136 ideas
| 12865 | Analytic philosophers may prefer formal systems because natural language is such mess [Simons] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 12815 | Classical mereology doesn't apply well to the objects around us [Simons] |
| 12832 | Complement: the rest of the Universe apart from some individual, written x-bar [Simons] |
| 12834 | Criticisms of mereology: parts? transitivity? sums? identity? four-dimensional? [Simons] |
| 12819 | A 'part' has different meanings for individuals, classes, and masses [Simons] |
| 12822 | Proper or improper part: x < y, 'x is (a) part of y' [Simons] |
| 12824 | Disjoint: two individuals are disjoint iff they do not overlap, written 'x | y' [Simons] |
| 12827 | Difference: the difference of individuals is the remainder of an overlap, written 'x - y' [Simons] |
| 12823 | Overlap: two parts overlap iff they have a part in common, expressed as 'x o y' [Simons] |
| 12825 | Product: the product of two individuals is the sum of all of their overlaps, written 'x · y' [Simons] |
| 12826 | Sum: the sum of individuals is what is overlapped if either of them are, written 'x + y' [Simons] |
| 12828 | General sum: the sum of objects satisfying some predicate, written σx(Fx) [Simons] |
| 12829 | General product: the nucleus of all objects satisfying a predicate, written πx(Fx) [Simons] |
| 12830 | Universe: the mereological sum of all objects whatever, written 'U' [Simons] |
| 12831 | Atom: an individual with no proper parts, written 'At x' [Simons] |
| 12844 | Dissective: stuff is dissective if parts of the stuff are always the stuff [Simons] |
| 12813 | Two standard formalisations of part-whole theory are the Calculus of Individuals, and Mereology [Simons] |
| 12821 | The part-relation is transitive and asymmetric (and thus irreflexive) [Simons] |
| 18847 | Each wheel is part of a car, but the four wheels are not a further part [Simons] |
| 12816 | Classical mereology doesn't handle temporal or modal notions very well [Simons] |
| 12846 | A 'group' is a collection with a condition which constitutes their being united [Simons] |
| 12848 | The same members may form two groups [Simons] |
| 12861 | 'The wolves' are the matter of 'the pack'; the latter is a group, with different identity conditions [Simons] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 12876 | Philosophy is stuck on the Fregean view that an individual is anything with a proper name [Simons] |
| 12845 | Some natural languages don't distinguish between singular and plural [Simons] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 21382 | Things get smaller without end [Anaxagoras] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 481 | Nothing is created or destroyed; there is only mixing and separation [Anaxagoras] |
| 21822 | Anaxagoras's concept of supreme Mind has a simple First and a multiple One [Anaxagoras, by Plotinus] |
| 12838 | Four-dimensional ontology has no change, since that needs an object, and time to pass [Simons] |
| 12842 | There are real relational changes, as well as bogus 'Cambridge changes' [Simons] |
| 12841 | I don't believe in processes [Simons] |
| 12836 | Fans of process ontology cheat, since river-stages refer to 'rivers' [Simons] |
| 12880 | Moments are things like smiles or skids, which are founded on other things [Simons] |
| 12883 | Moving disturbances are are moments which continuously change their basis [Simons] |
| 12881 | A smiling is an event with causes, but the smile is a continuant without causes [Simons] |
| 12882 | A wave is maintained by a process, but it isn't a process [Simons] |
| 12840 | I do not think there is a general identity condition for events [Simons] |
| 12839 | Relativity has an ontology of things and events, not on space-time diagrams [Simons] |
| 12879 | Independent objects can exist apart, and maybe even entirely alone [Simons] |
| 17995 | Basic is the potentially perceptible, then comes the contrary qualities, and finally the 'elements' [Anaxagoras] |
| 12847 | Mass nouns admit 'much' and 'a little', and resist 'many' and 'few'. [Simons] |
| 12863 | Mass terms (unlike plurals) are used with indifference to whether they can exist in units [Simons] |
| 12862 | Gold is not its atoms, because the atoms must be all gold, but gold contains neutrons [Simons] |
| 12858 | Mixtures disappear if nearly all of the mixture is one ingredient [Simons] |
| 12859 | A mixture can have different qualities from its ingredients. [Simons] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 12850 | To individuate something we must pick it out, but also know its limits of variation [Simons] |
