87 ideas
12330 | In ontology, logic dominated language, until logic was mathematized [Badiou] |
Full Idea: From Aristotle to Hegel, logic was the philosophical category of ontology's dominion over language. The mathematization of logic has authorized language to become that which seizes philosophy for itself. | |
From: Alain Badiou (Briefings on Existence [1998], 8) |
9808 | Philosophy aims to reveal the grandeur of mathematics [Badiou] |
Full Idea: Philosophy's role consists in informing mathematics of its own speculative grandeur. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.11) | |
A reaction: Revealing the grandeur of something sounds more like a rhetorical than a rational exercise. How would you reveal the grandeur of a sunset to someone? |
12318 | The female body, when taken in its entirety, is the Phallus itself [Badiou] |
Full Idea: The female body, when taken in its entirety, is the Phallus itself. | |
From: Alain Badiou (Briefings on Existence [1998]) | |
A reaction: Too good to pass over, too crazy to file sensibly, too creepy to have been filed under humour, my candidate for the weirdest remark I have ever read in a serious philosopher, but no doubt if you read Lacan etc for long enough it looks deeply wise. |
18835 | Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics [Rumfitt] |
Full Idea: There is surely no metaphysical basis for logic, but equally there is no logical basis for metaphysics, if that implies that we can settle the choice of logic in advance of settling any seriously contested metaphysical-cum-semantic issues. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.5) | |
A reaction: Is this aimed at Tim Williamson's book on treating modal logic as metaphysics? I agree with the general idea that logic won't deliver a metaphysics. I might want to defend a good metaphysics giving rise to a good logic. |
12325 | Philosophy has been relieved of physics, cosmology, politics, and now must give up ontology [Badiou] |
Full Idea: Philosophy has been released from, even relieved of, physics, cosmology, and politics, as well as many other things. It is important for it to be released from ontology per se. | |
From: Alain Badiou (Briefings on Existence [1998], 3) | |
A reaction: A startling proposal, for anyone who thought that ontology was First Philosophy. Badiou wants to hand ontology over to mathematicians, but I am unclear what remains for the philosophers to do. |
12324 | Consensus is the enemy of thought [Badiou] |
Full Idea: Consensus is the enemy of thought. | |
From: Alain Badiou (Briefings on Existence [1998], 2) | |
A reaction: A nice slogan for bringing Enlightenment optimists to a halt. I am struck. Do I allow my own thinking to always be diverted towards something which might result in a consensus? Do I actually (horror!) prefer consensus to truth? |
18819 | The idea that there are unrecognised truths is basic to our concept of truth [Rumfitt] |
Full Idea: The realist principle that a statement may be true even though no one is able to recognise its truth is so deeply embedded in our ordinary conception of truth that any account that flouts it is liable to engender confusion. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.1) |
18826 | 'True at a possibility' means necessarily true if what is said had obtained [Rumfitt] |
Full Idea: A statement is 'true at a possibility' if, necessarily, things would have been as the statement (actually) says they are, had the possibility obtained. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.6) | |
A reaction: This is deliberately vague about what a 'possibility' is, but it is intended to be more than a property instantiation, and less than a possible world. |
18803 | Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables [Rumfitt] |
Full Idea: The classical semantics of natural language propositions says 1) valid arguments preserve truth, 2) no statement is both true and false, 3) each statement is either true or false, 4) the familiar truth tables. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) |
12204 | The logic of metaphysical necessity is S5 [Rumfitt] |
Full Idea: It is a widely accepted thesis that the logic of metaphysical necessity is S5. | |
From: Ian Rumfitt (Logical Necessity [2010], §5) | |
A reaction: Rumfitt goes on to defend this standard view (against Dummett's defence of S4). The point, I take it, is that one can only assert that something is 'true in all possible worlds' only when the worlds are all accessible to one another. |
18814 | 'Absolute necessity' would have to rest on S5 [Rumfitt] |
Full Idea: If there is such a notion as 'absolute necessity', its logic is surely S5. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3) | |
A reaction: There are plenty of people (mainly in the strict empiricist tradition) who don't believe in 'absolute' necessity. |
18798 | It is the second-order part of intuitionistic logic which actually negates some classical theorems [Rumfitt] |
Full Idea: Although intuitionistic propositional and first-order logics are sub-systems of the corresponding classical systems, intuitionistic second-order logic affirms the negations of some classical theorems. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) |
