6 ideas
| 15380 | Barcan:nothing comes into existence; Converse:nothing goes out; Both:domain is unchanging [Vervloesem] |
| Full Idea: Intuitively, the Barcan formula says that nothing comes into existence when moving from a possible world to an alternative world. The converse says that nothing goes out of existence. Together they say the domain of quantification is fixed for all worlds. | |
| From: Koen Vervloesem (Barcan Formulae [2010]) | |
| A reaction: Stated so clearly, they sound absurd. The sensible idea, I suppose, is that you can refer to all the things from any world, but that doesn't mean they are possible. Shades of Meinong. 'Square circles' are not possible. |
| 8758 | We could talk of open sentences, instead of sets [Chihara, by Shapiro] |
| Full Idea: Chihara's programme is to replace talk of sets with talk of open sentences. Instead of speaking of the set of all cats, we talk about the open sentence 'x is a cat'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: As Shapiro points out, this is following up Russell's view that sets should be replaced with talk of properties. Chihara is expressing it more linguistically. I'm in favour of any attempt to get rid of sets. |
| 10265 | Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro] |
| Full Idea: Chihara's system is a version of type theory. Translate thus: replace variables of sets of type n with level n variables over open sentences, replace membership/predication with satisfaction, and high quantifiers with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 |
| 8759 | We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro] |
| Full Idea: Chihara's system is similar to simple type theory; he replaces each type with variables over open sentences, replaces membership (or predication) with satisfaction, and replaces quantifiers over level 1+ variables with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: This is interesting for showing that type theory may not be dead. The revival of supposedly dead theories is the bread-and-butter of modern philosophy. |
| 10264 | Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro] |
| Full Idea: Chihara has proposal a modal primitive, a 'constructability quantifier'. Syntactically it behaves like an ordinary quantifier: Φ is a formula, and x a variable. Then (Cx)Φ is a formula, read as 'it is possible to construct an x such that Φ'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 | |
| A reaction: We only think natural numbers are infinite because we see no barrier to continuing to count, i.e. to construct new numbers. We accept reals when we know how to construct them. Etc. Sounds promising to me (though not to Shapiro). |
| 17555 | 'One' can mean undivided and not a multitude, or it can add measurement, giving number [Aquinas] |
| Full Idea: There are two sorts of one. There is the one which is convertible with being, which adds nothing to being except being undivided; and this deprives of multitude. Then there is the principle of number, which to the notion of being adds measurement. | |
| From: Thomas Aquinas (Quaestiones de Potentia Dei [1269], q3 a16 ad 3-um) | |
| A reaction: [From a lecture handout] I'm not sure I understand this. We might say, I suppose, that insofar as water is water, it is all one, but you can't count it. Perhaps being 'unified' and being a 'unity' are different? |