| 12860 | Sortal nouns for continuants tell you their continuance- and cessation-conditions [Simons] |
| 12886 | A whole requires some unique relation which binds together all of the parts [Simons] |
| 12857 | Tibbles isn't Tib-plus-tail, because Tibbles can survive its loss, but the sum can't [Simons] |
| 12835 | Does Tibbles remain the same cat when it loses its tail? [Simons] |
| 12820 | Without extensional mereology two objects can occupy the same position [Simons] |
| 12866 | Composition is asymmetric and transitive [Simons] |
| 12867 | A hand constitutes a fist (when clenched), but a fist is not composed of an augmented hand [Simons] |
| 12864 | We say 'b is part of a', 'b is a part of a', 'b are a part of a', or 'b are parts of a'. [Simons] |
| 12814 | Classical mereology says there are 'sums', for whose existence there is no other evidence [Simons] |
| 12817 | 'Mereological extensionality' says objects with the same parts are identical [Simons] |
| 12833 | If there are c atoms, this gives 2^c - 1 individuals, so there can't be just 2 or 12 individuals [Simons] |
| 12849 | Sums are more plausible for pluralities and masses than they are for individuals [Simons] |
| 12877 | Sums of things in different categories are found within philosophy. [Simons] |
| 12888 | The wholeness of a melody seems conventional, but of an explosion it seems natural [Simons] |
| 12871 | Objects have their essential properties because of the kind of objects they are [Simons] |
| 12870 | We must distinguish the de dicto 'must' of propositions from the de re 'must' of essence [Simons] |
| 12873 | Original parts are the best candidates for being essential to artefacts [Simons] |
| 12874 | An essential part of an essential part is an essential part of the whole [Simons] |
| 12837 | Four dimensional-objects are stranger than most people think [Simons] |
| 12856 | Intermittent objects would be respectable if they occurred in nature, as well as in artefacts [Simons] |
| 12885 | Objects like chess games, with gaps in them, are thereby less unified [Simons] |
| 12854 | An entrepreneur and a museum curator would each be happy with their ship at the end [Simons] |
| 12855 | The 'best candidate' theories mistakenly assume there is one answer to 'Which is the real ship?' [Simons] |
| 12872 | The zygote is an essential initial part, for a sexually reproduced organism [Simons] |
| 12889 | The limits of change for an individual depend on the kind of individual [Simons] |
| 20802 | Snow is not white, and doesn't even appear white, because it is made of black water [Anaxagoras, by Cicero] |
| 13257 | The senses are too feeble to determine the truth [Anaxagoras] |
| 22761 | We reveal unreliability in the senses when we cannot discriminate a slow change of colour [Anaxagoras, by Sext.Empiricus] |
| 13256 | Nous is unlimited, self-ruling and pure; it is the finest thing, with great discernment and strength [Anaxagoras] |
| 13784 | Mind is self-ruling, pure, ordering and ubiquitous [Anaxagoras, by Plato] |
| 5118 | Anaxagoras says mind remains pure, and so is not affected by what it changes [Anaxagoras, by Aristotle] |
| 12843 | With activities if you are doing it you've done it, with performances you must finish to have done it [Simons] |
| 12875 | One false note doesn't make it a performance of a different work [Simons] |
| 18231 | Anaxagoras said a person would choose to be born to contemplate the ordered heavens [Anaxagoras] |
| 631 | For Anaxagoras the Good Mind has no opposite, and causes all movement, for a higher reason [Anaxagoras, by Aristotle] |
| 22727 | Mind creates the world from a mixture of pure substances [Anaxagoras, by ] |
| 550 | Anaxagoras said that the number of principles was infinite [Anaxagoras, by Aristotle] |
| 21383 | The ultimate constituents of reality are the homoeomeries [Anaxagoras, by Vlastos] |
| 13208 | Anaxagoreans regard the homoeomeries as elements, which compose earth, air, fire and water [Anaxagoras, by Aristotle] |
| 367 | Anaxagoras says mind produces order and causes everything [Anaxagoras, by Plato] |
| 21381 | Germs contain microscopic organs, which become visible as they grow [Anaxagoras] |
| 22726 | When things were unified, Mind set them in order [Anaxagoras] |
| 2629 | Anaxagoras was the first to say that the universe is directed by an intelligence [Anaxagoras, by Cicero] |
| 480 | Past, present and future, and the movements of the heavens, were arranged by Mind [Anaxagoras] |
| 5956 | Anaxagoras was charged with impiety for calling the sun a lump of stone [Anaxagoras, by Plutarch] |
| 7488 | Anaxagoras was the first recorded atheist [Anaxagoras, by Watson] |