18799 | Intuitionists can accept Double Negation Elimination for decidable propositions [Rumfitt] |
Full Idea: Double Negation Elimination is a rule of inference which the classicist accepts without restriction, but which the intuitionist accepts only for decidable propositions. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |
A reaction: This cures me of my simplistic understanding that intuitionists just reject the rules about double negation. |
18830 | Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic [Rumfitt] |
Full Idea: Many set theorists doubt if the Generalised Continuum Hypothesis must be either true or false; certainly, its bivalence is far from obvious. All the same, almost all set theorists use classical logic in their proofs. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2) | |
A reaction: His point is that classical logic is usually taken to rest on bivalence. He offers the set theorists a helping hand, by defending classical logic without resorting to bivalence. |
12337 | There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou] |
Full Idea: 'Transitivity' signifies that all of the elements of the set are also parts of the set. If you have α∈Β, you also have α⊆Β. This correlation of membership and inclusion gives a stability which is the sets' natural being. | |
From: Alain Badiou (Briefings on Existence [1998], 11) |
18843 | The iterated conception of set requires continual increase in axiom strength [Rumfitt] |
Full Idea: We are doomed to postulate an infinite sequence of successively stronger axiom systems as we try to spell out what is involved in iterating the power set operation 'as far as possible'. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.3) | |
A reaction: [W.W. Tait is behind this idea] The problem with set theory, then, especially as a foundation of mathematics, is that it doesn't just expand, but has to keep reinventing itself. The 'large cardinal axioms' are what is referred to. |
18836 | A set may well not consist of its members; the empty set, for example, is a problem [Rumfitt] |
Full Idea: There seem strong grounds for rejecting the thesis that a set consists of its members. For one thing, the empty set is a perpetual embarrassment for the thesis. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4) | |
A reaction: Rumfitt also says that if 'red' has an extension, then membership of that set must be vague. Extensional sets are precise because their objects are decided in advance, but intensional (or logical) sets, decided by a predicate, can be vague. |
18837 | A set can be determinate, because of its concept, and still have vague membership [Rumfitt] |
Full Idea: Vagueness in respect of membership is consistent with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of the concept A. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4) | |
A reaction: To be determinate, it must be presumed that there is some test which will decide what falls under the concept. The rule can say 'if it is vague, reject it' or 'if it is vague, accept it'. Without one of those, how could the set have a clear identity? |
18845 | If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom [Rumfitt] |
Full Idea: Someone who is sympathetic to the thesis that the totality of sets is not well-defined ought to concede that we have no reason to think that the Power Set Axiom is true. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6) | |
A reaction: The point is that it is only this Axiom which generates the vast and expanding totality. In principle it is hard, though, to see what is intrinsically wrong with the operation of taking the power set of a set. Hence 'limitation of size'? |
12321 | The axiom of choice must accept an indeterminate, indefinable, unconstructible set [Badiou] |
Full Idea: The axiom of choice actually amounts to admitting an absolutely indeterminate infinite set whose existence is asserted albeit remaining linguistically indefinable. On the other hand, as a process, it is unconstructible. | |
From: Alain Badiou (Briefings on Existence [1998], 2) | |
A reaction: If only constructible sets are admitted (see 'V = L') then there is a contradiction. |
12342 | Topos theory explains the plurality of possible logics [Badiou] |
Full Idea: Topos theory explains the plurality of possible logics. | |
From: Alain Badiou (Briefings on Existence [1998], 14) | |
A reaction: This will because logic will have a distinct theory within each 'topos'. |
11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
Full Idea: If a designated conclusion follows from the premisses, but the argument involves two howlers which cancel each other out, then the moral is that the path an argument takes from premisses to conclusion does matter to its logical evaluation. | |
From: Ian Rumfitt ("Yes" and "No" [2000], II) | |
A reaction: The drift of this is that our view of logic should be a little closer to the reasoning of ordinary language, and we should rely a little less on purely formal accounts. |
18815 | Logic is higher-order laws which can expand the range of any sort of deduction [Rumfitt] |
