111 ideas
| 3807 | Reason is and ought to be the slave of the passions [Hume] |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 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] |
| 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] |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| 13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
| 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] |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| 13670 | Categoricity can't be reached in a first-order language [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] |
| 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] |
| 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] |
| 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] |
| 8649 | Two numbers are equal if all of their units correspond to one another [Hume] |
| 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] |
| 21291 | There is no medium state between existence and non-existence [Hume] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 11942 | Power is the possibility of action, as discovered by experience [Hume] |
| 11949 | There may well be powers in things, with which we are quite unacquainted [Hume] |
| 11950 | We have no idea of powers, because we have no impressions of them [Hume] |
| 11941 | The distinction between a power and its exercise is entirely frivolous [Hume] |
| 11098 | Momentary impressions are wrongly identified with one another on the basis of resemblance [Hume, by Quine] |
| 7954 | If we see a resemblance among objects, we apply the same name to them, despite their differences [Hume] |
| 21293 | Individuation is only seeing that a thing is stable and continuous over time [Hume] |
| 12048 | The only meaning we have for substance is a collection of qualities [Hume] |
| 13424 | Aristotelians propose accidents supported by substance, but they don't understand either of them [Hume] |
| 21299 | Changing a part can change the whole, not absolutely, but by its proportion of the whole [Hume] |
| 21300 | A change more obviously destroys an identity if it is quick and observed [Hume] |
| 1321 | If identity survives change or interruption, then resemblance, contiguity or causation must unite the parts of it [Hume] |
| 1330 | If a republic can retain identity through many changes, so can an individual [Hume] |
| 21302 | If a ruined church is rebuilt, its relation to its parish makes it the same church [Hume] |
| 21303 | We accept the identity of a river through change, because it is the river's nature [Hume] |
| 21301 | The purpose of the ship makes it the same one through all variations [Hume] |
| 1207 | Both number and unity are incompatible with the relation of identity [Hume] |
| 21290 | Multiple objects cannot convey identity, because we see them as different [Hume] |
| 21289 | 'An object is the same with itself' is meaningless; it expresses unity, not identity [Hume] |
| 21292 | Saying an object is the same with itself is only meaningful over a period of time [Hume] |
| 9428 | Nothing we clearly imagine is absolutely impossible [Hume] |
| 4766 | Necessity only exists in the mind, and not in objects [Hume] |
| 6526 | Hume says objects are not a construction, but an imaginative leap [Hume, by Robinson,H] |
| 6489 | Associationism results from having to explain intentionality just with sense-data [Robinson,H on Hume] |
| 6182 | Even Hume didn't include mathematics in his empiricism [Hume, by Kant] |
| 12417 | Mathematicians only accept their own proofs when everyone confims them [Hume] |
| 5548 | Hume became a total sceptic, because he believed that reason was a deception [Hume, by Kant] |
| 7446 | The idea of inductive evidence, around 1660, made Hume's problem possible [Hume, by Hacking] |
| 21806 | Memory, senses and understanding are all founded on the imagination [Hume] |
| 3819 | Hume's 'bundle' won't distinguish one mind with ten experiences from ten minds [Searle on Hume] |
| 1317 | A person is just a fast-moving bundle of perceptions [Hume] |
| 1331 | The parts of a person are always linked together by causation [Hume] |
| 1388 | Hume gives us an interesting sketchy causal theory of personal identity [Perry on Hume] |
| 21297 | A person is simply a bundle of continually fluctuating perceptions [Hume] |
| 1316 | Introspection always discovers perceptions, and never a Self without perceptions [Hume] |
| 1333 | Memory only reveals personal identity, by showing cause and effect [Hume] |
| 1332 | We use memory to infer personal actions we have since forgotten [Hume] |
| 21305 | Memory not only reveals identity, but creates it, by producing resemblances [Hume] |
| 21307 | Who thinks that because you have forgotten an incident you are no longer that person? [Hume] |
| 21306 | Causation unites our perceptions, by producing, destroying and modifying each other [Hume] |
| 21294 | A continuous lifelong self must be justified by a single sustained impression, which we don't have [Hume] |
| 21295 | When I introspect I can only observe my perceptions, and never a self which has them [Hume] |
| 21298 | We pretend our perceptions are continuous, and imagine a self to fill the gaps [Hume] |
| 21304 | Identity in the mind is a fiction, like that fiction that plants and animals stay the same [Hume] |
| 20030 | If one event causes another, the two events must be wholly distinct [Hume, by Wilson/Schpall] |
| 6692 | For Hume, practical reason has little force, because we can always modify our desires [Hume, by Graham] |
| 8257 | Reason alone can never be a motive to any action of the will [Hume] |
| 22374 | You can only hold people responsible for actions which arise out of their character [Hume] |
| 22382 | We cannot discover vice by studying a wilful murder; that only arises from our own feelings [Hume] |
| 4008 | Modern science has destroyed the Platonic synthesis of scientific explanation and morality [Hume, by Taylor,C] |
| 8067 | The problem of getting to 'ought' from 'is' would also apply in getting to 'owes' or 'needs' [Anscombe on Hume] |
| 4578 | You can't move from 'is' to 'ought' without giving some explanation or reason for the deduction [Hume] |
| 3650 | Total selfishness is not irrational [Hume] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 14301 | We have no good concept of solidity or matter, because accounts of them are all circular [Hume] |
| 8382 | For Hume a constant conjunction is both necessary and sufficient for causation [Hume, by Crane] |
| 19274 | Hume seems to presuppose necessary connections between mental events [Kripke on Hume] |
| 21296 | If all of my perceptions were removed by death, nothing more is needed for total annihilation [Hume] |