16 ideas
| 8868 | Objective truth arises from interpersonal communication [Davidson] |
| Full Idea: The source of the concept of objective truth is interpersonal communication. | |
| From: Donald Davidson (Three Varieties of Knowledge [1991], p.209) | |
| A reaction: This is a distinctively Davidsonian idea, arising out of Wittgenstein's Private Language Argument. We could go a step further, and just say that 'objectivity is a social concept'. Davidson more or less pleads guilty to pragmatism in this essay. |
| 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) |
| 13191 | The properties of a thing flow from its essence [Leibniz] |
| Full Idea: It is the same to look for perfection in an essence and in the properties that flow from an essence. | |
| From: Gottfried Leibniz (Letters to Wolff [1715], 1715.05.18) | |
| A reaction: It is helpful to have Leibniz spelling out his commitment to the traditional view of essence, as that from which the more evident properties flow. |
| 8867 | A belief requires understanding the distinctions of true-and-false, and appearance-and-reality [Davidson] |
| Full Idea: Having a belief demands in addition appreciating the contrast between true belief and false, between appearance and reality, mere seeming and being. | |
| From: Donald Davidson (Three Varieties of Knowledge [1991], p.209) | |
| A reaction: This sets the bar very high for belief (never mind knowledge), and seems to imply that animals don't have beliefs. How should we describe their cognitive states then? I would say these criteria only apply to actual knowledge. |
| 10347 | Objectivity is intersubjectivity [Davidson] |
| Full Idea: An entity is objective in so far as it is intersubjective. | |
| From: Donald Davidson (Three Varieties of Knowledge [1991]), quoted by Martin Kusch - Knowledge by Agreement Ch.10 | |
| A reaction: This thought baffled me until I saw it in the context of socialised epistemology. Effectively objectivity is subsumed under justification, which in turn is seen in a social context, not private to individuals. |
| 8866 | If we know other minds through behaviour, but not our own, we should assume they aren't like me [Davidson] |
| Full Idea: If the mental states of others are known only through their behavioral and other outward manifestations, while this is not true of our own mental states, why should we think our own mental states are anything like those of others? | |
| From: Donald Davidson (Three Varieties of Knowledge [1991], p.207) | |
| A reaction: His point is that if you seriously doubt other minds, you should follow through on the implications. But that is to treat it as a theory about other minds, rather an a sceptical worry. Descartes didn't walk into walls while writing Meditation 1. |
| 10346 | Knowing other minds rests on knowing both one's own mind and the external world [Davidson, by Dummett] |
| Full Idea: Davidson argues that knowledge of other minds presupposes knowledge of one's own mind, and that there is no knowledge of other minds without knowledge of the external world. | |
| From: report of Donald Davidson (Three Varieties of Knowledge [1991]) by Michael Dummett - Common Sense and Physics Ch.10 | |
| A reaction: Davidson't argument is actually hard to swallow because it is so long and complex. Compressing the point makes it begin to sound like a variant of the argument from analogy. |
| 8870 | Content of thought is established through communication, so knowledge needs other minds [Davidson] |
| Full Idea: Until a baseline has been established by communication with someone else, there is no point is saying one's own thoughts have a propositional content. Hence knowledge of another mind is essential all thought and all knowledge. | |
| From: Donald Davidson (Three Varieties of Knowledge [1991], p.213) | |
| A reaction: This really is building a skyscraper on the slightly shaky claims of the Private Language Argument (e.g. Idea 4158). Animals are so important in discussions of this kind. Is an albatross more or less devoid of thought and belief? |
| 8869 | The principle of charity attributes largely consistent logic and largely true beliefs to speakers [Davidson] |
| Full Idea: Concerning charity, the Principle of Coherence seeks logical consistency in the thought of the speaker, and the Principle of Correspondence seeks a similar response to features of the world to that of an interpreter. The speaker has logic and true belief. | |
| From: Donald Davidson (Three Varieties of Knowledge [1991], p.211) | |
| A reaction: Davidson adds a Kantian commitment to pure and universal reason to the very sceptical framework created by Quine. I agree with Davidson, but it seems more like faith than like an argument or an empirical observation. |