Full Idea: On the conception of logic recommended here, logical laws are higher-order laws that can be applied to expand the range of any deductive principles. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3) | |
A reaction: You need the concept of a 'deductive principle' to get this going, but I take it that might be directly known, rather than derived from a law. |
9390 | Logic guides thinking, but it isn't a substitute for it [Rumfitt] |
Full Idea: Logic is part of a normative theory of thinking, not a substitute for thinking. | |
From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.13) | |
A reaction: There is some sort of logicians' dream, going back to Leibniz, of a reasoning engine, which accepts propositions and outputs inferences. I agree with this idea. People who excel at logic are often, it seems to me, modest at philosophy. |
18805 | Classical logic rules cannot be proved, but various lines of attack can be repelled [Rumfitt] |
Full Idea: There is not the slightest prospect of proving that the rules of classical logic are sound. ….All that the defender of classical logic can do is scrutinize particular attacks and try to repel them. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |
A reaction: This is the agenda for Rumfitt's book. |
18804 | The case for classical logic rests on its rules, much more than on the Principle of Bivalence [Rumfitt] |
Full Idea: I think it is a strategic mistake to rest the case for classical logic on the Principle of Bivalence: the soundness of the classical logic rules is far more compelling than the truth of Bivalence. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |
A reaction: The 'rules' to which he is referring are those of 'natural deduction', which make very few assumptions, and are intended to be intuitively appealing. |
18827 | If truth-tables specify the connectives, classical logic must rely on Bivalence [Rumfitt] |
Full Idea: If we specify the senses of the connectives by way of the standard truth-tables, then we must justify classical logic only by appeal to the Principle of Bivalence. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7) | |
A reaction: Rumfitt proposes to avoid the truth-tables, and hence not to rely on Bivalence for his support of classical logic. He accepts that Bivalence is doubtful, citing the undecidability of the Continuum Hypothesis as a problem instance. |
12195 | Soundness in argument varies with context, and may be achieved very informally indeed [Rumfitt] |
Full Idea: Our ordinary standards for deeming arguments to be sound vary greatly from context to context. Even the package tourist's syllogism ('It's Tuesday, so this is Belgium') may meet the operative standards for soundness. | |
From: Ian Rumfitt (Logical Necessity [2010], Intro) | |
A reaction: No doubt one could spell out the preconceptions of package tourist reasoning, and arrive at the logical form of the implication which is being offered. |
12199 | There is a modal element in consequence, in assessing reasoning from suppositions [Rumfitt] |
Full Idea: There is a modal element in consequence, in its applicability to assessing reasoning from suppositions. | |
From: Ian Rumfitt (Logical Necessity [2010], §2) |
12201 | We reject deductions by bad consequence, so logical consequence can't be deduction [Rumfitt] |
Full Idea: A rule is to be rejected if it enables us to deduce from some premisses a purported conclusion that does not follow from them in the broad sense. The idea that deductions answer to consequence is incomprehensible if consequence consists in deducibility. | |
From: Ian Rumfitt (Logical Necessity [2010], §2) |
18813 | Logical consequence is a relation that can extended into further statements [Rumfitt] |
Full Idea: Logical consequence, I argue, is distinguished from other implication relations by the fact that logical laws may be applied in extending any implication relation so that it applies among some complex statements involving logical connectives. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3) | |
A reaction: He offers implication in electronics as an example of a non-logical implication relation. This seems to indicate that logic must be monotonic, that consequence is transitive, and that the Cut Law always applies. |
18808 | Normal deduction presupposes the Cut Law [Rumfitt] |
Full Idea: Our deductive practices seem to presuppose the Cut Law. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3) | |
A reaction: That is, if you don't believe that deductions can be transitive (and thus form a successful chain of implications), then you don't really believe in deduction. It remains a well known fact that you can live without the Cut Law. |
12341 | Logic is a mathematical account of a universe of relations [Badiou] |
Full Idea: Logic should first and foremost be a mathematical thought of what a universe of relations is. | |
From: Alain Badiou (Briefings on Existence [1998], 14) |
18840 | When faced with vague statements, Bivalence is not a compelling principle [Rumfitt] |
Full Idea: I do not regard Bivalence, when applied to vague statements, as an intuitively compelling principle which we ought to try to preserve. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.7) | |
