38 ideas
| 17729 | Examining concepts can recover information obtained through the senses [Jenkins] |
| 17740 | Instead of correspondence of proposition to fact, look at correspondence of its parts [Jenkins] |
| 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] |
| 17730 | Combining the concepts of negation and finiteness gives the concept of infinity [Jenkins] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 17719 | Arithmetic concepts are indispensable because they accurately map the world [Jenkins] |
| 17717 | Senses produce concepts that map the world, and arithmetic is known through these concepts [Jenkins] |
| 17724 | It is not easy to show that Hume's Principle is analytic or definitive in the required sense [Jenkins] |
| 17727 | We can learn about the world by studying the grounding of our concepts [Jenkins] |
| 17720 | There's essential, modal, explanatory, conceptual, metaphysical and constitutive dependence [Jenkins, by PG] |
| 17728 | The concepts we have to use for categorising are ones which map the real world well [Jenkins] |
| 17726 | Examining accurate, justified or grounded concepts brings understanding of the world [Jenkins] |
| 17734 | It is not enough that intuition be reliable - we need to know why it is reliable [Jenkins] |
| 17723 | Knowledge is true belief which can be explained just by citing the proposition believed [Jenkins] |
| 17718 | Grounded concepts are trustworthy maps of the world [Jenkins] |
| 17739 | The physical effect of world on brain explains the concepts we possess [Jenkins] |
| 17731 | Verificationism is better if it says meaningfulness needs concepts grounded in the senses [Jenkins] |
| 17732 | Success semantics explains representation in terms of success in action [Jenkins] |
| 17725 | 'Analytic' can be conceptual, or by meaning, or predicate inclusion, or definition... [Jenkins] |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| 6610 | I believe because it is absurd [Tertullian] |