12 ideas
| 9987 | An aggregate in which order does not matter I call a 'set' [Bolzano] |
| Full Idea: An aggregate whose basic conception renders the arrangement of its members a matter of indifference, and whose permutation therefore produces no essential difference, I call a 'set'. | |
| From: Bernard Bolzano (Paradoxes of the Infinite [1846], §4), quoted by William W. Tait - Frege versus Cantor and Dedekind IX | |
| A reaction: The idea of 'sets' was emerging before Cantor formalised it, and clarified it by thinking about infinite sets. Nowadays we also have 'ordered' sets, which rather contradicts Bolzano, and we also expect the cardinality to be determinate. |
| 10856 | A truly infinite quantity does not need to be a variable [Bolzano] |
| Full Idea: A truly infinite quantity (for example, the length of a straight line, unbounded in either direction) does not by any means need to be a variable. | |
| From: Bernard Bolzano (Paradoxes of the Infinite [1846]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable §10 | |
| A reaction: This is an important idea, followed up by Cantor, which relegated to the sidelines the view of infinity as simply something that could increase without limit. Personally I like the old view, but there is something mathematically stable about infinity. |
| 15990 | Every individual thing which exists has an essence, which is its internal constitution [Locke] |
| Full Idea: I take essences to be in everything that internal constitution or frame for the modification of substance, which God in his wisdom gives to every particular creature, when he gives it a being; and such essences I grant there are in all things that exist. | |
| From: John Locke (Letters to Edward Stillingfleet [1695], Letter 1), quoted by Simon Blackburn - Quasi-Realism no Fictionalism | |
| A reaction: This is the clearest statement I have found of Locke's commitment to essences, for all his doubts about whether we can know such things. Alexander says (ch.13) Locke was reacting against scholastic essence, as pertaining to species. |
| 15086 | Absolute necessity might be achievable either logically or metaphysically [Hale] |
| Full Idea: Maybe peaceful co-existence between absolute logical necessity and absolute metaphysical necessity can be secured, ..and absolute necessity is their union. ...However, a truth would then qualify as absolutely necessary in two quite different ways. | |
| From: Bob Hale (Absolute Necessities [1996], 4) | |
| A reaction: Hale is addressing a really big question for metaphysic (absolute necessity) which others avoid. In the end he votes for rejecting 'metaphysical' necessity. I am tempted to vote for rejecting logical necessity (as being relative). 'Absolute' is an ideal. |
| 8261 | Maybe not-p is logically possible, but p is metaphysically necessary, so the latter is not absolute [Hale] |
| Full Idea: It might be metaphysically necessary that p but logically possible that not-p, so that metaphysical necessity is not, after all, absolute. | |
| From: Bob Hale (Absolute Necessities [1996]), quoted by E.J. Lowe - The Possibility of Metaphysics 1.5 | |
| A reaction: Lowe presents this as dilemma, but it sounds fine to me. Flying pigs etc. have no apparent logical problems, but I can't conceive of a possible world where pigs like ours fly in a world like ours. Earthbound pigs may be metaphysically necessary. |
| 15081 | A strong necessity entails a weaker one, but not conversely; possibilities go the other way [Hale] |
| Full Idea: One type of necessity may be said to be 'stronger' than another when the first always entails the second, but not conversely. This will obtain only if the possibility of the first is weaker than the possibility of the second. | |
| From: Bob Hale (Absolute Necessities [1996], 1) | |
| A reaction: Thus we would normally say that if something is logically necessary (a very strong claim) then it will have to be naturally necessary. If something is naturally possible, then clearly it will have to be logically possible. Sounds OK. |
| 15080 | 'Relative' necessity is just a logical consequence of some statements ('strong' if they are all true) [Hale] |
| Full Idea: Necessity is 'relative' if a claim of φ-necessary that p just claims that it is a logical consequence of some statements Φ that p. We have a 'strong' version if we add that the statements in Φ are all true, and a 'weak' version if not. | |
| From: Bob Hale (Absolute Necessities [1996], 1) | |
| A reaction: I'm not sure about 'logical' consequence here. It may be necessary that a thing be a certain way in order to qualify for some category (which would be 'relative'), but that seems like 'sortal' necessity rather than logical. |
| 15082 | Metaphysical necessity says there is no possibility of falsehood [Hale] |
| Full Idea: Friends of metaphysical necessity would want to hold that when it is metaphysically necessary that p, there is no good sense of 'possible' (except, perhaps, an epistemic one) in which it is possible that not-p. | |
| From: Bob Hale (Absolute Necessities [1996], 2) | |
| A reaction: We might want to say which possible worlds this refers to (and presumably it won't just be in the actual world). The normal claim would refer to all possible worlds. Adding a '...provided that' clause moves it from absolute to relative necessity. |
| 15085 | 'Broadly' logical necessities are derived (in a structure) entirely from the concepts [Hale] |
| Full Idea: 'Broadly' logical necessities are propositions whose truth derives entirely from the concepts involved in them (together, of course, with relevant structure). | |
| From: Bob Hale (Absolute Necessities [1996], 3) | |
| A reaction: Is the 'logical' part of this necessity bestowed by the concepts, or by the 'structure' (which I take to be a logical structure)? |
| 15088 | Logical necessities are true in virtue of the nature of all logical concepts [Hale] |
| Full Idea: The logical necessities can be taken to be the propositions which are true in virtue of the nature of all logical concepts. | |
| From: Bob Hale (Absolute Necessities [1996], p.10) | |
| A reaction: This is part of his story of essences giving rise to necessities. His proposal sounds narrow, but logical concepts may have the highest degree of generality which it is possible to have. It must be how the concepts connect that causes the necessities. |
| 15087 | Conceptual necessities are made true by all concepts [Hale] |
| Full Idea: Conceptual necessities can be taken to be propositions which are true in virtue of the nature of all concepts. | |
| From: Bob Hale (Absolute Necessities [1996], p.9) | |
| A reaction: Fine endorse essences for these concepts. Could we then come up with a new concept which contradicted all the others, and destroyed the necessity? Yes, presumably. Presumably witchcraft and astrology are full of 'conceptual necessities'. |
| 15994 | If it is knowledge, it is certain; if it isn't certain, it isn't knowledge [Locke] |
| Full Idea: What reaches to knowledge, I think may be called certainty; and what comes short of certainty, I think cannot be knowledge. | |
| From: John Locke (Letters to Edward Stillingfleet [1695], Letter 2), quoted by Simon Blackburn - Quasi-Realism no Fictionalism | |
| A reaction: I much prefer that fallibilist approach offered by the pragmatists. Knowledge is well-supported belief which seems (and is agreed) to be true, but there is a small shadow of doubt hanging over all of it. |