57 ideas
| 21360 | Unobservant thinkers tend to dogmatise using insufficient facts [Aristotle] |
| 9376 | A sentence may simultaneously define a term, and also assert a fact [Boghossian] |
| 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] |
| 9375 | Conventionalism agrees with realists that logic has truth values, but not over the source [Boghossian] |
| 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] |
| 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] |
| 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] |
| 13212 | Infinity is only potential, never actual [Aristotle] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 13221 | Existence is either potential or actual [Aristotle] |
| 16100 | True change is in a thing's logos or its matter, not in its qualities [Aristotle] |
| 16101 | A change in qualities is mere alteration, not true change [Aristotle] |
| 12133 | If the substratum persists, it is 'alteration'; if it doesn't, it is 'coming-to-be' or 'passing-away' [Aristotle] |
| 13213 | All comings-to-be are passings-away, and vice versa [Aristotle] |
| 12134 | Matter is the substratum, which supports both coming-to-be and alteration [Aristotle] |
| 16572 | Does the pure 'this' come to be, or the 'this-such', or 'so-great', or 'somewhere'? [Aristotle] |
| 16573 | Philosophers have worried about coming-to-be from nothing pre-existing [Aristotle] |
| 13214 | The substratum changing to a contrary is the material cause of coming-to-be [Aristotle] |
| 13215 | If a perceptible substratum persists, it is 'alteration'; coming-to-be is a complete change [Aristotle] |
| 9369 | 'Snow is white or it isn't' is just true, not made true by stipulation [Boghossian] |
| 9367 | The a priori is explained as analytic to avoid a dubious faculty of intuition [Boghossian] |
| 9373 | That logic is a priori because it is analytic resulted from explaining the meaning of logical constants [Boghossian] |
| 9380 | We can't hold a sentence true without evidence if we can't agree which sentence is definitive of it [Boghossian] |
| 9384 | We may have strong a priori beliefs which we pragmatically drop from our best theory [Boghossian] |
| 16717 | Which of the contrary features of a body are basic to it? [Aristotle] |
| 9374 | If we learn geometry by intuition, how could this faculty have misled us for so long? [Boghossian] |
| 9378 | If meaning depends on conceptual role, what properties are needed to do the job? [Boghossian] |
| 9377 | 'Conceptual role semantics' says terms have meaning from sentences and/or inferences [Boghossian] |
| 9372 | Could expressions have meaning, without two expressions possibly meaning the same? [Boghossian] |
| 17721 | There are no truths in virtue of meaning, but there is knowability in virtue of understanding [Boghossian, by Jenkins] |
| 9368 | Epistemological analyticity: grasp of meaning is justification; metaphysical: truth depends on meaning [Boghossian] |
| 13216 | Matter is the limit of points and lines, and must always have quality and form [Aristotle] |
| 17994 | The primary matter is the substratum for the contraries like hot and cold [Aristotle] |
| 13224 | There couldn't be just one element, which was both water and air at the same time [Aristotle] |
| 16594 | The Four Elements must change into one another, or else alteration is impossible [Aristotle] |
| 13223 | Fire is hot and dry; Air is hot and moist; Water is cold and moist; Earth is cold and dry [Aristotle] |
| 13220 | Bodies are endlessly divisible [Aristotle] |
| 13210 | Wood is potentially divided through and through, so what is there in the wood besides the division? [Aristotle] |
| 13211 | If a body is endlessly divided, is it reduced to nothing - then reassembled from nothing? [Aristotle] |
| 13228 | There is no time without movement [Aristotle] |
| 16595 | If each thing can cease to be, why hasn't absolutely everything ceased to be long ago? [Aristotle] |
| 13227 | Being is better than not-being [Aristotle] |
| 13226 | An Order controls all things [Aristotle] |