129 ideas
16281 | Honesty requires philosophical theories we can commit to with our ordinary commonsense [Lewis] |
16288 | Analysis reduces primitives and makes understanding explicit (without adding new knowledge) [Lewis] |
9651 | Verisimilitude might be explained as being close to the possible world where the truth is exact [Lewis] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
15731 | Quantification sometimes commits to 'sets', but sometimes just to pluralities (or 'classes') [Lewis] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
10470 | There are only two kinds: sets, and possibilia (actual and possible particulars) [Lewis, by Oliver] |
9650 | Supervenience concerns whether things could differ, so it is a modal notion [Lewis] |
8909 | Abstractions may well be verbal fictions, in which we ignore some features of an object [Lewis] |
9057 | Vagueness is semantic indecision: we haven't settled quite what our words are meant to express [Lewis] |
9671 | Whether or not France is hexagonal depends on your standards of precision [Lewis] |
15751 | Surely 'slept in by Washington' is a property of some bed? [Lewis] |
15735 | Properties don't have degree; they are determinate, and things have varying relations to them [Lewis] |
9656 | The 'abundant' properties are just any bizarre property you fancy [Lewis] |
15737 | To be a 'property' is to suit a theoretical role [Lewis] |
15742 | A disjunctive property can be unnatural, but intrinsic if its disjuncts are intrinsic [Lewis] |
15397 | If a global intrinsic never varies between possible duplicates, all necessary properties are intrinsic [Cameron on Lewis] |
15398 | Global intrinsic may make necessarily coextensive properties both intrinsic or both extrinsic [Cameron on Lewis] |
15741 | All of the natural properties are included among the intrinsic properties [Lewis] |
15752 | We might try defining the natural properties by a short list of them [Lewis] |
14996 | Natural properties give similarity, joint carving, intrinsicness, specificity, homogeneity... [Lewis] |
15744 | We can't define natural properties by resemblance, if they are used to explain resemblance [Lewis] |
15743 | Defining natural properties by means of laws of nature is potentially circular [Lewis] |
15740 | I don't take 'natural' properties to be fixed by the nature of one possible world [Lewis] |
16262 | Sparse properties rest either on universals, or on tropes, or on primitive naturalness [Lewis, by Maudlin] |
15739 | There is the property of belonging to a set, so abundant properties are as numerous as the sets [Lewis] |
10723 | A property is the set of its actual and possible instances [Lewis, by Oliver] |
15399 | The property of being F is identical with the set of objects, in all possible worlds, which are F [Lewis, by Cameron] |
15732 | Properties don't seem to be sets, because different properties can have the same set [Lewis] |
15733 | Accidentally coextensive properties come apart when we include their possible instances [Lewis] |
15734 | If a property is relative, such as being a father or son, then set membership seems relative too [Lewis] |
9655 | Trilateral and triangular seem to be coextensive sets in all possible worlds [Lewis] |
16290 | I believe in properties, which are sets of possible individuals [Lewis] |
9653 | It would be easiest to take a property as the set of its instances [Lewis] |
9657 | You must accept primitive similarity to like tropes, but tropes give a good account of it [Lewis] |
15750 | Tropes need a similarity primitive, so they cannot be used to explain similarity [Lewis] |
15749 | Trope theory (unlike universals) needs a primitive notion of being duplicates [Lewis] |
15748 | Trope theory needs a primitive notion for what unites some tropes [Lewis] |
15745 | Universals recur, are multiply located, wholly present, make things overlap, and are held in common [Lewis] |
15746 | If particles were just made of universals, similar particles would be the same particle [Lewis] |
15747 | Universals aren't parts of things, because that relationship is transitive, and universals need not be [Lewis] |
9667 | Mereological composition is unrestricted: any class of things has a mereological sum [Lewis] |
13268 | There are no restrictions on composition, because they would be vague, and composition can't be vague [Lewis, by Sider] |
13793 | An essential property is one possessed by all counterparts [Lewis, by Elder] |
9663 | A thing 'perdures' if it has separate temporal parts, and 'endures' if it is wholly present at different times [Lewis] |
14737 | Properties cannot be relations to times, if there are temporary properties which are intrinsic [Lewis, by Sider] |
9664 | Endurance is the wrong account, because things change intrinsic properties like shape [Lewis] |
9665 | There are three responses to the problem that intrinsic shapes do not endure [Lewis] |
19280 | I can ask questions which create a context in which origin ceases to be essential [Lewis] |
