24 ideas
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |
| A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |
| A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |
| A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |
| A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
| 14352 | '¬', '&', and 'v' are truth functions: the truth of the compound is fixed by the truth of the components [Jackson] |
| Full Idea: It is widely agreed that '¬', '&', and 'v' are 'truth functions': the truth value of a compound sentence formed using them is fully determined by the truth value or values of the component sentences. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: A candidate for not being a truth function might be a conditional →, where the arrow adds something over and above the propositions it connects. The relationship has an additional truth value? Does A depend on B? |
| 14360 | Possible worlds for subjunctives (and dispositions), and no-truth for indicatives? [Jackson] |
| Full Idea: Subjunctive conditionals are intimately connected with dispositional properties and causation. ...Consequently, a position some find attractive is that possible worlds theory applies to subjunctives, while the no-truth theory applies to indicatives. | |
| From: Frank Jackson (Conditionals [2006], 'Indicative') | |
| A reaction: My intuitions are to reject this and favour a unified account, where both sorts of conditionals are mappings of the relationships among the facts of actuality. Nice slogan! |
| 14353 | Modus ponens requires that A→B is F when A is T and B is F [Jackson] |
| Full Idea: Modus ponens is intuitively valid, but in A,A→B|B if A is true and B is false that must be because A→B is false. So A→B is false when A is true and B is false. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: This is his first step in showing how the truth functional account of A→B acquires its truth table. If you are giving up the truth functional view of conditionals, presumably you are not also going to give up modus ponens? |
| 14354 | When A and B have the same truth value, A→B is true, because A→A is a logical truth [Jackson] |
| Full Idea: (A→A) is a logical truth, so some conditionals with antecedent and consequent the same truth value are true. But if '→' is a truth function, that will be true for all cases. Hence whenever A and B are alike in truth value, (A→B) is true. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: His second step in demonstrating the truth table for →, assuming it is truth functional. |
| 14355 | (A&B)→A is a logical truth, even if antecedent false and consequent true, so it is T if A is F and B is T [Jackson] |
| Full Idea: (A&B)→A is a logical truth, but A can be true and B false, so that (A&B) is false. So some conditionals with false antecedent and true consequent are true. If → is a truth function, then whenever A is false and B is true (A→B) is true. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: This is his third and final step in showing the truth table of → if it is truth functional. |
| 14358 | In the possible worlds account of conditionals, modus ponens and modus tollens are validated [Jackson] |
| Full Idea: In the possible worlds account modus ponens is validated (the closest world, the actual, is a B-world just if B is true), and modus tollens is validated (if B is false, the actual world is not an A-world, so A is false). | |
| From: Frank Jackson (Conditionals [2006], 'Famous') | |
| A reaction: [see Jackson for slightly fuller versions] This looks like a minimal requirement for a decent theory of conditionals, so Jackson explains the attractions of the possible worlds view very persuasively. |
| 14359 | Only assertions have truth-values, and conditionals are not proper assertions [Jackson] |
| Full Idea: In the no-truth theory of conditionals they have justified assertion or acceptability conditions but not truth conditions. ...The motivation is that only assertions have truth values, and conditionals are arguments, not proper assertions. | |
| From: Frank Jackson (Conditionals [2006], 'No-truth') | |
| A reaction: Once I trim this idea down to its basics, it suddenly looks very persuasive. Except that I am inclined to think that conditional truths do state facts about the world - perhaps as facts about how more basic truths are related to each other. |
| 14357 | Possible worlds account, unlike A⊃B, says nothing about when A is false [Jackson] |
| Full Idea: In the possible worlds account of conditionals A⊃B is not sufficient for A→B. If A is false then A⊃B is true, but here nothing is implied about whether the world most like the actual world except that A is true is or is not a B-world. | |
| From: Frank Jackson (Conditionals [2006], 'Possible') | |
| A reaction: The possible worlds account seems to be built on Ramsey's idea of just holding A true and seeing what you get. Being committed to B being automatically true if A is false seems highly counterintuitive. |
| 14356 | We can't insist that A is relevant to B, as conditionals can express lack of relevance [Jackson] |
| Full Idea: One addition to the truth functional account of conditionals is that A be somehow relevant to B. However, sometimes we use conditionals to express lack of relevance, as in 'If Fred works he will fail, and if Fred doesn't work he will fail'. | |
| From: Frank Jackson (Conditionals [2006], 'Possible') | |
| A reaction: This certainly seems to put paid to an attractive instant solution to the problem. |
| 21833 | Research suggest that we overrate conscious experience [Flanagan] |
| Full Idea: The emerging consensus is that we probably overrate the power of conscious experience in our lives. Freud, of course, said the same thing for different reasons. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 3 'Ontology') | |
| A reaction: [He cites Pockett, Banks and Gallagher 2006]. Freud was concerned with big deep secrets, but the modern view concerns ordinary decisions and perceptions. An important idea, which should incline us all to become Nietzscheans. |
| 21834 | Sensations may be identical to brain events, but complex mental events don't seem to be [Flanagan] |
| Full Idea: There is still some hope for something like identity theory for sensations. But almost no one believes that strict identity theory will work for more complex mental states. Strict identity is stronger than type neurophysicalism. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 3 'Ontology') | |
| A reaction: It is so hard to express the problem. What needs to be explained? How can one bunch of neurons represent many different things? It's not like computing. That just transfers the data to brains, where the puzzling stuff happens. |
| 21837 | Morality is normative because it identifies best practices among the normal practices [Flanagan] |
| Full Idea: Morality is 'normative' in the sense that it consists of the extraction of ''good' or 'excellent' practices from common practices. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 4 'Naturalism') |
| 21830 | For Darwinians, altruism is either contracts or genetics [Flanagan] |
| Full Idea: Two explanations came forward in the neo-Darwinian synthesis. Altruism is either 1) person-based reciprocal altruism, or 2) gene-based kin altruism. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 2 'Darwin') | |
| A reaction: Flanagan obviously thinks there is also 'genuine psychological atruism'. Presumably we don't explain mathematics or music or the desire to travel as either contracts or genetics, so we have other explanations available. |
| 21835 | We need Eudaimonics - the empirical study of how we should flourish [Flanagan] |
| Full Idea: It would be nice if I could advance the case for Eudaimonics - empirical enquiry into the nature, causes, and constituents of flourishing, …and the case for some ways of living and being as better than others. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 4 'Normative') | |
| A reaction: Things seem to be moving in that direction. Lots of statistics about happiness have been appearing. |
| 21831 | Alienation is not finding what one wants, or being unable to achieve it [Flanagan] |
| Full Idea: What Marx called 'alienation' is the widespread condition of not being able to discover what one wants, or not being remotely positioned to achieve. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 2 'Expanding') | |
| A reaction: I took alienation to concern people's relationship to the means of production in their trade. On Flanagan's definition I would expect almost everyone aged under 20 to count as alienated. |
| 21832 | Buddhists reject God and the self, and accept suffering as key, and liberation through wisdom [Flanagan] |
| Full Idea: Buddhism rejected the idea of a creator God, and the unchanging self [atman]. They accept the appearance-reality distinction, reward for virtue [karma], suffering defining our predicament, and that liberation [nirvana] is possible through wisdom. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 3 'Buddhism') | |
| A reaction: [Compressed] Flanagan is an analytic philosopher and a practising Buddhist. Looking at a happiness map today which shows Europeans largely happy, and Africans largely miserable, I can see why they thought suffering was basic. |