38 ideas
| 3123 | Science is in the business of carving nature at the joints [Segal] |
| 3125 | Psychology studies the way rationality links desires and beliefs to causality [Segal] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 10247 | We have no adequate logic at the moment, so mathematicians must create one [Veblen] |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 3105 | Is 'Hesperus = Phosphorus' metaphysically necessary, but not logically or epistemologically necessary? [Segal] |
| 3106 | If claims of metaphysical necessity are based on conceivability, we should be cautious [Segal] |
| 3113 | The success and virtue of an explanation do not guarantee its truth [Segal] |
| 3112 | Folk psychology is ridiculously dualist in its assumptions [Segal] |
| 3110 | Humans are made of H2O, so 'twins' aren't actually feasible [Segal] |
| 3124 | Externalists can't assume old words refer to modern natural kinds [Segal] |
| 3108 | If 'water' has narrow content, it refers to both H2O and XYZ [Segal] |
| 3109 | If content is external, so are beliefs and desires [Segal] |
| 3117 | Concepts can survive a big change in extension [Segal] |
| 3116 | Maybe experts fix content, not ordinary users [Segal] |
| 3104 | Must we relate to some diamonds to understand them? [Segal] |
| 3111 | Externalism can't explain concepts that have no reference [Segal] |
| 3103 | Maybe content involves relations to a language community [Segal] |
| 3121 | If content is narrow, my perfect twin shares my concepts [Segal] |
| 3118 | If thoughts ARE causal, we can't explain how they cause things [Segal] |
| 3119 | Even 'mass' cannot be defined in causal terms [Segal] |