A reaction: The point of Rumfitt's book is to defend classical logic despite failures of bivalence. He also cites undecidable concepts such as the Continuum Hypothesis. |
12194 | Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B' [Rumfitt] |
Full Idea: Overt contradictions include formal contradictions of form 'B and not B', but I also take them to include 'This is red all over and green all over' and 'This is red and not coloured'. | |
From: Ian Rumfitt (Logical Necessity [2010], Intro) |
11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
Full Idea: A connective will possess the sense that it has by virtue of its competent users' finding certain rules of inference involving it to be primitively obvious. | |
From: Ian Rumfitt ("Yes" and "No" [2000], III) | |
A reaction: Rumfitt cites Peacocke as endorsing this view, which characterises the logical connectives by their rules of usage rather than by their pure semantic value. |
11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
Full Idea: If 'and' and 'but' really are alike in sense, in what might that likeness consist? Some philosophers of classical logic will reply that they share a sense by virtue of sharing a truth table. | |
From: Ian Rumfitt ("Yes" and "No" [2000]) | |
A reaction: This is the standard view which Rumfitt sets out to challenge. |
18802 | In specifying a logical constant, use of that constant is quite unavoidable [Rumfitt] |
Full Idea: There is no prospect whatever of giving the sense of a logical constant without using that very constant, and much else besides, in the metalinguistic principle that specifies that sense. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) |
12198 | Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths) [Rumfitt] |
Full Idea: The geometrical style of formalization of logic is now little more than a quaint anachronism, largely because it fails to show logical truths for what they are: simply by-products of rules of inference that are applicable to suppositions. | |
From: Ian Rumfitt (Logical Necessity [2010], §1) | |
A reaction: This is the rejection of Russell-style axiom systems in favour of Gentzen-style natural deduction systems (starting from rules). Rumfitt quotes Dummett in support. |
18800 | Introduction rules give deduction conditions, and Elimination says what can be deduced [Rumfitt] |
Full Idea: 'Introduction rules' state the conditions under which one may deduce a conclusion whose dominant logical operator is the connective. 'Elimination rules' state what may be deduced from some premises, where the major premise is dominated by the connective. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |
A reaction: So Introduction gives conditions for deduction, and Elimination says what can actually be deduced. If my magic wand can turn you into a frog (introduction), and so I turn you into a frog, how does that 'eliminate' the wand? |
18809 | Logical truths are just the assumption-free by-products of logical rules [Rumfitt] |
Full Idea: Gentzen's way of formalising logic has accustomed people to the idea that logical truths are simply the by-products of logical rules, that arise when all the assumptions on which a conclusion rests have been discharged. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.5) | |
A reaction: This is the key belief of those who favour the natural deduction account of logic. If you really believe in separate logic truths, then you can use them as axioms. |
18807 | Monotonicity means there is a guarantee, rather than mere inductive support [Rumfitt] |
Full Idea: Monotonicity seems to mark the difference between cases in which a guarantee obtains and those where the premises merely provide inductive support for a conclusion. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3) | |
A reaction: Hence it is plausible to claim that 'non-monotonic logic' is a contradiction in terms. |
9812 | In mathematics, if a problem can be formulated, it will eventually be solved [Badiou] |
Full Idea: Only in mathematics can one unequivocally maintain that if thought can formulate a problem, it can and will solve it, regardless of how long it takes. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.17) | |
A reaction: I hope this includes proving the Continuum Hypothesis, and Goldbach's Conjecture. It doesn't seem quite true, but it shows why philosophers of a rationalist persuasion are drawn to mathematics. |
12334 | There is no single unified definition of number [Badiou] |
Full Idea: Apparently - and this is quite unlike old Greek times - there is no single unified definition of number. | |
From: Alain Badiou (Briefings on Existence [1998], 11) |
12335 | Numbers are for measuring and for calculating (and the two must be consistent) [Badiou] |
Full Idea: Number is an instance of measuring (distinguishing the more from the less, and calibrating data), ..and a figure for calculating (one counts with numbers), ..and it ought to be a figure of consistency (the compatibility of order and calculation). | |
From: Alain Badiou (Briefings on Existence [1998], 11) |
