104 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] |
| 8558 | One system has properties, powers, events, similarity and substance [Shoemaker] |
| 8559 | Analysis aims at internal relationships, not reduction [Shoemaker] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 8594 | People have had good reasons for thinking that the circle has been squared [Shoemaker] |
| 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] |
| 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] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 15091 | Restrict 'logical truth' to formal logic, rather than including analytic and metaphysical truths [Shoemaker] |
| 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] |
| 9117 | The smallest heap has four objects: three on the bottom, one on the top [Hart,WD, by Sorensen] |
| 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] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 8596 | Inability to measure equality doesn't make all lengths unequal [Shoemaker] |
| 8597 | We couldn't verify the earth's rotation if everyone simultaneously fell asleep [Shoemaker] |
| 15092 | Formerly I said properties are individuated by essential causal powers and causing instantiation [Shoemaker, by Shoemaker] |
| 15095 | A property's causal features are essential, and only they fix its identity [Shoemaker] |
| 15097 | I claim that a property has its causal features in all possible worlds [Shoemaker] |
| 8543 | Genuine properties are closely related to genuine changes [Shoemaker] |
| 8551 | Properties must be essentially causal if we can know and speak about them [Shoemaker] |
| 8557 | To ascertain genuine properties, examine the object directly [Shoemaker] |
| 15761 | We should abandon the idea that properties are the meanings of predicate expressions [Shoemaker] |
| 15756 | Some truths are not because of a thing's properties, but because of the properties of related things [Shoemaker] |
| 15758 | Things have powers in virtue of (which are entailed by) their properties [Shoemaker] |
| 8547 | One power can come from different properties; a thing's powers come from its properties [Shoemaker] |
| 8549 | Properties are functions producing powers, and powers are functions producing effects [Shoemaker] |
| 15094 | I now deny that properties are cluster of powers, and take causal properties as basic [Shoemaker] |
| 12678 | Shoemaker says all genuine properties are dispositional [Shoemaker, by Ellis] |
| 8545 | A causal theory of properties focuses on change, not (say) on abstract properties of numbers [Shoemaker] |
| 15757 | 'Square', 'round' and 'made of copper' show that not all properties are dispositional [Shoemaker] |
| 15759 | The identity of a property concerns its causal powers [Shoemaker] |
| 15760 | Properties are clusters of conditional powers [Shoemaker] |
| 15762 | Could properties change without the powers changing, or powers change without the properties changing? [Shoemaker] |
| 8552 | If properties are separated from causal powers, this invites total elimination [Shoemaker] |
| 4040 | The notions of property and of causal power are parts of a single system of related concepts [Shoemaker] |
| 15765 | Actually, properties are individuated by causes as well as effects [Shoemaker] |
| 14534 | Shoemaker moved from properties as powers to properties bestowing powers [Shoemaker, by Mumford/Anjum] |
| 8548 | Dispositional predicates ascribe powers, and the rest ascribe properties [Shoemaker] |
| 9485 | Universals concern how things are, and how they could be [Shoemaker, by Bird] |
| 8550 | Triangular and trilateral are coextensive, but different concepts; but powers and properties are the same [Shoemaker] |
| 594 | Speusippus suggested underlying principles for every substance, and ended with a huge list [Speussipus, by Aristotle] |
| 8555 | There is no subset of properties which guarantee a thing's identity [Shoemaker] |
| 15099 | If something is possible, but not nomologically possible, we need metaphysical possibility [Shoemaker] |
| 8554 | Possible difference across worlds depends on difference across time in the actual world [Shoemaker] |
| 15101 | Once you give up necessity as a priori, causal necessity becomes the main type of necessity [Shoemaker] |
| 15764 | 'Conceivable' is either not-provably-false, or compatible with what we know? [Shoemaker] |
| 15098 | Empirical evidence shows that imagining a phenomenon can show it is possible [Shoemaker] |
| 15100 | Imagination reveals conceptual possibility, where descriptions avoid contradiction or incoherence [Shoemaker] |
| 8562 | It is possible to conceive what is not possible [Shoemaker] |
| 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] |
| 5691 | The adverbial account of sensation says not 'see a red image' but be 'appeared to redly' [Shoemaker] |
| 8593 | Maybe billions of changeless years have elapsed since my last meal [Shoemaker] |
| 15096 | 'Grue' only has causal features because of its relation to green [Shoemaker] |
| 8556 | Grueness is not, unlike green and blue, associated with causal potential [Shoemaker] |
| 5687 | For true introspection, must we be aware that we are aware of our mental events? [Shoemaker] |
| 5688 | Empirical foundationalism says basic knowledge is self-intimating, and incorrigible or infallible [Shoemaker] |
| 1389 | If memory is the sole criterion of identity, we ought to use it for other people too [Shoemaker] |
| 1390 | Bodily identity is one criterion and memory another, for personal identity [Shoemaker, by PG] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 8542 | If causality is between events, there must be reference to the properties involved [Shoemaker] |
| 8598 | If things turn red for an hour and then explode, we wouldn't say the redness was the cause [Shoemaker] |
| 15093 | We might say laws are necessary by combining causal properties with Armstrong-Dretske-Tooley laws [Shoemaker] |
| 8560 | If causal laws describe causal potentialities, the same laws govern properties in all possible worlds [Shoemaker] |
| 15763 | If properties are causal, then causal necessity is a species of logical necessity [Shoemaker] |
| 8561 | If a world has different causal laws, it must have different properties [Shoemaker] |
| 8553 | It looks as if the immutability of the powers of a property imply essentiality [Shoemaker] |
| 4226 | If three regions 'freeze' every three, four and five years, after sixty years everything stops for a year [Shoemaker, by Lowe] |
| 8595 | If three regions freeze every 3rd, 4th and 5th year, they all freeze together every 60 years [Shoemaker] |
| 2632 | Speusippus said things were governed by some animal force rather than the gods [Speussipus, by Cicero] |