77 ideas
| 13395 | If an analysis shows the features of a concept, it doesn't seem to 'reduce' the concept [Jubien] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 9967 | 'Impure' sets have a concrete member, while 'pure' (abstract) sets do not [Jubien] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 13378 | It is a mistake to think that the logic developed for mathematics can clarify language and philosophy [Jubien] |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| 13402 | We only grasp a name if we know whether to apply it when the bearer changes [Jubien] |
| 13405 | The baptiser picks the bearer of a name, but social use decides the category [Jubien] |
| 13399 | Examples show that ordinary proper names are not rigid designators [Jubien] |
| 13398 | We could make a contingent description into a rigid and necessary one by adding 'actual' to it [Jubien] |
| 11115 | 'All horses' either picks out the horses, or the things which are horses [Jubien] |
| 13392 | Philosophers reduce complex English kind-quantifiers to the simplistic first-order quantifier [Jubien] |
| 9968 | A model is 'fundamental' if it contains only concrete entities [Jubien] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 9965 | There couldn't just be one number, such as 17 [Jubien] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 9966 | The subject-matter of (pure) mathematics is abstract structure [Jubien] |
| 9963 | If we all intuited mathematical objects, platonism would be agreed [Jubien] |
| 9962 | How can pure abstract entities give models to serve as interpretations? [Jubien] |
| 9964 | Since mathematical objects are essentially relational, they can't be picked out on their own [Jubien] |
| 13404 | To exist necessarily is to have an essence whose own essence must be instantiated [Jubien] |
| 13386 | If objects are just conventional, there is no ontological distinction between stuff and things [Jubien] |
| 13403 | The category of Venus is not 'object', or even 'planet', but a particular class of good-sized object [Jubien] |
| 11116 | Being a physical object is our most fundamental category [Jubien] |
| 9969 | The empty set is the purest abstract object [Jubien] |
| 13375 | The idea that every entity must have identity conditions is an unfortunate misunderstanding [Jubien] |
| 11117 | Haecceities implausibly have no qualities [Jubien] |
| 13393 | Any entity has the unique property of being that specific entity [Jubien] |
| 13388 | It is incoherent to think that a given entity depends on its kind for its existence [Jubien] |
| 13384 | Objects need conventions for their matter, their temporal possibility, and their spatial possibility [Jubien] |
| 13385 | Basically, the world doesn't have ready-made 'objects'; we carve objects any way we like [Jubien] |
| 13383 | If the statue is loved and the clay hated, that is about the object first qua statue, then qua clay [Jubien] |
| 13400 | If one entity is an object, a statue, and some clay, these come apart in at least three ways [Jubien] |
| 13401 | The idea of coincident objects is a last resort, as it is opposed to commonsense naturalism [Jubien] |
| 13380 | Parts seem to matter when it is just an object, but not matter when it is a kind of object [Jubien] |
| 13376 | We should not regard essentialism as just nontrivial de re necessity [Jubien] |
| 13381 | Thinking of them as 'ships' the repaired ship is the original, but as 'objects' the reassembly is the original [Jubien] |
| 13382 | Rearranging the planks as a ship is confusing; we'd say it was the same 'object' with a different arrangement [Jubien] |
| 13379 | If two objects are indiscernible across spacetime, how could we decide whether or not they are the same? [Jubien] |
| 13394 | Entailment does not result from mutual necessity; mutual necessity ensures entailment [Jubien] |
| 11119 | De re necessity is just de dicto necessity about object-essences [Jubien] |
| 13391 | Modality concerns relations among platonic properties [Jubien] |
| 13374 | To analyse modality, we must give accounts of objects, properties and relations [Jubien] |
| 11118 | Modal propositions transcend the concrete, but not the actual [Jubien] |
| 11108 | Your properties, not some other world, decide your possibilities [Jubien] |
| 11111 | Modal truths are facts about parts of this world, not about remote maximal entities [Jubien] |
| 11105 | We have no idea how many 'possible worlds' there might be [Jubien] |
| 11107 | If there are no other possible worlds, do we then exist necessarily? [Jubien] |
| 11106 | If all possible worlds just happened to include stars, their existence would be necessary [Jubien] |
| 11112 | Possible worlds just give parallel contingencies, with no explanation at all of necessity [Jubien] |
| 11109 | If other worlds exist, then they are scattered parts of the actual world [Jubien] |
| 11113 | Worlds don't explain necessity; we use necessity to decide on possible worlds [Jubien] |
| 13389 | The love of possible worlds is part of the dream that technical logic solves philosophical problems [Jubien] |
| 13390 | Possible worlds don't explain necessity, because they are a bunch of parallel contingencies [Jubien] |
| 11110 | We mustn't confuse a similar person with the same person [Jubien] |
| 13396 | Analysing mental concepts points to 'inclusionism' - that mental phenomena are part of the physical [Jubien] |
| 13377 | First-order logic tilts in favour of the direct reference theory, in its use of constants for objects [Jubien] |
| 2854 | Prescriptivism says 'ought' without commitment to act is insincere, or weakly used [Hooker,B] |
| 2856 | Universal moral judgements imply the Golden Rule ('do as you would be done by') [Hooker,B] |
| 20883 | Modern utilitarians value knowledge, friendship, autonomy, and achievement, as well as pleasure [Hooker,B] |
| 20884 | Rule-utilitarians prevent things like torture, even on rare occasions when it seems best [Hooker,B] |
| 20885 | Euthanasia is active or passive, and voluntary, non-voluntary or involuntary [Hooker,B] |
| 20882 | Euthanasia may not involve killing, so it is 'killing or not saving, out of concern for that person' [Hooker,B] |