146 ideas
| 9593 | Progress in philosophy is incremental, not an immature seeking after drama [Williamson] |
| Full Idea: The incremental progress which I envisage for philosophy lacks the drama after which some philosophers still hanker, and that hankering is itself a symptom of the intellectual immaturity that helps hold philosophy back. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], Intro) | |
| A reaction: This could stand as a motto for the whole current profession of analytical philosophy. It means that if anyone attempts to be dramatic they can make their own way out. They'll find Kripke out there, smoking behind the dustbins. |
| 9184 | We can't presume that all interesting concepts can be analysed [Williamson] |
| Full Idea: We have no prior reason to suppose that philosophically significant concepts have interesting analyses into necessary and sufficient conditions. | |
| From: Timothy Williamson (Review of Bob Hale's 'Abstract Objects' [1988]) | |
| A reaction: We might think that they are either analysable or primitive, and that failure of analysis invites us to take a concept as primitive. But maybe God can analyse it and we can't. |
| 6859 | Analytic philosophy has much higher standards of thinking than continental philosophy [Williamson] |
| Full Idea: Certain advances in philosophical standards have been made within analytic philosophy, and there would be a serious loss of integrity involved in abandoning them in the way required to participate in current continental philosophy. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.151) | |
| A reaction: The reply might be to concede the point, but say that the precision and rigour achieved are precisely what debar analytical philosophy from thinking about the really interesting problems. One might as well switch to maths and have done with it. |
| 21616 | Truth and falsity apply to suppositions as well as to assertions [Williamson] |
| Full Idea: The notion of truth and falsity apply to suppositions as well as to assertions. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This may not be obvious to those who emphasise pragmatics and ordinary language, but it is self-evident to anyone who emphasises logic. |
| 21623 | True and false are not symmetrical; false is more complex, involving negation [Williamson] |
| Full Idea: The concepts of truth and falsity are not symmetrical. The asymmetry is visible in the fundamental principles governing them, for F is essentially more complex than T, by its use of negation. | |
| From: Timothy Williamson (Vagueness [1994], 7.5) | |
| A reaction: If T and F are primitives, controlled by axioms, then they might be symmetrical in nature, but asymmetrical in use. However, if forced to choose just one primitive, I presume it would be T. |
| 15134 | The truthmaker principle requires some specific named thing to make the difference [Williamson] |
| Full Idea: The truthmaker principle seems compelling, because if a proposition is true, something must be different from a world in which it is false. The principle makes this specific, by treating 'something' as a quantifier binding a variable in name position. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: See Williamson for an examination of the logical implications of this. The point is that the principle seems to require some very specific 'thing', which may be asking too much. For a start, it might be the absence of a thing. |
| 15141 | Truthmaker is incompatible with modal semantics of varying domains [Williamson] |
| Full Idea: Friends of the truthmaker principle should reject the Kripke semantics of varying domains. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: See other ideas from this paper to get a sense of what that is about. |
| 15140 | The converse Barcan formula will not allow contingent truths to have truthmakers [Williamson] |
| Full Idea: The converse Barcan formula does not allow any contingent truths at all to have a truthmaker. Once cannot combine the converse Barcan formula with any truthmaker principle worth having. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: One might reply, so much the worse for the converse Barcan formula, but Williamson doesn't think that. |
| 9594 | Correspondence to the facts is a bad account of analytic truth [Williamson] |
| Full Idea: Even if talk of truth as correspondence to the facts is metaphorical, it is a bad metaphor for analytic truth in a way that it is not for synthetic truth. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], 3.1) | |
| A reaction: A very simple and rather powerful point. Maybe the word 'truth' should be withheld from such cases. You might say that accepted analytic truths are 'conventional'. If that is wrong, then they correspond to natural facts at a high level of abstraction. |
| 15131 | If metaphysical possibility is not a contingent matter, then S5 seems to suit it best [Williamson] |
| Full Idea: In S5, necessity and possibility are not themselves contingent matters. This is plausible for metaphysical modality, since metaphysical possibility, unlike practical possibility, does not depend on the contingencies of one's situation. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §1) | |
| A reaction: This is the clearest statement I have found of why S5 might be preferable for metaphysics. See Nathan Salmon for the rival view. Williamson's point sounds pretty persuasive to me. |
| 14626 | In S5 matters of possibility and necessity are non-contingent [Williamson] |
| Full Idea: In system S5 matters of possibility and necessity are always non-contingent. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 3) | |
| A reaction: This will be because if something is possible in one world (because it can be seen to be true in some possible world) it will be possible for all worlds (since they can all see that world in S5). |
| 15135 | If the domain of propositional quantification is constant, the Barcan formulas hold [Williamson] |
| Full Idea: If the domain of propositional quantification is constant across worlds, the Barcan formula and its converse hold. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: So the issue is whether we should take metaphysics to be dealing with a constant or varying domains. Williamson seems to favour the former, but my instincts incline towards the latter. |
| 15130 | If a property is possible, there is something which can have it [Williamson] |
| Full Idea: Barcan's axiom says if there can be something that has a certain property, then there is something that can have that property. It and its converse are not obviously correct or incorrect. They claim that it is non-contingent what individuals there are. | |
| From: Timothy Williamson (Laudatio: Prof Ruth Barcan Marcus [2011], p.1) | |
| A reaction: Williamson defends the two Barcan formulas, but the more I understand them the less plausible they sound to me. |
| 15139 | Converse Barcan: could something fail to meet a condition, if everything meets that condition? [Williamson] |
