43 ideas
| 12619 | We have no successful definitions, because they all use indefinable words [Fodor] |
| 8820 | Rules of reasoning precede the concept of truth, and they are what characterize it [Pollock] |
| 8819 | We need the concept of truth for defeasible reasoning [Pollock] |
| 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] |
| 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] |
| 12620 | If 'exist' is ambiguous in 'chairs and numbers exist', that mirrors the difference between chairs and numbers [Fodor] |
| 12613 | Empiricists use dispositions reductively, as 'possibility of sensation' or 'possibility of experimental result' [Fodor] |
| 8822 | Statements about necessities need not be necessarily true [Pollock] |
| 8818 | Defeasible reasoning requires us to be able to think about our thoughts [Pollock] |
| 12617 | Associationism can't explain how truth is preserved [Fodor] |
| 8811 | What we want to know is - when is it all right to believe something? [Pollock] |
| 8817 | Logical entailments are not always reasons for beliefs, because they may be irrelevant [Pollock] |
| 8814 | Epistemic norms are internalised procedural rules for reasoning [Pollock] |
| 8823 | Reasons are always for beliefs, but a perceptual state is a reason without itself being a belief [Pollock] |
| 8813 | If we have to appeal explicitly to epistemic norms, that will produce an infinite regress [Pollock] |
| 8812 | Norm Externalism says norms must be internal, but their selection is partly external [Pollock] |
| 8816 | Externalists tend to take a third-person point of view of epistemology [Pollock] |
| 8815 | Belief externalism is false, because external considerations cannot be internalized for actual use [Pollock] |
| 12615 | Mental representations are the old 'Ideas', but without images [Fodor] |
| 6650 | Fodor is now less keen on the innateness of concepts [Fodor, by Lowe] |
| 12618 | It is essential to the concept CAT that it be satisfied by cats [Fodor] |
| 12614 | I prefer psychological atomism - that concepts are independent of epistemic capacities [Fodor] |
| 12621 | Definable concepts have constituents, which are necessary, individuate them, and demonstrate possession [Fodor] |
| 12622 | Many concepts lack prototypes, and complex prototypes aren't built from simple ones [Fodor] |
| 12623 | The theory theory can't actually tell us what concepts are [Fodor] |
| 12616 | English has no semantic theory, just associations between sentences and thoughts [Fodor] |