93 ideas
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 15842 | An ad hominem refutation is reasonable, if it uses the opponent's assumptions [Harte,V] |
| 21616 | Truth and falsity apply to suppositions as well as to assertions [Williamson] |
| 21623 | True and false are not symmetrical; false is more complex, involving negation [Williamson] |
| 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
| 13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
| 13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
| 13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
| 13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
| 21602 | Many-valued logics don't solve vagueness; its presence at the meta-level is ignored [Williamson] |
| 13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
| 13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
| 13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
| 13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
| 13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
| 13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
| 13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
| 13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
| 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
| 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| 13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
| 13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
| 15841 | Mereology began as a nominalist revolt against the commitments of set theory [Harte,V] |
| 21611 | Formal semantics defines validity as truth preserved in every model [Williamson] |
| 21606 | 'Bivalence' is the meta-linguistic principle that 'A' in the object language is true or false [Williamson] |
| 21605 | Excluded Middle is 'A or not A' in the object language [Williamson] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 21612 | Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B' [Williamson] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
| 13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
| 13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
| 13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
| 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
| 13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
| 13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
| 13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
| 21599 | A sorites stops when it collides with an opposite sorites [Williamson] |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| 13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 15858 | Traditionally, the four elements are just what persists through change [Harte,V] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 21589 | When bivalence is rejected because of vagueness, we lose classical logic [Williamson] |
| 21596 | Vagueness undermines the stable references needed by logic [Williamson] |
| 21601 | A vague term can refer to very precise elements [Williamson] |
| 21629 | Equally fuzzy objects can be identical, so fuzziness doesn't entail vagueness [Williamson] |
| 21591 | Vagueness is epistemic. Statements are true or false, but we often don't know which [Williamson] |
| 21619 | If a heap has a real boundary, omniscient speakers would agree where it is [Williamson] |
| 21620 | The epistemic view says that the essence of vagueness is ignorance [Williamson] |
| 21622 | If there is a true borderline of which we are ignorant, this drives a wedge between meaning and use [Williamson] |
| 9120 | Vagueness in a concept is its indiscriminability from other possible concepts [Williamson] |
| 21625 | The vagueness of 'heap' can remain even when the context is fixed [Williamson] |
| 21614 | The 'nihilist' view of vagueness says that 'heap' is not a legitimate concept [Williamson] |
| 21617 | We can say propositions are bivalent, but vague utterances don't express a proposition [Williamson] |
| 21618 | If the vague 'TW is thin' says nothing, what does 'TW is thin if his perfect twin is thin' say? [Williamson] |
| 21590 | Asking when someone is 'clearly' old is higher-order vagueness [Williamson] |
| 21592 | Supervaluation keeps classical logic, but changes the truth in classical semantics [Williamson] |
| 21603 | You can't give a precise description of a language which is intrinsically vague [Williamson] |
| 21604 | Supervaluation assigns truth when all the facts are respected [Williamson] |
| 21607 | Supervaluation has excluded middle but not bivalence; 'A or not-A' is true, even when A is undecided [Williamson] |
| 21608 | Truth-functionality for compound statements fails in supervaluation [Williamson] |
| 21609 | Supervaluationism defines 'supertruth', but neglects it when defining 'valid' [Williamson] |
| 21610 | Supervaluation adds a 'definitely' operator to classical logic [Williamson] |
| 21613 | Supervaluationism cannot eliminate higher-order vagueness [Williamson] |
| 21633 | Nominalists suspect that properties etc are our projections, and could have been different [Williamson] |
| 21630 | If fuzzy edges are fine, then why not fuzzy temporal, modal or mereological boundaries? [Williamson] |
| 15848 | Mereology treats constitution as a criterion of identity, as shown in the axiom of extensionality [Harte,V] |
| 15837 | What exactly is a 'sum', and what exactly is 'composition'? [Harte,V] |
| 15839 | If something is 'more than' the sum of its parts, is the extra thing another part, or not? [Harte,V] |
| 15838 | The problem with the term 'sum' is that it is singular [Harte,V] |
| 21632 | A river is not just event; it needs actual and counterfactual boundaries [Williamson] |
| 21621 | We can't infer metaphysical necessities to be a priori knowable - or indeed knowable in any way [Williamson] |
| 21627 | We have inexact knowledge when we include margins of error [Williamson] |
| 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
| 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
| 21626 | Knowing you know (KK) is usually denied if the knowledge concept is missing, or not considered [Williamson] |
| 21631 | To know, believe, hope or fear, one must grasp the thought, but not when you fail to do them [Williamson] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 21600 | 'Blue' is not a family resemblance, because all the blues resemble in some respect [Williamson] |
| 21615 | References to the 'greatest prime number' have no reference, but are meaningful [Williamson] |
| 18038 | The 't' and 'f' of formal semantics has no philosophical interest, and may not refer to true and false [Williamson] |
| 21624 | It is known that there is a cognitive loss in identifying propositions with possible worlds [Williamson] |