15968 | Identity is simple - absolutely everything is self-identical, and nothing is identical to another thing [Lewis] |
15969 | Two things can never be identical, so there is no problem [Lewis] |
9660 | The impossible can be imagined as long as it is a bit vague [Lewis] |
9669 | There are no free-floating possibilia; they have mates in a world, giving them extrinsic properties [Lewis] |
16133 | Possible worlds can contain contradictions if such worlds are seen as fictions [Lewis] |
16132 | On mountains or in worlds, reporting contradictions is contradictory, so no such truths can be reported [Lewis] |
16283 | For me, all worlds are equal, with each being actual relative to itself [Lewis] |
12255 | For Lewis there is no real possibility, since all possibilities are actual [Oderberg on Lewis] |
9219 | Lewis posits possible worlds just as Quine says that physics needs numbers and sets [Lewis, by Sider] |
15022 | If possible worlds really exist, then they are part of actuality [Sider on Lewis] |
10469 | A world is a maximal mereological sum of spatiotemporally interrelated things [Lewis] |
16441 | Lewis rejects actualism because he identifies properties with sets [Lewis, by Stalnaker] |
16282 | Ersatzers say we have one world, and abstract representations of how it might have been [Lewis] |
16284 | Ersatz worlds represent either through language, or by models, or magically [Lewis] |
16286 | Linguistic possible worlds need a complete supply of unique names for each thing [Lewis] |
16287 | Maximal consistency for a world seems a modal distinction, concerning what could be true together [Lewis] |
9662 | Linguistic possible worlds have problems of inconsistencies, no indiscernibles, and vocabulary [Lewis] |
7690 | If sets exist, then defining worlds as proposition sets implies an odd distinction between existing and actual [Jacquette on Lewis] |
14404 | The counterpart relation is sortal-relative, so objects need not be a certain way [Lewis, by Merricks] |
5440 | A counterpart in a possible world is sufficiently similar, and more similar than anything else [Lewis, by Mautner] |
5441 | Why should statements about what my 'counterpart' could have done interest me? [Mautner on Lewis] |
16291 | In counterpart theory 'Humphrey' doesn't name one being, but a mereological sum of many beings [Lewis] |
11903 | Extreme haecceitists could say I might have been a poached egg, but it is too remote to consider [Lewis, by Mackie,P] |
15129 | Haecceitism implies de re differences but qualitative identity [Lewis] |
9670 | Extreme haecceitism says you might possibly be a poached egg [Lewis] |
16279 | General causal theories of knowledge are refuted by mathematics [Lewis] |
9661 | Induction is just reasonable methods of inferring the unobserved from the observed [Lewis] |
9652 | To just expect unexamined emeralds to be grue would be totally unreasonable [Lewis] |
9658 | An explanation tells us how an event was caused [Lewis] |
16280 | Often explanaton seeks fundamental laws, rather than causal histories [Lewis] |
16274 | If the well-ordering of a pack of cards was by shuffling, the explanation would make it more surprising [Lewis] |
8901 | Abstraction is usually explained either by example, or conflation, or abstraction, or negatively [Lewis] |
8904 | The Way of Abstraction says an incomplete description of a concrete entity is the complete abstraction [Lewis] |
8938 | The Way of Example compares donkeys and numbers, but what is the difference, and what are numbers? [Lewis] |
8903 | Abstracta can be causal: sets can be causes or effects; there can be universal effects; events may be sets [Lewis] |
8902 | If abstractions are non-spatial, then both sets and universals seem to have locations [Lewis] |
8905 | If universals or tropes are parts of things, then abstraction picks out those parts [Lewis] |
8906 | If we can abstract the extrinsic relations and features of objects, abstraction isn't universals or tropes [Lewis] |
8907 | The abstract direction of a line is the equivalence class of it and all lines parallel to it [Lewis] |
8908 | For most sets, the concept of equivalence is too artificial to explain abstraction [Lewis] |
16289 | We can't account for an abstraction as 'from' something if the something doesn't exist [Lewis] |
16278 | A particular functional role is what gives content to a thought [Lewis] |
9654 | A proposition is a set of entire possible worlds which instantiate a particular property [Lewis] |
15736 | A proposition is the property of being a possible world where it holds true [Lewis] |
15738 | Propositions can't have syntactic structure if they are just sets of worlds [Lewis] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
9659 | Causation is when at the closest world without the cause, there is no effect either [Lewis] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
9666 | It is quite implausible that the future is unreal, as that would terminate everything [Lewis] |