| Full Idea: The converse Barcan is at least plausible, since its denial says there is something that could fail to meet a condition when everything met that condition; but how could everything meet that condition if that thing did not? | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: Presumably the response involves a discussion of domains, since everything in a given domain might meet a condition, but something in a different domain might fail it. |
| 21602 | Many-valued logics don't solve vagueness; its presence at the meta-level is ignored [Williamson] |
| Full Idea: It is an illusion that many-valued logic constitutes a well-motivated and rigorously worked out theory of vagueness. ...[top] There has been a reluctance to acknowledge higher-order vagueness, or to abandon classical logic in the meta-language. | |
| From: Timothy Williamson (Vagueness [1994], 4.12) |
| 6862 | Fuzzy logic uses a continuum of truth, but it implies contradictions [Williamson] |
| Full Idea: Fuzzy logic is based on a continuum of degrees of truth, but it is committed to the idea that it is half-true that one identical twin is tall and the other twin is not, even though they are the same height. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.154) | |
| A reaction: Maybe to be shocked by a contradiction is missing the point of fuzzy logic? Half full is the same as half empty. The logic does not say the twins are different, because it is half-true that they are both tall, and half-true that they both aren't. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| Full Idea: A set is any collection into a whole M of definite, distinct objects m ... of our intuition or thought. | |
| From: George Cantor (The Theory of Transfinite Numbers [1897], p.85), quoted by James Robert Brown - Philosophy of Mathematics Ch.2 | |
| A reaction: This is the original conception of a set, which hit trouble with Russell's Paradox. Cantor's original definition immediately invites thoughts about the status of vague objects. |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
| Full Idea: Cantor gives informal versions of the axioms of ZF as ways of getting from one set to another. | |
| From: report of George Cantor (Later Letters to Dedekind [1899]) by John Lake - Approaches to Set Theory 1.6 | |
| A reaction: Lake suggests that it should therefore be called CZF. |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 6858 | Formal logic struck me as exactly the language I wanted to think in [Williamson] |
| Full Idea: As soon as I started learning formal logic, that struck me as exactly the language that I wanted to think in. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001]) | |
| A reaction: It takes all sorts… It is interesting that formal logic might be seen as having the capacity to live up to such an aspiration. I don't think the dream of an ideal formal language is dead, though it will never encompass all of reality. Poetic truth. |
| 21611 | Formal semantics defines validity as truth preserved in every model [Williamson] |
| Full Idea: An aim of formal semantics is to define in mathematical terms a set of models such that an argument is valid if and only if it preserves truth in every model in the set, for that will provide us with a precise standard of validity. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) |
| 21606 | 'Bivalence' is the meta-linguistic principle that 'A' in the object language is true or false [Williamson] |
| Full Idea: The meta-logical law of excluded middle is the meta-linguistic principle that any statement 'A' in the object language is either truth or false; it is now known as the principle of 'bivalence'. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: [He cites Henryk Mehlberg 1958] See also Idea 21605. Without this way of distinguishing bivalence from excluded middle, most discussions of them strikes me as shockingly lacking in clarity. Personally I would cut the normativity from this one. |
| 21605 | Excluded Middle is 'A or not A' in the object language [Williamson] |
| Full Idea: The logical law of excluded middle (now the standard one) is the schema 'A or not A' in the object-language. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: [He cites Henryk Mehlberg 1958] See Idea 21606. The only sensible way to keep Excluded Middle and Bivalence distinct. I would say: (meta-) only T and F are available, and (object) each proposition must have one of them. Are they both normative? |
| 18492 | Not all quantification is either objectual or substitutional [Williamson] |
| Full Idea: We should not assume that all quantification is either objectual or substitutional. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], p.262) | |
| A reaction: [see Prior 1971:31-4] He talks of quantifying into sentence position. |
| 15136 | Substitutional quantification is metaphysical neutral, and equivalent to a disjunction of instances [Williamson] |
| Full Idea: If quantification into sentence position is substitutional, then it is metaphysically neutral. A substitutionally interpreted 'existential' quantification is semantically equivalent to the disjunction (possibly infinite) of its substitution instances. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: Is it not committed to the disjunction, just as the objectual reading commits to objects? Something must make the disjunction true. Or is it too verbal to be about reality? |
| 15138 | Not all quantification is objectual or substitutional [Williamson] |
| Full Idea: We should not assume that all quantification is objectual or substitutional. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) |
| 21612 | Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B' [Williamson] |
| Full Idea: Argument by Cases (or or-elimination) is the standard way of using disjunctive premises. If one can argue from A and some premises to C, and from B and some premises to C, one can argue from 'A or B' and the combined premises to C. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 21599 | A sorites stops when it collides with an opposite sorites [Williamson] |
| Full Idea: A sorites paradox is stopped when it collides with a sorites paradox going in the opposite direction. That account will not strike a logician as solving the sorites paradox. | |
| From: Timothy Williamson (Vagueness [1994], 3.3) |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| Full Idea: Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections. | |
| From: report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I | |
| A reaction: I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| Full Idea: The author entirely overlooks the fact that the 'extension of a concept' in general may be quantitatively completely indeterminate. Only in certain cases is the 'extension of a concept' quantitatively determinate. | |
| From: George Cantor (Review of Frege's 'Grundlagen' [1885], 1932:440), quoted by William W. Tait - Frege versus Cantor and Dedekind | |