12333 | Each type of number has its own characteristic procedure of introduction [Badiou] |
Full Idea: There is a heterogeneity of introductory procedures of different classical number types: axiomatic for natural numbers, structural for ordinals, algebraic for negative and rational numbers, topological for reals, mainly geometric for complex numbers. | |
From: Alain Badiou (Briefings on Existence [1998], 11) |
12322 | Must we accept numbers as existing when they no longer consist of units? [Badiou] |
Full Idea: Do we have to confer existence on numbers whose principle is to no longer consist of units? | |
From: Alain Badiou (Briefings on Existence [1998], 2) | |
A reaction: This very nicely expresses what seems to me perhaps the most important question in the philosophy of mathematics. I am reluctant to accept such 'unitless' numbers, but I then feel hopelessly old-fashioned and naïve. What to do? |
18842 | Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set [Rumfitt] |
Full Idea: Menzel proposes that an ordinal is something isomorphic well-ordered sets have in common, so while an ordinal can be represented as a set, it is not itself a set, but a 'property' of well-ordered sets. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.2) | |
A reaction: [C.Menzel 1986] This is one of many manoeuvres available if you want to distance mathematics from set theory. |
17462 | A single object must not be counted twice, which needs knowledge of distinctness (negative identity) [Rumfitt] |
Full Idea: One requirement for a successful count is that double counting should be avoided: a single object should not be counted twice. ...but that is to make a knowledgeable judgement of distinctness - to resolve a question of identity in the negative. | |
From: Ian Rumfitt (Concepts and Counting [2002], III) | |
A reaction: He also notes later (p.65) that you must count all and only the right things. |
9813 | Mathematics shows that thinking is not confined to the finite [Badiou] |
Full Idea: Mathematics teaches us that there is no reason whatsoever to confne thinking within the ambit of finitude. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.19) | |
A reaction: This would perhaps make Cantor the greatest thinker who ever lived. It is an exhilarating idea, but we should ward the reader against romping of into unrestrained philosophical thought about infinities. You may be jumping without your Cantorian parachute. |
12327 | The undecidability of the Continuum Hypothesis may have ruined or fragmented set theory [Badiou] |
Full Idea: As we have known since Paul Cohen's theorem, the Continuum Hypothesis is intrinsically undecidable. Many believe Cohen's discovery has driven the set-theoretic project into ruin, or 'pluralized' what was once presented as a unified construct. | |
From: Alain Badiou (Briefings on Existence [1998], 6) | |
A reaction: Badiou thinks the theorem completes set theory, by (roughly) finalising its map. |
18834 | Infinitesimals do not stand in a determinate order relation to zero [Rumfitt] |
Full Idea: Infinitesimals do not stand in a determinate order relation to zero: we cannot say an infinitesimal is either less than zero, identical to zero, or greater than zero. ….Infinitesimals are so close to zero as to be theoretically indiscriminable from it. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.4) |
18846 | Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry) [Rumfitt] |
Full Idea: One of the motivations behind Cantor's and Dedekind's pioneering explorations in the field was the ambition to give real analysis a new foundation in set theory - and hence a foundation independent of geometry. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6) | |
A reaction: Rumfitt is inclined to think that the project has failed, although a weaker set theory than ZF might do the job (within limits). |
17461 | Some 'how many?' answers are not predications of a concept, like 'how many gallons?' [Rumfitt] |
Full Idea: We hit trouble if we hear answers to some 'How many?' questions as predications about concepts. The correct answer to 'how many gallons of water are in the tank?' may be 'ten', but that doesn''t mean ten things instantiate 'gallon of water in the tank'. | |
From: Ian Rumfitt (Concepts and Counting [2002], I) | |
A reaction: Rumfitt makes the point that a huge number of things instantiate that concept in a ten gallon tank of water. No problem, says Rumfitt, because Frege wouldn't have counted that as a statement of number. |
12329 | If mathematics is a logic of the possible, then questions of existence are not intrinsic to it [Badiou] |
Full Idea: If mathematics is a logic of the possible, then questions of existence are not intrinsic to it (as they are for the Platonist). | |
From: Alain Badiou (Briefings on Existence [1998], 7) | |
A reaction: See also Idea 12328. I file this to connect it with Hellman's modal (and nominalist) version of structuralism. Could it be that mathematics and modal logic are identical? |
12328 | Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic [Badiou] |
Full Idea: A Platonist's interest focuses on axioms in which the decision of thought is played out, where an Aristotelian or Leibnizian interest focuses on definitions laying out the representation of possibilities (...and the essence of mathematics is logic). | |
