119 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] |
| 9414 | Metaphysics is the mapping of possibilities [Lowe, by Mumford] |
| 16414 | Science needs metaphysics to weed out its presuppositions [Lowe, by Hofweber] |
| 8282 | Only metaphysics can decide whether identity survives through change [Lowe] |
| 16127 | Metaphysics tells us what there could be, rather than what there is [Lowe] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 8262 | How can a theory of meaning show the ontological commitments of two paraphrases of one idea? [Lowe] |
| 15842 | An ad hominem refutation is reasonable, if it uses the opponent's assumptions [Harte,V] |
| 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
| 8315 | Maybe facts are just true propositions [Lowe] |
| 8319 | One-to-one correspondence would need countable, individuable items [Lowe] |
| 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] |
| 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] |
| 8309 | A set is a 'number of things', not a 'collection', because nothing actually collects the members [Lowe] |
| 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] |
| 8322 | I don't believe in the empty set, because (lacking members) it lacks identity-conditions [Lowe] |
| 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] |
| 13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [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] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [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] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 8312 | It is better if the existential quantifier refers to 'something', rather than a 'thing' which needs individuation [Lowe] |
| 13513 | Models are ways the world might be from a first-order point of view [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] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [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] |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [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] |
| 8297 | Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe] |
| 8266 | Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe] |
| 8302 | Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe] |
| 8298 | Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe] |
| 8311 | If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe] |
| 8310 | Does the existence of numbers matter, in the way space, time and persons do? [Lowe] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 8321 | All possible worlds contain abstracta (e.g. numbers), which means they contain concrete objects [Lowe] |
| 8300 | Perhaps possession of causal power is the hallmark of existence (and a reason to deny the void) [Lowe] |
| 8281 | Heraclitus says change is new creation, and Spinoza that it is just phases of the one substance [Lowe] |
| 15858 | Traditionally, the four elements are just what persists through change [Harte,V] |
| 8270 | Events are changes or non-changes in properties and relations of persisting objects [Lowe] |
| 8308 | Events are ontologically indispensable for singular causal explanations [Lowe] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 8314 | Are facts wholly abstract, or can they contain some concrete constituents? [Lowe] |
| 8316 | Facts cannot be wholly abstract if they enter into causal relations [Lowe] |
| 8318 | The problem with the structured complex view of facts is what binds the constituents [Lowe] |
| 8323 | It is whimsical to try to count facts - how many facts did I learn before breakfast? [Lowe] |
| 8313 | Facts are needed for truth-making and causation, but they seem to lack identity criteria [Lowe] |
| 8258 | Two of the main rivals for the foundations of ontology are substances, and facts or states-of-affairs [Lowe] |
| 8301 | Some abstractions exist despite lacking causal powers, because explanation needs them [Lowe] |
| 8283 | Ontological categories are not natural kinds: the latter can only be distinguished using the former [Lowe] |
| 8284 | The top division of categories is either abstract/concrete, or universal/particular, or necessary/contingent [Lowe] |
| 13122 | Lowe divides things into universals and particulars, then kinds and properties, and abstract/concrete [Lowe, by Westerhoff] |
| 8273 | Is 'the Thames is broad in London' relational, or adverbial, or segmental? [Lowe] |
| 8285 | I prefer 'modes' to 'tropes', because it emphasises their dependence [Lowe] |
| 8286 | Tropes cannot have clear identity-conditions, so they are not objects [Lowe] |
| 8294 | How can tropes depend on objects for their identity, if objects are just bundles of tropes? [Lowe] |
| 8295 | Why cannot a trope float off and join another bundle? [Lowe] |
| 8296 | Does a ball snug in plaster have one trope, or two which coincide? [Lowe] |
| 8288 | Sortal terms for universals involve a substance, whereas adjectival terms do not [Lowe] |
| 8293 | Real universals are needed to explain laws of nature [Lowe] |
| 8307 | Particulars are instantiations, and universals are instantiables [Lowe] |
| 8267 | Perhaps concrete objects are entities which are in space-time and subject to causality [Lowe] |
| 8265 | Our commitment to the existence of objects should depend on their explanatory value [Lowe] |
| 8275 | Objects are entities with full identity-conditions, but there are entities other than objects [Lowe] |
| 16130 | To be an object at all requires identity-conditions [Lowe] |
| 8263 | An object is an entity which has identity-conditions [Lowe] |
| 8268 | Some things (such as electrons) can be countable, while lacking proper identity [Lowe] |
| 8303 | Criteria of identity cannot individuate objects, because they are shared among different types [Lowe] |
| 8292 | Diversity of two tigers is their difference in space-time; difference of matter is a consequence [Lowe] |
| 8291 | Individuation principles identify what kind it is; identity criteria distinguish items of the same kind [Lowe] |
| 16128 | A 'substance' is an object which doesn't depend for existence on other objects [Lowe] |
| 8279 | The identity of composite objects isn't fixed by original composition, because how do you identify the origin? [Lowe] |
| 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] |
| 8271 | An object 'endures' if it is always wholly present, and 'perdures' if different parts exist at different times [Lowe] |
| 8272 | How can you identify temporal parts of tomatoes without referring to tomatoes? [Lowe] |
| 8305 | A clear idea of the kind of an object must precede a criterion of identity for it [Lowe] |
| 8290 | One view is that two objects of the same type are only distinguished by differing in matter [Lowe] |
| 15079 | 'Conceptual' necessity is narrow logical necessity, true because of concepts and logical laws [Lowe] |
| 16063 | Metaphysical necessity is logical necessity 'broadly construed' [Lowe, by Lynch/Glasgow] |
| 8260 | Logical necessity can be 'strict' (laws), or 'narrow' (laws and definitions), or 'broad' (all logical worlds) [Lowe] |
| 16131 | The metaphysically possible is what acceptable principles and categories will permit [Lowe] |
| 8320 | Does every abstract possible world exist in every possible world? [Lowe] |
| 8280 | While space may just be appearance, time and change can't be, because the appearances change [Lowe] |
| 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] |
| 8276 | Properties or qualities are essentially adjectival, not objectual [Lowe] |
| 8289 | The idea that Cartesian souls are made of some ghostly 'immaterial' stuff is quite unwarranted [Lowe] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 8299 | Abstractions are non-spatial, or dependent, or derived from concepts [Lowe] |
| 8306 | You can think of a direction without a line, but a direction existing with no lines is inconceivable [Lowe] |
| 8317 | To cite facts as the elements in causation is to confuse states of affairs with states of objects [Lowe] |
| 8269 | Points are limits of parts of space, so parts of space cannot be aggregates of them [Lowe] |