display all the ideas for this combination of texts
3 ideas
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| Full Idea: First-order logic is not decidable. That is, there is no test which can be applied to any arbitrary formula of that logic and which will tell one whether the formula is or is not valid (as proved by Church in 1936). | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| Full Idea: From the fact that the usual rules for first-level logic are complete (as proved by Gödel 1930), it follows that this logic is 'compact'. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) | |
| A reaction: The point is that the completeness requires finite proofs. |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
| A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |