82 ideas
| 18996 | A statement S is 'partly true' if it has some wholly true parts [Yablo] |
| 19006 | An 'enthymeme' is an argument with an indispensable unstated assumption [Yablo] |
| 8859 | The main modal logics disagree over three key formulae [Yablo] |
| 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] |
| 18999 | y is only a proper part of x if there is a z which 'makes up the difference' between them [Yablo] |
| 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] |
| 19001 | 'Pegasus doesn't exist' is false without Pegasus, yet the absence of Pegasus is its truthmaker [Yablo] |
| 9138 | An infinite series of sentences asserting falsehood produces the paradox without self-reference [Yablo, by Sorensen] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 8865 | If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise? [Yablo] |
| 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] |
| 19002 | A nominalist can assert statements about mathematical objects, as being partly true [Yablo] |
| 8863 | We must treat numbers as existing in order to express ourselves about the arrangement of planets [Yablo] |
| 10580 | Mathematics is both necessary and a priori because it really consists of logical truths [Yablo] |
| 8862 | Platonic objects are really created as existential metaphors [Yablo] |
| 10579 | Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 10577 | Concrete objects have few essential properties, but properties of abstractions are mostly essential [Yablo] |
| 10578 | We are thought to know concreta a posteriori, and many abstracta a priori [Yablo] |
| 19489 | For me, fictions are internally true, without a significant internal or external truth-value [Yablo] |
| 19490 | Make-believe can help us to reason about facts and scientific procedures [Yablo] |
| 19491 | 'The clouds are angry' can only mean '...if one were attributing emotions to clouds' [Yablo] |
| 8864 | We quantify over events, worlds, etc. in order to make logical possibilities clearer [Yablo] |
| 19494 | Fictionalism allows that simulated beliefs may be tracking real facts [Yablo] |
| 8858 | Philosophers keep finding unexpected objects, like models, worlds, functions, numbers, events, sets, properties [Yablo] |
| 14381 | A statue is essentially the statue, but its lump is not essentially a statue, so statue isn't lump [Yablo, by Rocca] |
| 18998 | Parthood lacks the restriction of kind which most relations have [Yablo] |
| 19493 | Governing possible worlds theory is the fiction that if something is possible, it happens in a world [Yablo] |
| 19004 | Gettier says you don't know if you are confused about how it is true [Yablo] |
| 19007 | A theory need not be true to be good; it should just be true about its physical aspects [Yablo] |
| 18993 | If sentences point to different evidence, they must have different subject-matter [Yablo] |
| 19003 | Most people say nonblack nonravens do confirm 'all ravens are black', but only a tiny bit [Yablo] |
| 2584 | Lobotomised patients can cease to care about a pain [Block] |
| 6172 | The Inverted Earth example shows that phenomenal properties are not representational [Block, by Rowlands] |
| 2582 | A brain looks no more likely than anything else to cause qualia [Block] |
| 2574 | Behaviour requires knowledge as well as dispositions [Block] |
| 2576 | In functionalism, desires are internal states with causal relations [Block] |
| 2575 | Functionalism is behaviourism, but with mental states as intermediaries [Block] |
| 2583 | You might invert colours, but you can't invert beliefs [Block] |
| 2578 | Could a creature without a brain be in the right functional state for pain? [Block] |
| 2585 | Not just any old functional network will have mental states [Block] |
| 2586 | In functionalism, what are the special inputs and outputs of conscious creatures? [Block] |
| 2579 | Physicalism is prejudiced in favour of our neurology, when other systems might have minds [Block] |
| 2577 | Simple machine-functionalism says mind just is a Turing machine [Block] |
| 2580 | A Turing machine, given a state and input, specifies an output and the next state [Block] |
| 3178 | A fast machine could pass all behavioural tests with a vast lookup table [Block, by Rey] |
| 10805 | A sentence should be recarved to reveal its content or implication relations [Yablo] |
| 18992 | Sentence-meaning is the truth-conditions - plus factors responsible for them [Yablo] |
| 18033 | The meaning of a representation is its role in thought, perception or decisions [Block] |
| 2581 | Intuition may say that a complex sentence is ungrammatical, but linguistics can show that it is not [Block] |
| 18994 | The content of an assertion can be quite different from compositional content [Yablo] |
| 18997 | Truth-conditions as subject-matter has problems of relevance, short cut, and reversal [Yablo] |
| 19005 | Not-A is too strong to just erase an improper assertion, because it actually reverses A [Yablo] |
| 8861 | Hardly a word in the language is devoid of metaphorical potential [Yablo] |