From: Alain Badiou (Briefings on Existence [1998], 7) | |
A reaction: See Idea 12323 for the significance of the Platonist approach. So logicism is an Aristotelian project? Frege is not a true platonist? I like the notion of 'the representation of possibilities', so will vote for the Aristotelians, against Badiou. |
12331 | Logic is definitional, but real mathematics is axiomatic [Badiou] |
Full Idea: Logic is definitional, whereas real mathematics is axiomatic. | |
From: Alain Badiou (Briefings on Existence [1998], 10) |
12340 | There is no Being as a whole, because there is no set of all sets [Badiou] |
Full Idea: The fundamental theorem that 'there does not exist a set of all sets' designates the inexistence of Being as a whole. ...A crucial consequence of this property is that any ontological investigation is irremediably local. | |
From: Alain Badiou (Briefings on Existence [1998], 14) | |
A reaction: The second thought pushes Badiou into Topos Theory, where the real numbers (for example) have a separate theory in each 'topos'. |
9809 | Mathematics inscribes being as such [Badiou] |
Full Idea: Mathematics inscribes being as such. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.12) | |
A reaction: I don't pretend to understand that, but there is something about the purity and certainty of mathematics that makes us feel we are grappling with the core of existence. Perhaps. The same might be said of stubbing your toe on a bedpost. |
12323 | Existence is Being itself, but only as our thought decides it [Badiou] |
Full Idea: Existence is precisely Being itself in as much as thought decides it. And that decision orients thought essentially. ...It is when you decide upon what exists that you bind your thought to Being. | |
From: Alain Badiou (Briefings on Existence [1998], 2) | |
A reaction: [2nd half p.57] Helpful for us non-Heideggerians to see what is going on. Does this mean that Being is Kant's noumenon? |
12326 | The primitive name of Being is the empty set; in a sense, only the empty set 'is' [Badiou] |
Full Idea: In Set Theory, the primitive name of Being is the void, the empty set. The whole hierarchy takes root in it. In a certain sense, it alone 'is'. | |
From: Alain Badiou (Briefings on Existence [1998], 6) | |
A reaction: This is the key to Badiou's view that ontology is mathematics. David Lewis pursued interesting enquiries in this area. |
12332 | The modern view of Being comes when we reject numbers as merely successions of One [Badiou] |
Full Idea: The saturation and collapse of the Euclidean idea of the being of number as One's procession signs the entry of the thought of Being into modern times. | |
From: Alain Badiou (Briefings on Existence [1998], 11) | |
A reaction: That is, by allowing that not all numbers are built of units, numbers expand widely enough to embrace everything we think of as Being. The landmark event is the acceptance of the infinite as a number. |
9811 | It is of the essence of being to appear [Badiou] |
Full Idea: It is of the essence of being to appear. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.16) | |
A reaction: Nice slogan. In my humble opinion 'continental' philosophy is well worth reading because, despite the fluffy rhetoric and the shameless egotism and the desire to shock the bourgeoisie, they occasionally make wonderfully thought-provoking remarks. |
12320 | Ontology is (and always has been) Cantorian mathematics [Badiou] |
Full Idea: Enlightened by the Cantorian grounding of mathematics, we can assert ontology to be nothing other than mathematics itself. This has been the case ever since its Greek origin. | |
From: Alain Badiou (Briefings on Existence [1998], 1) | |
A reaction: There seems to be quite a strong feeling among mathematicians that new 'realms of being' are emerging from their researches. Only a Platonist, of course, is likely to find this idea sympathetic. |
9389 | Vague membership of sets is possible if the set is defined by its concept, not its members [Rumfitt] |
Full Idea: Vagueness in respect of membership is consistency with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of a concept. | |
From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.5) | |
A reaction: I find this view of sets much more appealing than the one that identifies a set with its members. The empty set is less of a problem, as well as non-existents. Logicians prefer the extensional view because it is tidy. |
18839 | An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases [Rumfitt] |
Full Idea: A borderline red-orange object satisfies the disjunctive predicate 'red or orange', even though it satisfies neither 'red' or 'orange'. When applied to adjacent bands of colour, the disjunction 'sweeps up' objects which are reddish-orange. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5) | |
A reaction: Rumfitt offers a formal principle in support of this. There may be a problem with 'adjacent'. Different colour systems will place different colours adjacent to red. In other examples the idea of 'adjacent' may make no sense. Rumfitt knows this! |
