48 ideas
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] |
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] |
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] |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [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] |
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] |
17377 | All descriptive language is classificatory [Dupré] |
17376 | We should aim for a classification which tells us as much as possible about the object [Dupré] |
17390 | Natural kinds don't need essentialism to be explanatory [Dupré] |
17388 | It seems that species lack essential properties, so they can't be natural kinds [Dupré] |
17389 | A species might have its essential genetic mechanism replaced by a new one [Dupré] |
17374 | The possibility of prediction rests on determinism [Dupré] |
17378 | Presumably molecular structure seems important because we never have the Twin Earth experience [Dupré] |
5901 | Is 'productive of happiness' the definition of 'right', or the cause of it? [Ross on Bentham] |
5934 | Of Bentham's 'dimensions' of pleasure, only intensity and duration matter [Ross on Bentham] |
5271 | Prejudice apart, push-pin has equal value with music and poetry [Bentham] |
3777 | Pleasure and pain control all human desires and duties [Bentham] |
3554 | Bentham thinks happiness is feeling good, but why use morality to achieve that? [Annas on Bentham] |
3781 | The value of pleasures and pains is their force [Bentham] |
20977 | Natural rights are nonsense, and unspecified natural rights is nonsense on stilts [Bentham] |
3778 | The community's interest is a sum of individual interests [Bentham] |
21003 | Only laws can produce real rights; rights from 'law of nature' are imaginary [Bentham] |
20280 | Large mature animals are more rational than babies. But all that really matters is - can they suffer? [Bentham] |
3779 | Unnatural, when it means anything, means infrequent [Bentham] |
17381 | Phylogenetics involves history, and cladism rests species on splits in lineage [Dupré] |
17385 | Kinds don't do anything (including evolve) because they are abstract [Dupré] |
17380 | Wales may count as fish [Dupré] |
17375 | Natural kinds are decided entirely by the intentions of our classification [Dupré] |
17379 | Borders between species are much less clear in vegetables than among animals [Dupré] |
17382 | Cooks, unlike scientists, distinguish garlic from onions [Dupré] |
17384 | Even atoms of an element differ, in the energy levels of their electrons [Dupré] |
17387 | Ecologists favour classifying by niche, even though that can clash with genealogy [Dupré] |
17386 | The theory of evolution is mainly about species [Dupré] |
17383 | Species are the lowest-level classification in biology [Dupré] |
3780 | We must judge a thing morally to know if it conforms to God's will [Bentham] |