| A reaction: Cantor presumably has in mind various infinite sets. Tait is drawing our attention to the fact that this objection long precedes Russell's paradox, which made the objection more formal (a language Frege could understand!). |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 9183 | Platonism claims that some true assertions have singular terms denoting abstractions, so abstractions exist [Williamson] |
| Full Idea: The Fregean argument for platonism is that some true assertions contain singular terms which denote abstract objects if they denote anything; since the assertions are true, the singular terms denote. | |
| From: Timothy Williamson (Review of Bob Hale's 'Abstract Objects' [1988]) | |
| A reaction: I am perplexed that anyone would rest their view of reality on such an argument. The obvious comparison would be with true remarks about blatantly fictional characters, or blatantly invented concepts such as 'checkmate'. |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 9601 | The realist/anti-realist debate is notoriously obscure and fruitless [Williamson] |
| Full Idea: The debate between realism and anti-realism has become notorious in the rest of philosophy for its obscurity, convolution, and lack of progress. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], After) | |
| A reaction: I find this reassuring, because fairly early on I decided that this problem was not of great interest, and quietly tiptoed away. I take the central issue to be whether nature has 'joints', to which the answer appears to be 'yes'. End of story. |
| 15137 | If 'fact' is a noun, can we name the fact that dogs bark 'Mary'? [Williamson] |
| Full Idea: If one uses 'fact' as a noun, the question arises why one cannot name the fact that dogs bark 'Mary'. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2 n10) | |
| A reaction: What an intriguing thought! Must all nouns pass this test? 'The courage of the regiment was called Alfred'? |
| 21589 | When bivalence is rejected because of vagueness, we lose classical logic [Williamson] |
| Full Idea: The principle of bivalence (that every statement is either true or false) has been rejected for vague languages. To reject bivalence is to reject classical logic or semantics. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: His example is specifying a moment when Rembrandt became 'old'. This is the number one reason why the problem of vagueness is seen as important. Is the rejection of classical logic a loss of our grip on the world? |
| 21596 | Vagueness undermines the stable references needed by logic [Williamson] |
| Full Idea: Logic requires expressions to have the same referents wherever they occur; vague natural languages violate this contraint. | |
| From: Timothy Williamson (Vagueness [1994], 2.2) | |
| A reaction: This doesn't mean that logic has to win. Maybe it is important for philosophers who see logic as central to be always aware of vagueness as the gulf between their precision and the mess of reality. Precision is worth trying for, though. |
| 21601 | A vague term can refer to very precise elements [Williamson] |
| Full Idea: Both 30° and 60° are clearly acute angles. 'Acute' is precise in all relevant respects. Nevertheless, 30° is acuter than 60°. | |
| From: Timothy Williamson (Vagueness [1994], 4.11) | |
| A reaction: A very nice example of something which is vague, despite involving precise ingredients. But then 'bald' is vague, while 'this is a hair on his head' is fairly precise. |
| 21629 | Equally fuzzy objects can be identical, so fuzziness doesn't entail vagueness [Williamson] |
| Full Idea: Fuzzy boundaries do not in any way require vague identity. Objects are identical only if their boundaries have exactly the same fuzziness. | |
| From: Timothy Williamson (Vagueness [1994], 9.2) | |
| A reaction: This all rests on the Fregean idea that determinate existence requires the ability to participate in an identity statement. |
| 9599 | There cannot be vague objects, so there may be no such thing as a mountain [Williamson] |
| Full Idea: It is sometimes argued that if there is such a thing as a mountain it would be a vague object, but it is logically impossible for an object to be vague, so there is no such thing as a mountain. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], 7.2) | |
| A reaction: I don't take this to be a daft view. No one is denying the existence of the solid rock that is involved, but allowing such a vague object may be a slippery slope to the acceptance of almost anything as an 'object'. |
| 21591 | Vagueness is epistemic. Statements are true or false, but we often don't know which [Williamson] |
| Full Idea: My thesis is that vagueness is an epistemic phenomenon. In cases of unclarity, statements remain true or false, but speakers of the language have no way of knowing which. Higher-order vagueness consists in ignorance about ignorance. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: He has plumped for the intuitively least plausible theory. It means that a hair dropping out of someone's head triggers a situation where they are 'bald', but none of us know when that was. And Rembrandt became 'old' in an instant. |
| 21619 | If a heap has a real boundary, omniscient speakers would agree where it is [Williamson] |
| Full Idea: If, in judging a heap as grains are removed, omniscient speakers all stop at the same point, it must does mark some sort of previously hidden boundary. ...If there is no hidden boundary, then different omniscient speakers would stop at different points. | |
| From: Timothy Williamson (Vagueness [1994], 7.3) | |
| A reaction: A very nice thought experiment, which obviously won't settle anything, but brings out nicely the view the vagueness is a sort of ignorance. God is never vague in the application of terms (though God might withhold the application if there is no boundary). |
| 21620 | The epistemic view says that the essence of vagueness is ignorance [Williamson] |
| Full Idea: The epistemic view is that ignorance is the real essence of the phenomenon ostensively identified as vagueness. ...[203] According to the epistemic view, I am either thin or not thin, ...and we have no idea how to find out out which. | |
| From: Timothy Williamson (Vagueness [1994], 7.4) | |
| A reaction: Presumably this implies that there is often a real border (of which we may be ignorant), but it doesn't seem to rule out cases where there just is no border. Where does the east Atlantic meet the west Atlantic? |
| 21622 | If there is a true borderline of which we are ignorant, this drives a wedge between meaning and use [Williamson] |
| Full Idea: A common complaint against the epistemic view is that to postulate a matter of fact in borderline cases is to suppose, incoherently, that the meanings of our words draw a line where our use of them does not. | |
| From: Timothy Williamson (Vagueness [1994], 7.5) | |
| A reaction: This doesn't necessarily seem to require the view that the meaning of words is their usage. Just that if there is one consensus on usage, it seems unlikely that there is a different underlying reality about the true meaning. Externalist meanings? |
| 9120 | Vagueness in a concept is its indiscriminability from other possible concepts [Williamson] |
