display all the ideas for this combination of texts
8 ideas
13010 | In order to select the logic justified by experience, we would need to use a lot of logic [Boghossian on Quine] |
Full Idea: Quine ends up with the logic that is maximally justified by experience, ...but a large number of the core principles of logic will have to be used to select the logic that is maximally justified by experience. | |
From: comment on Willard Quine (Carnap and Logical Truth [1954]) by Paul Boghossian - Knowledge of Logic p.233 | |
A reaction: In order to grasp some core principles of logic, you will probably need a certain amount of experience. I take logic to be an abstracted feature of reality (unless it is extended by pure fictions). Some basic logic may be hard wired in us. |
9002 | Elementary logic requires truth-functions, quantifiers (and variables), identity, and also sets of variables [Quine] |
Full Idea: Elementary logic, as commonly systematized nowadays, comprises truth-function theory (involving 'or', 'and', 'not' etc.), quantifiers (and their variables), and identity theory ('='). In addition, set theory requires classes among values of variables. | |
From: Willard Quine (Carnap and Logical Truth [1954], II) | |
A reaction: Quine is famous for trying to squeeze properties out of the picture, which would then block higher-order logics (which quantify over properties). Quine's list gives a nice programme for a student of the philosophy of logic to understand. |
13681 | Logical consequence is marked by being preserved under all nonlogical substitutions [Quine, by Sider] |
Full Idea: Quine's view of logical consequence is that it is when there is no way of uniformly substituting nonlogical expressions in the premises and consequences so that the premises all remain true but the consequence now becomes false. | |
From: report of Willard Quine (Carnap and Logical Truth [1954], p.103) by Theodore Sider - Logic for Philosophy 1.5 | |
A reaction: One might just say that the consequence holds if you insert consistent variables for the nonlogical terms, which looks like Aristotle's view of the matter. |
13829 | If logical truths essentially depend on logical constants, we had better define the latter [Hacking on Quine] |
Full Idea: Quine said a logical truth is a truth in which only logical constants occur essentially, ...but then a fruitful definition of 'logical constant' is called for. | |
From: comment on Willard Quine (Carnap and Logical Truth [1954]) by Ian Hacking - What is Logic? §02 |
10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
A reaction: [Smith summarises the diagonal argument] |
9003 | Set theory was struggling with higher infinities, when new paradoxes made it baffling [Quine] |
Full Idea: Unlike elementary logic, the truths of set theory are not obvious. Set theory was straining at the leash of intuition ever since Cantor discovered higher infinites; and with the added impetus of the paradoxes of set theory the leash snapped. | |
From: Willard Quine (Carnap and Logical Truth [1954], II) | |
A reaction: This problem seems to have forced Quine into platonism about sets, because he felt they were essential for mathematics and science, but couldn't be constructed with precision. So they must be real, but we don't quite understand them. |
13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
From: report of George Cantor (works [1880]) by Michčle Friend - Introducing the Philosophy of Mathematics | |
A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |