34 ideas
1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
13520 | A 'tautology' must include connectives [Wolf,RS] |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [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] |
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] |
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] |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [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] |
3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
19682 | Internalists are much more interested in evidence than externalists are [McGrew] |
19684 | Does spotting a new possibility count as evidence? [McGrew] |
19687 | Absence of evidence proves nothing, and weird claims need special evidence [McGrew] |
19688 | Every event is highly unlikely (in detail), but may be perfectly plausible [McGrew] |
19686 | Criminal law needs two separate witnesses, but historians will accept one witness [McGrew] |
19680 | Maybe all evidence consists of beliefs, rather than of facts [McGrew] |
19681 | If all evidence is propositional, what is the evidence for the proposition? Do we face a regress? [McGrew] |
19689 | Several unreliable witnesses can give good support, if they all say the same thing [McGrew] |
19683 | Narrow evidentialism relies wholly on propositions; the wider form includes other items [McGrew] |
19685 | Falsificationism would be naive if even a slight discrepancy in evidence killed a theory [McGrew] |
1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |