32 ideas
| 19306 | It is a principle of reasoning not to clutter your mind with trivialities [Harman] |
| 19304 | The rules of reasoning are not the rules of logic [Harman] |
| 19307 | If there is a great cost to avoiding inconsistency, we learn to reason our way around it [Harman] |
| 19309 | Logic has little relevance to reasoning, except when logical conclusions are immediate [Harman] |
| 19303 | Implication just accumulates conclusions, but inference may also revise our views [Harman] |
| 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] |
| 19305 | The Gambler's Fallacy (ten blacks, so red is due) overemphasises the early part of a sequence [Harman] |
| 19310 | High probability premises need not imply high probability conclusions [Harman] |
| 19308 | We strongly desire to believe what is true, even though logic does not require it [Harman] |
| 19311 | In revision of belief, we need to keep track of justifications for foundations, but not for coherence [Harman] |
| 19312 | Coherence is intelligible connections, especially one element explaining another [Harman] |
| 5049 | Intelligent pleasure is the perception of beauty, order and perfection [Leibniz] |
| 5048 | Perfection is simply quantity of reality [Leibniz] |
| 5050 | Evil serves a greater good, and pain is necessary for higher pleasure [Leibniz] |