124 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] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
13643 | Aristotelian logic is complete [Shapiro] |
1668 | An axiom is a principle which must be understood if one is to learn anything [Aristotle] |
13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
12376 | Demonstrations by reductio assume excluded middle [Aristotle] |
15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
12373 | Something holds universally when it is proved of an arbitrary and primitive case [Aristotle] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
12363 | Everything is either asserted or denied truly [Aristotle] |
13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
13644 | Semantics for models uses set-theory [Shapiro] |
13670 | Categoricity can't be reached in a first-order language [Shapiro] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
13004 | Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof [Aristotle, by Leibniz] |
13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
13628 | We can live well without completeness in logic [Shapiro] |
13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
13646 | Compactness is derived from soundness and completeness [Shapiro] |
13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
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] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
12369 | A unit is what is quantitatively indivisible [Aristotle] |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
18910 | To seek truth, study the real connections between subjects and attributes [Aristotle] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
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] |
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] |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
12375 | Whatever holds of a kind intrinsically holds of it necessarily [Aristotle] |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
1688 | Properties must be proved, but not essence; but existents are not a kind, so existence isn't part of essence [Aristotle] |