61 ideas
1695 | Without extensive examination firm statements are hard, but studying the difficulties is profitable [Aristotle] |
1697 | The contrary of good is bad, but the contrary of bad is either good or another evil [Aristotle] |
1698 | Both sides of contraries need not exist (as health without sickness, white without black) [Aristotle] |
11034 | The differentiae of genera which are different are themselves different in kind [Aristotle] |
18367 | A true existence statement has its truth caused by the existence of the thing [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] |
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] |
11033 | Predications of predicates are predications of their subjects [Aristotle] |
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] |
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] |
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] |
11044 | One is prior to two, because its existence is implied by two [Aristotle] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
11042 | Parts of a line join at a point, so it is continuous [Aristotle] |
11041 | Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
11286 | Primary being must be more than mere indeterminate ultimate subject of predication [Politis on Aristotle] |
1700 | There are six kinds of change: generation, destruction, increase, diminution, alteration, change of place [Aristotle] |
1699 | A thing is prior to another if it implies its existence [Aristotle] |
18366 | Of interdependent things, the prior one causes the other's existence [Aristotle] |
3311 | The categories (substance, quality, quantity, relation, action, passion, place, time) peter out inconsequentially [Benardete,JA on Aristotle] |
11035 | There are ten basic categories for thinking about things [Aristotle] |
13121 | Substance,Quantity,Quality,Relation,Place,Time,Being-in-a-position,Having,Doing,Being affected [Aristotle, by Westerhoff] |
16116 | Aristotle derived categories as answers to basic questions about nature, size, quality, location etc. [Aristotle, by Gill,ML] |
21345 | Aristotle said relations are not substances, so (if they exist) they must be accidents [Aristotle, by Heil] |
16155 | Aristotle promoted the importance of properties and objects (rather than general and particular) [Aristotle, by Frede,M] |
11032 | Some things said 'of' a subject are not 'in' the subject [Aristotle] |
11038 | We call them secondary 'substances' because they reveal the primary substances [Aristotle] |
16739 | Four species of quality: states, capacities, affects, and forms [Aristotle, by Pasnau] |
11037 | Colour must be in an individual body, or it is not embodied [Aristotle] |
16154 | Aristotle gave up his earlier notion of individuals, because it relied on universals [Aristotle, by Frede,M] |
12351 | Genus and species are substances, because only they reveal the primary substance [Aristotle, by Wedin] |
16091 | Is primary substance just an ultimate subject, or some aspect of a complex body? [Aristotle, by Gill,ML] |
11280 | Primary being is 'that which lies under', or 'particular substance' [Aristotle, by Politis] |
1694 | Substances have no opposites, and don't come in degrees (including if the substance is a man) [Aristotle] |
11040 | A single substance can receive contrary properties [Aristotle] |
16140 | Secondary substances do have subjects, so they are not ultimate in the ontology [Aristotle, by Frede,M] |
10965 | In earlier Aristotle the substances were particulars, not kinds [Aristotle, by Lawson-Tancred] |
11036 | A 'primary' substance is in each subject, with species or genera as 'secondary' substances [Aristotle] |
8287 | Earlier Aristotle had objects as primary substances, but later he switched to substantial form [Aristotle, by Lowe] |
12350 | Things are called 'substances' because they are subjects for everything else [Aristotle] |
11039 | A primary substance reveals a 'this', which is an individual unit [Aristotle] |
12361 | Primary substances are ontological in 'Categories', and explanatory in 'Metaphysics' [Aristotle, by Wedin] |
3315 | Aristotle denigrates the category of relation, but for modern absolutists self-relation is basic [Benardete,JA on Aristotle] |
11837 | Some predicates signify qualification of a substance, others the substance itself [Aristotle] |
12349 | Only what can be said of many things is a predicable [Aristotle, by Wedin] |
19216 | Propositions (such as 'that dog is barking') only exist if their items exist [Williamson] |
11043 | It is not possible for fire to be cold or snow black [Aristotle] |
1696 | Change goes from possession to loss (as in baldness), but not the other way round [Aristotle] |