92 ideas
| 11283 | There is pure deductive reasoning, and explanatory demonstration reasoning [Aristotle, by Politis] |
| 1672 | Maybe everything could be demonstrated, if demonstration can be reciprocal or circular [Aristotle] |
| 1684 | Two falsehoods can be contrary to one another [Aristotle] |
| 12145 | Definitions are of what something is, and that is universal [Aristotle] |
| 12384 | Definition by division needs predicates, which are well ordered and thorough [Aristotle] |
| 12075 | An Aristotelian definition is causal [Aristotle, by Witt] |
| 9066 | You can define objects by progressively identifying what is the same and what is different [Aristotle] |
| 12382 | What it is and why it is are the same; screening defines and explains an eclipse [Aristotle] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 1668 | An axiom is a principle which must be understood if one is to learn anything [Aristotle] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [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] |
| 12376 | Demonstrations by reductio assume excluded middle [Aristotle] |
| 12373 | Something holds universally when it is proved of an arbitrary and primitive case [Aristotle] |
| 12363 | Everything is either asserted or denied truly [Aristotle] |
| 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] |
| 13004 | Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof [Aristotle, by Leibniz] |
| 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] |
| 12377 | Mathematics is concerned with forms, not with superficial properties [Aristotle] |
| 12372 | The essence of a triangle comes from the line, mentioned in any account of triangles [Aristotle] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 12369 | A unit is what is quantitatively indivisible [Aristotle] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 18910 | To seek truth, study the real connections between subjects and attributes [Aristotle] |
| 1675 | Separate Forms aren't needed for logic, but universals (one holding of many) are essential [Aristotle] |
| 1677 | We can forget the Forms, as they are irrelevant, and not needed in giving demonstrations [Aristotle] |
| 1687 | Why are being terrestrial and a biped combined in the definition of man, but being literate and musical aren't? [Aristotle] |
| 1681 | Units are positionless substances, and points are substances with position [Aristotle] |
| 12146 | Definitions recognise essences, so are not themselves essences [Aristotle] |
| 17039 | The predicates of a thing's nature are necessary to it [Aristotle] |
| 11994 | Aristotelian essences are properties mentioned at the starting point of a science [Aristotle, by Kung] |
| 12381 | What is necessary cannot be otherwise [Aristotle] |
| 1690 | A stone travels upwards by a forced necessity, and downwards by natural necessity [Aristotle] |
| 3016 | Even the gods cannot strive against necessity [Pittacus, by Diog. Laertius] |
| 12072 | For Aristotle knowledge is explanatory, involving understanding, and principles or causes [Aristotle, by Witt] |
| 12073 | 'Episteme' means grasping causes, universal judgments, explanation, and teaching [Aristotle, by Witt] |
| 12378 | The reason why is the key to knowledge [Aristotle] |
| 12364 | We understand a thing when we know its explanation and its necessity [Aristotle] |
| 12370 | Some understanding, of immediate items, is indemonstrable [Aristotle] |
| 12366 | We only understand something when we know its explanation [Aristotle] |
| 1685 | No one has mere belief about something if they think it HAS to be true [Aristotle] |
| 1673 | Knowledge proceeds from principles, so it is hard to know if we know [Aristotle] |
| 12379 | You cannot understand anything through perception [Aristotle] |
| 16725 | Some knowledge is lost if you lose a sense, and there is no way the knowledge can be replaced [Aristotle] |
| 1693 | Animals may have some knowledge if they retain perception, but understanding requires reasons to be given [Aristotle] |
| 23309 | Aristotle's concepts of understanding and explanation mean he is not a pure empiricist [Aristotle, by Frede,M] |
| 9067 | Many memories of the same item form a single experience [Aristotle] |
| 1671 | Sceptics say justification is an infinite regress, or it stops at the unknowable [Aristotle] |
| 1670 | When you understand basics, you can't be persuaded to change your mind [Aristotle] |
| 12147 | The principles of demonstrations are definitions [Aristotle] |
| 12383 | There must be definitions before demonstration is possible [Aristotle] |
| 24068 | Demonstration is more than entailment, as the explanatory order must match the causal order [Aristotle, by Koslicki] |
| 17310 | Aristotle gets asymmetric consequence from demonstration, which reflects real causal priority [Aristotle, by Koslicki] |
| 21359 | Aristotle doesn't actually apply his theory of demonstration to his practical science [Leroi on Aristotle] |
| 12365 | We can know by demonstration, which is a scientific deduction leading to understanding [Aristotle] |
| 1667 | Premises must be true, primitive and immediate, and prior to and explanatory of conclusions [Aristotle] |
| 10918 | Demonstrative understanding rests on necessary features of the thing in itself [Aristotle] |
| 12374 | Demonstrations must be necessary, and that depends on the middle term [Aristotle] |
| 12148 | Demonstrations are syllogisms which give explanations [Aristotle] |
| 1680 | Demonstration is better with fewer presuppositions, and it is quicker if these are familiar [Aristotle] |
| 1674 | All demonstration is concerned with existence, axioms and properties [Aristotle] |
| 1679 | Universal demonstrations are about thought; particular demonstrations lead to perceptions [Aristotle] |
| 1691 | Aim to get definitions of the primitive components, thus establishing the kind, and work towards the attributes [Aristotle] |
| 12371 | A demonstration is a deduction which proceeds from necessities [Aristotle] |
| 1683 | We learn universals from many particulars [Aristotle] |
| 12380 | Universals are valuable because they make the explanations plain [Aristotle] |
| 12367 | What is most universal is furthest away, and the particulars are nearest [Aristotle] |
| 12385 | Are particulars explained more by universals, or by other particulars? [Aristotle] |
| 1689 | Explanation is of the status of a thing, inferences to it, initiation of change, and purpose [Aristotle] |
| 1686 | What we seek and understand are facts, reasons, existence, and identity [Aristotle] |
| 12357 | Explanation and generality are inseparable [Aristotle, by Wedin] |
| 1669 | The foundation or source is stronger than the thing it causes [Aristotle] |
| 1678 | Universals give better explanations, because they are self-explanatory and primitive [Aristotle] |
| 9068 | Perception creates primitive immediate principles by building a series of firm concepts [Aristotle] |
| 9069 | A perception lodging in the soul creates a primitive universal, which becomes generalised [Aristotle] |
| 9070 | We learn primitives and universals by induction from perceptions [Aristotle] |
| 12368 | Negation takes something away from something [Aristotle] |
| 1692 | If you shouldn't argue in metaphors, then you shouldn't try to define them either [Aristotle] |
| 12375 | Whatever holds of a kind intrinsically holds of it necessarily [Aristotle] |
| 1688 | Properties must be proved, but not essence; but existents are not a kind, so existence isn't part of essence [Aristotle] |