18838 | The extension of a colour is decided by a concept's place in a network of contraries [Rumfitt] |
Full Idea: On Sainsbury's picture, a colour has an extension that it has by virtue of its place in a network of contrary colour classifications. Something is determined to be 'red' by being a colour incompatible with orange, yellow, green, blue, indigo and violet. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5) | |
A reaction: Along with Idea 18839, this gives quite a nice account of vagueness, by requiring a foil to the vague predicate, and using the disjunction of the predicate and its foil to handle anything caught in between them. |
14532 | A distinctive type of necessity is found in logical consequence [Rumfitt, by Hale/Hoffmann,A] |
Full Idea: Rumfitt argues that there is a distinctive notion of necessity implicated in the notion of logical consequence. | |
From: report of Ian Rumfitt (Logical Necessity [2010]) by Bob Hale/ Aviv Hoffmann - Introduction to 'Modality' 2 |
18816 | Metaphysical modalities respect the actual identities of things [Rumfitt] |
Full Idea: The central characteristic mark of metaphysical necessity is that a metaphysical possibility respects the actual identities of things - in a capacious sense of 'thing'. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.4) | |
A reaction: He contrast this with logical necessity, and concludes that some truths are metaphysically but not logically necessary, such as 'Hesperus is identical with Phosphorus'. Personally I like the idea of a 'necessity-maker', so that fits. |
12200 | A logically necessary statement need not be a priori, as it could be unknowable [Rumfitt] |
Full Idea: There is no reason to suppose that any statement that is logically necessary (in the present sense) is knowable a priori. ..If a statement is logically necessary, its negation will yield a contradiction, but that does not imply that someone could know it. | |
From: Ian Rumfitt (Logical Necessity [2010], §2) | |
A reaction: This remark is aimed at Dorothy Edgington, who holds the opposite view. Rumfitt largely defends McFetridge's view (q.v.). |
12193 | Logical necessity is when 'necessarily A' implies 'not-A is contradictory' [Rumfitt] |
Full Idea: By the notion of 'logical necessity' I mean that there is a sense of 'necessary' for which 'It is necessary that A' implies and is implied by 'It is logically contradictory that not A'. ...From this, logical necessity is implicated in logical consequence. | |
From: Ian Rumfitt (Logical Necessity [2010], Intro) | |
A reaction: Rumfitt expresses a commitment to classical logic at this point. We will need to be quite sure what we mean by 'contradiction', which will need a clear notion of 'truth'.... |
12202 | Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not [Rumfitt] |
Full Idea: While Fine suggests defining a narrow notion of logical necessity in terms of metaphysical necessity by 'restriction' (to logical truths that can be defined in non-modal terms), this seems unpromising for broad logical necessity, which is modal. | |
From: Ian Rumfitt (Logical Necessity [2010], §2) | |
A reaction: [compressed] He cites Kit Fine 2002. Rumfitt glosses the non-modal definitions as purely formal. The metaphysics lurks somewhere in the proof. |
18825 | S5 is the logic of logical necessity [Rumfitt] |
Full Idea: I accept the widely held thesis that S5 is the logic of logical necessity. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4 n16) | |
A reaction: It seems plausible that S5 is also the logic of metaphysical necessity, but that does not make them the same thing. The two types of necessity have two different grounds. |
18828 | If two possibilities can't share a determiner, they are incompatible [Rumfitt] |
Full Idea: Two possibilities are incompatible when no possibility determines both. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1) | |
A reaction: This strikes me as just the right sort of language for building up a decent metaphysical picture of the world, which needs to incorporate possibilities as well as actualities. |
18824 | Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right [Rumfitt] |
Full Idea: Some philosophers describe the colour scarlet as a determination of the determinable red; since the ways the world might be are naturally taken to be properties of the world, it helps to bear this analogy in mind. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4) | |
A reaction: This fits nicely with the disposition accounts of modality which I favour. Hence being 'coloured' is a real property of objects, even in the absence of the name of its specific colour. |
12203 | If a world is a fully determinate way things could have been, can anyone consider such a thing? [Rumfitt] |
Full Idea: A world is usually taken to be a fully determinate way that things could have been; but then one might seriously wonder whether anyone is capable of 'considering' such a thing at all. | |
From: Ian Rumfitt (Logical Necessity [2010], §4) | |
A reaction: This has always worried me. If I say 'maybe my coat is in the car', I would hate to think that I had to be contemplating some entire possible world (including all the implications of my coat not being on the hat stand). |
