| 9627 | Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR] |
| 16308 | Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach] |
| 16345 | That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction [Halbach] |
| 16346 | Maybe necessity is a predicate, not the usual operator, to make it more like truth [Halbach] |
| 6595 | If we need a criterion of truth, we need to know whether it is the correct criterion [Pyrrho, by Fogelin] |
| 6593 | The Pyrrhonians attacked the dogmas of professors, not ordinary people [Pyrrho, by Fogelin] |
| 6592 | Academics said that Pyrrhonians were guilty of 'negative dogmatism' [Pyrrho, by Fogelin] |
| 1808 | Perception of things depends on their size or quantity (Mode 8) [Pyrrho, by Diog. Laertius] |
| 1802 | Individuals vary in responses and feelings (Mode 2) [Pyrrho, by Diog. Laertius] |
| 1807 | Perception varies with viewing distance and angle (Mode 7) [Pyrrho, by Diog. Laertius] |
| 1801 | Animals vary in their feelings and judgements (Mode 1) [Pyrrho, by Diog. Laertius] |
| 1804 | Perception varies with madness or disease (Mode 4) [Pyrrho, by Diog. Laertius] |
| 1810 | Perception and judgement depend on comparison (Mode 10) [Pyrrho, by Diog. Laertius] |
| 1805 | Judgements vary according to local culture and law (Mode 5) [Pyrrho, by Diog. Laertius] |
| 1809 | Perception is affected by expectations (Mode 9) [Pyrrho, by Diog. Laertius] |
| 1803 | Objects vary according to which sense perceives them (Mode 3) [Pyrrho, by Diog. Laertius] |
| 1806 | Perception of objects depends on surrounding conditions (Mode 6) [Pyrrho, by Diog. Laertius] |
| 17613 | We should judge principles by the science, not science by some fixed principles [Zermelo] |
| 16298 | We need propositions to ascribe the same beliefs to people with different languages [Halbach] |
| 3062 | There are no causes, because they are relative, and alike things can't cause one another [Pyrrho, by Diog. Laertius] |
| 3063 | Motion can't move where it is, and can't move where it isn't, so it can't exist [Pyrrho, by Diog. Laertius] |