30 ideas
| 12204 | The logic of metaphysical necessity is S5 [Rumfitt] |
| 12195 | Soundness in argument varies with context, and may be achieved very informally indeed [Rumfitt] |
| 12199 | There is a modal element in consequence, in assessing reasoning from suppositions [Rumfitt] |
| 12201 | We reject deductions by bad consequence, so logical consequence can't be deduction [Rumfitt] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 12194 | Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B' [Rumfitt] |
| 12198 | Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths) [Rumfitt] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| 14532 | A distinctive type of necessity is found in logical consequence [Rumfitt, by Hale/Hoffmann,A] |
| 12193 | Logical necessity is when 'necessarily A' implies 'not-A is contradictory' [Rumfitt] |
| 12200 | A logically necessary statement need not be a priori, as it could be unknowable [Rumfitt] |
| 12202 | Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not [Rumfitt] |
| 12203 | If a world is a fully determinate way things could have been, can anyone consider such a thing? [Rumfitt] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 22375 | Moral judgements need more than the relevant facts, if the same facts lead to 'x is good' and 'x is bad' [Foot] |
| 22378 | We can't affirm a duty without saying why it matters if it is not performed [Foot] |
| 22377 | Whether someone is rude is judged by agreed criteria, so the facts dictate the value [Foot] |
| 22376 | Facts and values are connected if we cannot choose what counts as evidence of rightness [Foot] |