| Full Idea: Vagueness in a concept is its indiscriminability from other possible concepts; this can be reconciled with our knowledge of vague terms. | |
| From: Timothy Williamson (Vagueness [1994], 8.1) | |
| A reaction: Sorensen objects that this makes vagueness too relative to members of a speech community. He prefers 'absolute borderline cases'. If you like the epistemic view, then Williamson seems more plausible. My 'vague' might differ from yours. |
| 6863 | Close to conceptual boundaries judgement is too unreliable to give knowledge [Williamson] |
| Full Idea: If one is very close to a conceptual boundary, then one's judgement will be too unreliable to constitute knowledge, and therefore one will be ignorant. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.156) | |
| A reaction: This is the epistemological rather than ontological interpretation of vagueness. It sounds very persuasive, but I am reluctant to accept that reality is full of very precise boundaries which we cannot quite discriminate. |
| 21625 | The vagueness of 'heap' can remain even when the context is fixed [Williamson] |
| Full Idea: Vagueness remains even when the context is fixed. In principle, a vague word might exhibit no context dependence whatsoever. ...For example, a dispute over whether someone has left a 'heap' of sand on the floor. | |
| From: Timothy Williamson (Vagueness [1994], 7.7) | |
| A reaction: A fairly devastating rebuttal of what seems to be David Lewis's view. He talks of something being 'smooth' depending on context. |
| 21614 | The 'nihilist' view of vagueness says that 'heap' is not a legitimate concept [Williamson] |
| Full Idea: The 'nihilist' view is that no genuine distinction can be vaguely drawn; since vague expressions are not properly meaningful, there is nothing for sorites reasoning to betray; they are empty. | |
| From: Timothy Williamson (Vagueness [1994], 6.1) | |
| A reaction: He cites Frege as holding this view. The thought is that 'heap' is not a legitimate concept, so fussing over what qualifies as one is pointless. This seems to be a semantic view of vagueness, of which the main rival is the contextual view. |
| 21617 | We can say propositions are bivalent, but vague utterances don't express a proposition [Williamson] |
| Full Idea: A philosopher might endorse bivalence for propositions, while treating vagueness as the failure of an utterance to express a unique proposition. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This idea jumps at out me as an extremely promising approach to vagueness, because I am a fan of propositions (and have written a paper on them). The whole point of propositions is that they are not ambiguous (and probably not vague). |
| 21618 | If the vague 'TW is thin' says nothing, what does 'TW is thin if his perfect twin is thin' say? [Williamson] |
| Full Idea: If vague utterances in borderline cases fail to say anything, then if 'TW is thin' is vague, and TW has a twin of identical dimensions, it still seems that 'If TW is thin then his twin is thin' must be true, and so it must have said something. | |
| From: Timothy Williamson (Vagueness [1994], 7.2 (d)) | |
| A reaction: This an objection to the Fregean 'nihilistic' view of Idea 21614. I am inclined to a solution based on the proposition expressed, rather than the sentence. The first question is whether you are willing to assert 'TW is thin'. |
| 21590 | Asking when someone is 'clearly' old is higher-order vagueness [Williamson] |
| Full Idea: Difficulties of vagueness are presented by the question 'When did Rembrandt become clearly old?', and the iterating question 'When did he become clearly clearly old?'. This is the phenomenon of higher-order vagueness. The language of vagueness is vague. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: [compressed] I presume the bottom level is a question about Rembrandt, the second level is about this use of the word 'old', and the third level is about this particular application of the word 'clearly'. Meta-languages. |
| 21592 | Supervaluation keeps classical logic, but changes the truth in classical semantics [Williamson] |
| Full Idea: Supervaluationism preserves almost all of classical logic, at the expense of classical semantics, but giving a non-standard account of truth. I argue that its treatment of higher-order vagueness undermines the non-standard account of truth. | |
| From: Timothy Williamson (Vagueness [1994], Intro) |
| 21603 | You can't give a precise description of a language which is intrinsically vague [Williamson] |
| Full Idea: If a vague language is made precise, its expressions change in meaning, so an accurate semantic description of the precise language is inaccurate as a description of the vague one. | |
| From: Timothy Williamson (Vagueness [1994], 5.1) | |
| A reaction: Kind of obvious, really, but it clarifies the nature of any project (starting with Leibniz) to produce a wholly precise language. That is usually seen as a specialist language for science. |
| 21604 | Supervaluation assigns truth when all the facts are respected [Williamson] |
| Full Idea: 'Admissible' interpretations respect all the theoretical and ostensive connections. ...'Supervaluation' is the assignment of truth to the statements true on all admissible valuations, falsity to the false one, and neither to the rest. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: So 'he is bald' is true if when faced with all observations and definitions it is acceptable. Prima facie, that doesn't sound like a solution to the problem. Supervaluation started in philosophy of science. [p.156 'Admissible seems vague'] |
| 21607 | Supervaluation has excluded middle but not bivalence; 'A or not-A' is true, even when A is undecided [Williamson] |
| Full Idea: The supervaluationist denies bivalence but accepts excluded middle. The statement 'A or not-A' is true on each admissible interpretation, and therefore true, even if 'A' (and hence 'not-A') are true and some and false on others, so neither T nor F. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: See Ideas 21605 and 21606 for the distinction being used here. Denying bivalence allows 'A' to be neither true nor false. It seems common sense that 'he is either bald or not-bald' is true, without being sure about the disjuncts. |
| 21608 | Truth-functionality for compound statements fails in supervaluation [Williamson] |
| Full Idea: A striking fearure of supervaluations is the failure of truth-functionality for compound statements. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: Supervaluations has the initial appearance of enhancing classical logic, but turns out to somewhat undermine it. Hence Williamson's lack of sympathy. But see Idea 21610. |
| 21609 | Supervaluationism defines 'supertruth', but neglects it when defining 'valid' [Williamson] |
| Full Idea: Supervaluationists identify truth with 'supertruth'; since validity is necessary preservation of truth, they should identify it with necessary preservation of supertruth. But it plays no role in their definition of 'local' validity. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: [See text for 'local'] Generally Williamson's main concern with attempts to sort out vagueness is that higher-order and meta-language issues are neglected. |
| 21610 | Supervaluation adds a 'definitely' operator to classical logic [Williamson] |
| Full Idea: Supervaluation seems to inherit the power of classical logic, ...but also enables it to be extended. It makes room for a new operator 'definitely' to express supertruth in the object-language. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: Once you mention higher-order vagueness you can see a regress looming over the horizon. 'He is definitely definitely definitely bald'. [p.164 he says 'definitely' has no analysis, and is an uninteresting primitive] |
| 21613 | Supervaluationism cannot eliminate higher-order vagueness [Williamson] |
| Full Idea: Supervaluationism cannot eliminate higher-order vagueness. It must conduct its business in a vague meta-language. ...[162] All truth is at least disquotational, and supertruth is not. | |
| From: Timothy Williamson (Vagueness [1994], 5.6) | |
| A reaction: This is Williamson's final verdict on the supervaluation strategy for vagueness. Intuitively, it looks as if merely narrowing down the vagueness (by some sort of consensus) is no solution to the problem of vagueness. |
| 21633 | Nominalists suspect that properties etc are our projections, and could have been different [Williamson] |
| Full Idea: The nominalist suspects that properties, relations and states of affairs are mere projections onto the world of our forms of speech. One source of the suspicion is a sense that we could just as well have classified things differently. | |
| From: Timothy Williamson (Vagueness [1994], 9.3) | |
| A reaction: I know it is very wicked to say so, but I'm afraid I have some sympathy with this view. But I like the primary/secondary distinction, so there is more 'projection' in the latter case. Classification is not random; it is a response to reality. |
| 6861 | What sort of logic is needed for vague concepts, and what sort of concept of truth? [Williamson] |
| Full Idea: The problem of vagueness is the problem of what logic is correct for vague concepts, and correspondingly what notions of truth and falsity are applicable to vague statements (does one need a continuum of degrees of truth, for example?). | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.153) | |
| A reaction: This certainly makes vagueness sound like one of the most interesting problems in all of philosophy, though also one of the most difficult. Williamson's solution is that we may be vague, but the world isn't. |
| 9602 | Common sense and classical logic are often simultaneously abandoned in debates on vagueness [Williamson] |
| Full Idea: The constraints of common sense and classical logic are often simultaneously abandoned in debates on vagueness. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], After) | |
| A reaction: Wiliamson has described himself (in my hearing) as a 'rottweiller realist', but presumably the problem of vagueness interests a lot of people precisely because it pushes us away from common sense and classical logic. |
| 21630 | If fuzzy edges are fine, then why not fuzzy temporal, modal or mereological boundaries? [Williamson] |
| Full Idea: If objects can have fuzzy spatial boundaries, surely they can have fuzzy temporal, modal or mereological boundaries too. | |
| From: Timothy Williamson (Vagueness [1994], 9.2) | |
| A reaction: Fair point. I think there is a distinction between parts of the thing, such as its edges, being fuzzy, and the whole thing being fuzzy, in the temporal case. |
| 21632 | A river is not just event; it needs actual and counterfactual boundaries [Williamson] |
| Full Idea: A river is not just an event. One would need to specify counterfactual as well as actual boundaries. | |
| From: Timothy Williamson (Vagueness [1994], 9.3) | |
| A reaction: In other words the same river can change its course a bit, but it can't head off in the opposite direction. |
| 14625 | Necessity is counterfactually implied by its negation; possibility does not counterfactually imply its negation [Williamson] |
| Full Idea: Modal thinking is logically equivalent to a type of counterfactual thinking. ...The necessary is that which is counterfactually implied by its own negation; the possible is that which does not counterfactually imply its own negation. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 1) | |
| A reaction: I really like this, because it builds modality on ordinary imaginative thinking. He says you just need to grasp counterfactuals, and also negation and absurdity, and you can then understand necessity and possibility. We can all do that. |
| 14623 | Strict conditionals imply counterfactual conditionals: □(A⊃B)⊃(A□→B) [Williamson] |
| Full Idea: The strict conditional implies the counterfactual conditional: □(A⊃B) ⊃ (A□→B) - suppose that A would not have held without B holding too; then if A had held, B would also have held. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 1) | |
| A reaction: [He then adds a reading of his formula in terms of possible worlds] This sounds rather close to modus ponens. If A implies B, and A is actually the case, what have you got? B! |
| 14624 | Counterfactual conditionals transmit possibility: (A□→B)⊃(◊A⊃◊B) [Williamson] |
| Full Idea: The counterfactual conditional transmits possibility: (A□→B) ⊃ (◊A⊃◊B). Suppose that if A had held, B would also have held; the if it is possible for A to hold, it is also possible for B to hold. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 1) |
| 14531 | Rather than define counterfactuals using necessity, maybe necessity is a special case of counterfactuals [Williamson, by Hale/Hoffmann,A] |
| Full Idea: Instead of regarding counterfactuals as conditionals restricted to a range of possible worlds, we can define the necessity operator by means of counterfactuals. Metaphysical necessity is a special case of ordinary counterfactual thinking. | |
| From: report of Timothy Williamson (Modal Logic within Counterfactual Logic [2010]) by Bob Hale/ Aviv Hoffmann - Introduction to 'Modality' 2 | |
| A reaction: [compressed] I very much like Williamson's approach, of basing these things on the ordinary way that ordinary people think. To me it is a welcome inclusion of psychology into metaphysics, which has been out in the cold since Frege. |
| 21621 | We can't infer metaphysical necessities to be a priori knowable - or indeed knowable in any way [Williamson] |
| Full Idea: The inference from metaphysical necessity to a priori knowlability is, as Kripke has emphasized, fallacious. Indeed, metaphysical necessities cannot be assumed knowable in any way at all. | |
| From: Timothy Williamson (Vagueness [1994], 7.4) | |
| A reaction: The second sentence sounds like common sense. He cites Goldbach's Conjecture. A nice case of the procedural rule of keeping your ontology firmly separated from your epistemology. How is it? is not How do we know it? |
| 9598 | Modal thinking isn't a special intuition; it is part of ordinary counterfactual thinking [Williamson] |
