47 ideas
| 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] |
| 15845 | It seems absurd that seeing a person's limbs, the one is many, and yet the many are one [Plato] |
| 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] |
| 9867 | It is absurd to define a circle, but not be able to recognise a real one [Plato] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 15857 | Any mixture which lacks measure and proportion doesn't even count as a mixture at all [Plato] |
| 14503 | If a mixture does not contain measure and proportion, it is corrupted and destroyed [Plato] |
| 4447 | If the good is one, is it unchanged when it is in particulars, and is it then separated from itself? [Plato] |
| 15856 | A thing can become one or many, depending on how we talk about it [Plato] |
| 374 | If one object is divided into its parts, someone can then say that one are many and many is one [Plato] |
| 389 | How can you be certain about aspects of the world if they aren't constant? [Plato] |
| 390 | If goodness involves moderation and proportion, then it seems to be found in beauty [Plato] |
| 391 | The good involves beauty, proportion and truth [Plato] |
| 392 | Neither intellect nor pleasure are the good, because they are not perfect and self-sufficient [Plato] |
| 393 | Good first, then beauty, then reason, then knowledge, then pleasure [Plato, by PG] |
| 385 | Some of the pleasures and pains we feel are false [Plato] |
| 387 | A small pure pleasure is much finer than a large one contaminated with pain [Plato] |
| 373 | Pleasure is certainly very pleasant, but it doesn't follow that all pleasures are good [Plato] |
| 379 | The good must be sufficient and perfect, and neither intellect nor pleasure are that [Plato] |
| 371 | Reason, memory, truth and wisdom are far better than pleasure, for those who can attain them [Plato] |
| 376 | Would you prefer a life of pleasure without reason, or one of reason without pleasure? [Plato] |
| 382 | It is unlikely that the gods feel either pleasure or pain [Plato] |
| 381 | We feel pleasure when we approach our natural state of harmony [Plato] |
| 386 | Intense pleasure and pain are not felt in a good body, but in a worthless one [Plato] |
| 388 | Hedonists must say that someone in pain is bad, even if they are virtuous [Plato] |
| 377 | If you lived a life of maximum pleasure, would you still be lacking anything? [Plato] |
| 378 | A life of pure pleasure with no intellect is the life of a jellyfish [Plato] |
| 8412 | A causal interaction is when two processes intersect, and correlated modifications persist afterwards [Salmon] |
| 8413 | Cause must come first in propagations of causal interactions, but interactions are simultaneous [Salmon] |
| 8411 | Instead of localised events, I take enduring and extended processes as basic to causation [Salmon] |