18821 | Possibilities are like possible worlds, but not fully determinate or complete [Rumfitt] |
Full Idea: Possibilities are things of the same general character as possible worlds, on one popular conception of the latter. They differ from worlds, though, in that they are not required to be fully determinate or complete. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6) | |
A reaction: A rather promising approach to such things, even though a possibility is fairly determinate at its core, but very vague at the edges. It is possible that the UK parliament might be located in Birmingham, for example. Is this world 'complete'? |
18831 | Medieval logicians said understanding A also involved understanding not-A [Rumfitt] |
Full Idea: Mediaeval logicians had a principle, 'Eadem est scientia oppositorum': in order to attain a clear conception of what it is for A to be the case, one needs to attain a conception of what it is for A not to be the case. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2) | |
A reaction: Presumably 'understanding' has to be a fairly comprehensive grasp of the matter, so understanding the negation sounds like a reasonable requirement for the real thing. |
18820 | In English 'evidence' is a mass term, qualified by 'little' and 'more' [Rumfitt] |
Full Idea: In English, the word 'evidence' behaves as a mass term: we speak of someone's having little evidence for an assertion, and of one thinker's having more evidence than another for a claim. One the other hand, we also speak of 'pieces' of evidence. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.2) | |
A reaction: And having 'more' evidence does not mean having a larger number of pieces of evidence, so it really is like an accumulated mass. |
18817 | We understand conditionals, but disagree over their truth-conditions [Rumfitt] |
Full Idea: It is striking that our understanding of conditionals is not greatly impeded by widespread disagreement about their truth-conditions. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 4.2) | |
A reaction: Compare 'if you dig there you might find gold' with 'if you dig there you will definitely find gold'. The second but not the first invites 'how do you know that?', implying truth. Two different ifs. |
12338 | We must either assert or deny any single predicate of any single subject [Badiou] |
Full Idea: There can be nothing intermediate to an assertion and a denial. We must either assert or deny any single predicate of any single subject. | |
From: Alain Badiou (Briefings on Existence [1998], 1011b24) | |
A reaction: The first sentence seems to be bivalence, and the second sentence excluded middle. |
11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
Full Idea: The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A. | |
From: Ian Rumfitt ("Yes" and "No" [2000], IV) | |
A reaction: [compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role). |
18829 | The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A [Rumfitt] |
Full Idea: The truth-grounds of '¬A' are precisely those possibilities that are incompatible with any truth-ground of A. | |
From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1) | |
A reaction: This is Rumfitt's proposal for the semantics of 'not', based on the central idea of a possibility, rather than a possible world. The incompatibility tracks back to an absence of shared grounding. |
9814 | All great poetry is engaged in rivalry with mathematics [Badiou] |
Full Idea: Like every great poet, Mallarmé was engaged in a tacit rivalry with mathematics. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.20) | |
A reaction: I love these French pronouncements! Would Mallarmé have agreed? If poetry and mathematics are the poles, where is philosophy to be found? |
12316 | For Enlightenment philosophers, God was no longer involved in politics [Badiou] |
Full Idea: For the philosophers of the Enlightenment politics is strictly the affair of humankind, an immanent practice from which recourse to the All Mighty's providential organization had to be discarded. | |
From: Alain Badiou (Briefings on Existence [1998], Prol) |
9287 | Bruno said that ancient Egyptian magic was the true religion [Bruno, by Yates] |
Full Idea: Giordano Bruno maintained that the magical Egyptian religion of the world was not only the most ancient but also the only true religion, which both Judaism and Christianity had obscured and corrupted. | |
From: report of Giordano Bruno (works [1590]) by Frances A. Yates - Giordano Bruno and Hermetic Tradition Ch.1 | |
A reaction: His beliefs were based on the Hermetic writings. No wonder he was burned at the stake. Atheists now lay flowers at his memorial in Rome. The sixteenth century was when the hunt for alternatives to established religion began. |
12317 | The God of religion results from an encounter, not from a proof [Badiou] |
Full Idea: The God of metaphysics makes sense of existing according to a proof, while the God of religion makes sense of living according to an encounter | |
From: Alain Badiou (Briefings on Existence [1998], Prol) |