| Full Idea: The epistemology of metaphysical modality requires no dedicated faculty of intuition. It is simply a special case of the epistemology of counterfactual thinking, a kind of thinking tightly integrated with our thinking about the spatio-temporal world. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], 5.6) | |
| A reaction: This seems to me to be spot-on, though it puts the focus increasingly on the faculty of imagination, as arguably an even more extraordinary feature of brains than the much-vaunted normal consciousness. |
| 16536 | Williamson can't base metaphysical necessity on the psychology of causal counterfactuals [Lowe on Williamson] |
| Full Idea: The psychological mechanism that Williamson proposes as the supposedly reliable source of our knowledge of necessities only seems applicable to counterfactuals that are distinctively causal, not metaphysical, in character. | |
| From: comment on Timothy Williamson (The Philosophy of Philosophy [2007]) by E.J. Lowe - What is the Source of Knowledge of Modal Truths? 5 | |
| A reaction: My rough impression of Williamson's account is that it is correct but unilluminating. We have to assess necessities by counterfactual thinking, because nothing else is available (apart from evaluating the coherence of the findings). |
| 9596 | We scorn imagination as a test of possibility, forgetting its role in counterfactuals [Williamson] |
| Full Idea: The epistemology of modality often focuses on (and pours scorn on) imagination or conceivability as a test of possibility, while ignoring the role of the imagination in the assessment of mundane counterfactuals. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], 5.4) | |
| A reaction: Good point. I've been guilty of this easy scorn myself. Williamson gives our modal capacities an evolutionary context. What is needed is well-informed imagination, rather than wild fantasy. |
| 15142 | Our ability to count objects across possibilities favours the Barcan formulas [Williamson] |
| Full Idea: Consideration of our ability to count objects across possibilities strongly favour both the Barcan formula and its converse. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: I'm not sure that I can understand counting objects across possibilities. The objects themselves are possibilia, and possibilia seem to include unknowns. The unexpected is highly possible. |
| 18925 | If talking donkeys are possible, something exists which could be a talking donkey [Williamson, by Cameron] |
| Full Idea: Williamson's view on modality is that everything that could exist does exist: since there could exist a talking donkey there actually exists some thing that could be a talking donkey. | |
| From: report of Timothy Williamson (Modal Logic as Metaphysics [2013], n20) by Ross P. Cameron - Truthmaking for Presentists n20 | |
| A reaction: Well that thing certainly isn't me, or Tim Williamson. I'm guessing that the thing is an actual donkey, probably a rather bright one. Actually, I think this is one of those views that invites the incredulous stare. (Barcan formulae). |
| 21627 | We have inexact knowledge when we include margins of error [Williamson] |
| Full Idea: Inexact knowledge is a widespread and easily recognised cognitive phenomenon, whose underlying nature turns out to be characterised by the holding of margin of error principles. | |
| From: Timothy Williamson (Vagueness [1994], 8.3) | |
| A reaction: Williamson is invoking this as a tool in developing his epistemic view of vagueness. It obviously invites the question of how it can be knowledge if error is a possibility. A very large margin of error would obviously invalidate it. |
| 4760 | Belief aims at knowledge (rather than truth), and mere believing is a kind of botched knowing [Williamson] |
| Full Idea: Knowing is the best kind of believing. Mere believing is a kind of botched knowing. In short, belief aims at knowledge (not just truth). | |
| From: Timothy Williamson (Knowledge and its Limits [2000], §1.5) | |
| A reaction: The difference between aiming at truth and aiming at knowledge has to be in the justificiation, so beliefs aim to be justified. Believers always aim at truth, but they can be strikingly relaxed about justification. |
| 19527 | We don't acquire evidence and then derive some knowledge, because evidence IS knowledge [Williamson] |
| Full Idea: When we acquire new evidence in perception, we do not first acquire unknown evidence and then somehow base knowledge on it later. Rather, acquiring new is evidence IS acquiring new knowledge. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.4) | |
| A reaction: This makes his point much better than Idea 19526 does. |
| 19512 | Don't analyse knowledge; use knowledge to analyse other concepts in epistemology [Williamson, by DeRose] |
| Full Idea: Williamson says that instead of being viewed as a concept to be analysed, knowledge should be seen as something useful in the analysis of all sorts of other concepts to epistemology - and to philosophy of mind as well. | |
| From: report of Timothy Williamson (Knowledge and its Limits [2000]) by Keith DeRose - The Case for Contextualism 1.8 | |
| A reaction: I just don't believe this, because knowledge is obviously a complex state of mind, which invites breaking it down into ingredients. How could knowledge possibly be prior to truth? |
| 19528 | Knowledge is prior to believing, just as doing is prior to trying to do [Williamson] |
| Full Idea: Knowing corresponds to doing, believing to trying. Just as trying is naturally understood in relation to doing, so believing is naturally understood in relation to knowing. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.4) | |
| A reaction: An interesting analogy. You might infer that there can be no concept of 'belief' without the concept of 'knowledge', but we could say that it is 'truth' which is indispensible, and leave out knowledge entirely. Belief is to truth as trying is to doing? |
| 19529 | Belief explains justification, and knowledge explains belief, so knowledge explains justification [Williamson] |
| Full Idea: If justification is the fundamental epistemic norm of belief, and a belief ought to constitute knowledge, then justification should be understood in terms of knowledge too. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.5) | |
| A reaction: If we are looking for the primitive norm which motivates the whole epistemic game, then I am thinking that truth might well play that role better than knowledge. TW would have to reply that it is the 'grasped truth', rather than the 'theoretical truth'. |
| 19530 | A neutral state of experience, between error and knowledge, is not basic; the successful state is basic [Williamson] |
| Full Idea: A neutral state covering both perceiving and misperceiving (or remembering and misrembering) is not somehow more basic than perceiving, for what unifies the case of each neutral state is their relation to the successful state. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.5-6) | |
| A reaction: An alternative is Disjunctivism, which denies the existence of a single neutral state, so that there is nothing to unite the two states, and they don't have a dependence relation. Why can't there be a prior family of appearances, some of them successful? |
| 19531 | Internalism about mind is an obsolete view, and knowledge-first epistemology develops externalism [Williamson] |
| Full Idea: A postulated underlying layer of narrow mental states is a myth, whose plausibility derives from a comfortingly familiar but obsolescent philosophy of mind. Knowledge-first epistemology is a further step in the development of externalism. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.6) | |
| A reaction: Williamson is a real bruiser, isn't he? I don't take internalism about mind to be obsolescent at all, but now I feel so inferior for clinging to such an 'obsolescent' belief. ...But then I cling to Aristotle, who is (no doubt) an obsolete philosopher. |
| 19536 | Knowledge-first says your total evidence IS your knowledge [Williamson] |
| Full Idea: Knowledge-first equate one's total evidence with one's total knowledge. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.8) | |
| A reaction: Couldn't lots of evidence which merely had a high probability be combined together to give a state we would call 'knowledge'? Many dubious witnesses confirm the truth, as long as they are independent, and agree. |
| 19526 | Surely I am acquainted with physical objects, not with appearances? [Williamson] |
| Full Idea: When I ask myself what I am acquainted with, the physical objects in front of me are far more natural candidates than their appearances. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.3) | |
| A reaction: Not very impressive. The word 'acquainted' means the content of the experience, not the phenomena. Do I 'experience' the objects, or the appearances? The answer there is less obvious. If you apply it to colours, it is even less obvious. |
| 9597 | There are 'armchair' truths which are not a priori, because experience was involved [Williamson] |
| Full Idea: There is extensive 'armchair knowledge' in which experience plays no strictly evidential role, but it may not fit the stereotype of the a priori, because the contribution of experience was more than enabling, such as armchair truths about our environment. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], 5.5) | |
| A reaction: Once this point is conceded we have no idea where to draw the line. Does 'if it is red it can't be green' derive from experience? I think it might. |
| 6860 | How can one discriminate yellow from red, but not the colours in between? [Williamson] |
| Full Idea: If one takes a spectrum of colours from yellow to red, it might be that given a series of colour samples along that spectrum, each sample is indiscriminable by the naked eye from the next one, though samples at either end are blatantly different. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.151) | |
| A reaction: This seems like a nice variant of the Sorites paradox (Idea 6008). One could demonstrate it with just three samples, where A and C seemed different from each other, but other comparisons didn't. |
| 9592 | Intuition is neither powerful nor vacuous, but reveals linguistic or conceptual competence [Williamson] |
| Full Idea: Crude rationalists postulate a special knowledge-generating faculty of rational intuition. Crude empiricists regard intuition as an obscurantist term of folk psychology. Linguistic/conceptual philosophy says it reveals linguistic or conceptual competence. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], Intro) | |
| A reaction: Kripke seems to think that it is the basis of logical competence. I would use it as a blank term for any insight in which we have considerable confidence, and yet are unable to articulate its basis; roughly, for rational thought that evades logic. |
| 20181 | When analytic philosophers run out of arguments, they present intuitions as their evidence [Williamson] |
| Full Idea: 'Intuition' plays a major role in contemporary analytic philosophy's self-understanding. ...When contemporary analytic philosophers run out of arguments, they appeal to intuitions. ...Thus intuitions are presented as our evidence in philosophy. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], p.214-5), quoted by Herman Cappelen - Philosophy without Intuitions 01.1 | |
| A reaction: Williamson says we must investigate this 'scandal', but Cappelen's book says analytic philosophy does not rely on intuition. |
| 21626 | Knowing you know (KK) is usually denied if the knowledge concept is missing, or not considered [Williamson] |
| Full Idea: The failure of the KK principle is not news. The standard counterexamples involve knowing subjects who lack the concept of knowledge, or have not reflected on their knowledge, and therefore do not know that they know. | |
| From: Timothy Williamson (Vagueness [1994], 8.2) | |
| A reaction: There is also the timid but knowledgeable pupil, who can't believe they know so much. The simplest case would be if we accept that animals know lots of things, but are largely devoid of any metathinking. |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 14628 | Imagination is important, in evaluating possibility and necessity, via counterfactuals [Williamson] |
| Full Idea: Imagination can be made to look cognitively worthless. Once we recall its fallible but vital role in evaluating counterfactual conditionals, we should be more open to the idea that it plays such a role in evaluating claims of possibility and necessity. | |
| From: Timothy Williamson (Modal Logic within Counterfactual Logic [2010], 6) | |
| A reaction: I take this to be a really important idea, because it establishes the importance of imagination within the formal framework of modern analytic philosopher (rather than in the whimsy of poets and dreamers). |
| 21631 | To know, believe, hope or fear, one must grasp the thought, but not when you fail to do them [Williamson] |
| Full Idea: To know, believe, hope, or fear that A, one must grasp the thought that A. In contrast, to fail to know, believe, hope or fear that A, one need not grasp the thought that A. | |
| From: Timothy Williamson (Vagueness [1994], 9.3 c) | |
| A reaction: A simple point, which at least shows that propositional attitudes are a two-stage operation. |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 21600 | 'Blue' is not a family resemblance, because all the blues resemble in some respect [Williamson] |
| Full Idea: 'Blue' is vague by some standards, for it has borderline cases, but that does not make it a family resemblance term, for all the shades of blue resemble each other in some respect. | |
| From: Timothy Williamson (Vagueness [1994], 3.3) | |
| A reaction: Presumably the point of family resemblance is that fringe members as still linked to the family, despite having lost the main features. A bit of essentialism seems needed here. |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| Full Idea: We call 'cardinal number' the general concept which, by means of our active faculty of thought, arises when we make abstraction from an aggregate of its various elements, and of their order. From this double abstraction the number is an image in our mind. | |
| From: George Cantor (Beitrage [1915], §1), quoted by Kit Fine - Cantorian Abstraction: Recon. and Defence Intro | |
| A reaction: [compressed] This is the great Cantor, creator of set theory, endorsing the traditional abstractionism which Frege and his followers so despise. Fine offers a defence of it. The Frege view is platonist, because it refuses to connect numbers to the world. |
| 9595 | You might know that the word 'gob' meant 'mouth', but not be competent to use it [Williamson] |
| Full Idea: Someone who acquires the word 'gob' just by being reliably told that it is synonymous with 'mouth' knows what 'gob' means without being fully competent to use it. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], 4.7) | |
| A reaction: Not exactly an argument against meaning-as-use, but a very nice cautionary example to show that 'knowing the meaning' of a word may be a rather limited, and dangerous, achievement. |
| 21615 | References to the 'greatest prime number' have no reference, but are meaningful [Williamson] |
| Full Idea: The predicate 'is a prime number greater than all other prime numbers' is necessarily not true of anything, but it is not semantically defective, for it occurs in sentences that constitute a sound proof that there is no such number. | |
| From: Timothy Williamson (Vagueness [1994], 6.2) | |
| A reaction: One might reply that the description can be legitimately mentioned, but not legitimately used. |
| 18038 | The 't' and 'f' of formal semantics has no philosophical interest, and may not refer to true and false [Williamson] |
| Full Idea: In a formal semantics we can label two properties 't' and 'f' and suppose that some sentences have neither (or both). Such a manoeuvre shows nothing of philosophical interest. No connection has been made between 't' and 'f' and truth and falsity. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This is right, and means there is a huge gulf between 'formal' semantics (which could be implemented on a computer), and seriously interesting semantics about how language refers to and describes the world. |
| 19534 | How does inferentialism distinguish the patterns of inference that are essential to meaning? [Williamson] |
| Full Idea: Inferentialism faces the grave problem of separating patterns of inference that are to count as essential to the meaning of an expression from those that will count as accidental (a form of the analytic/synthetic distinction). | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.6) | |
| A reaction: This sounds like a rather persuasive objection to inferentialism, though I don't personally take that as a huge objection to all internalist semantics. |
| 19535 | Internalist inferentialism has trouble explaining how meaning and reference relate [Williamson] |
| Full Idea: The internalist version of inferentialist semantics has particular difficulty in establishing an adequate relation between meaning and reference. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.6) | |
| A reaction: I would have thought that this was a big problem for referentialist semantics too, though evidently Williamson doesn't think so. If he is saying that the meaning is in the external world, dream on. |
| 19533 | Inferentialist semantics relies on internal inference relations, not on external references [Williamson] |
| Full Idea: On internalist inferential (or conceptual role) semantics, the inferential relations of an expression do not depend on what, if anything, it refers to, ...rather, the meaning is something like its place in a web of inferential relations. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.6) | |
| A reaction: Williamson says the competition is between externalist truth-conditional referential semantics (which he favours), and this internalist inferential semantics. He is, like, an expert, of course, but I doubt whether that is the only internalist option. |
| 19532 | Truth-conditional referential semantics is externalist, referring to worldly items [Williamson] |
| Full Idea: Truth-conditional referential semantics is an externalist programme. In a context of utterance the atomic expressions of a language refer to worldly items, from which the truth-conditions of sentences are compositionally determined. | |
| From: Timothy Williamson (Knowledge First (and reply) [2014], p.6) | |
| A reaction: I just don't see how a physical object can be part of the contents of a sentence. 'Dragons fly' is atomic, and meaningful, but its reference fails. 'The cat is asleep' is just words - it doesn't contain a live animal. |
| 21624 | It is known that there is a cognitive loss in identifying propositions with possible worlds [Williamson] |
| Full Idea: It is well known that when a proposition is identified with the set of possible worlds at which it is true, a region in the space of possible worlds, cognitively significant distinctions are lost. | |
| From: Timothy Williamson (Vagueness [1994], 7.6) | |
| A reaction: Alas, he doesn't specify which distinctions get lost, so this is just a pointer. It would seem likely that two propositions could have identical sets of possible worlds, while not actually saying the same thing. Equilateral/equiangular. |
| 19216 | Propositions (such as 'that dog is barking') only exist if their items exist [Williamson] |
| Full Idea: A proposition about an item exists only if that item exists... how could something be the proposition that that dog is barking in circumstances in which that dog does not exist? | |
| From: Timothy Williamson (Necessary Existents [2002], p.240), quoted by Trenton Merricks - Propositions | |
| A reaction: This is a view of propositions I can't make sense of. If I'm under an illusion that there is a dog barking nearby, when there isn't one, can I not say 'that dog is barking'? If I haven't expressed a proposition, what have I done? |
| 9600 | If languages are intertranslatable, and cognition is innate, then cultures are all similar [Williamson] |
| Full Idea: Given empirical evidence for the approximate intertranslatability of all human languages, and a universal innate basis of human cognition, we may wonder how 'other' any human culture really is. | |
| From: Timothy Williamson (The Philosophy of Philosophy [2007], 8.1) | |
| A reaction: This seems to be a fairly accurate account of the situation. In recent centuries people seem to have been over-impressed by superficial differences in cultural behaviour, but we increasingly see the underlying identity. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |
| 15133 | A thing can't be the only necessary existent, because its singleton set would be as well [Williamson] |
| Full Idea: That there is just one necessary existent is surely false, for if x is a necessary, {x} is a distinct necessary existent. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §1) | |
| A reaction: You would have to believe that sets actually 'exist' to accept this, but it is a very neat point. |