37 ideas
| 4036 | What matters is not how many entities we postulate, but how many kinds of entities [Armstrong, by Mellor/Oliver] |
| 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] |
| 15754 | Without properties we would be unable to express the laws of nature [Armstrong] |
| 4034 | Whether we apply 'cold' or 'hot' to an object is quite separate from its change of temperature [Armstrong] |
| 8535 | To the claim that every predicate has a property, start by eliminating failure of application of predicate [Armstrong] |
| 8537 | Tropes fall into classes, because exact similarity is symmetrical and transitive [Armstrong] |
| 8538 | Trope theory needs extra commitments, to symmetry and non-transitivity, unless resemblance is exact [Armstrong] |
| 8539 | Universals are required to give a satisfactory account of the laws of nature [Armstrong] |
| 8529 | Deniers of properties and relations rely on either predicates or on classes [Armstrong] |
| 8532 | Resemblances must be in certain 'respects', and they seem awfully like properties [Armstrong] |
| 8530 | Change of temperature in objects is quite independent of the predicates 'hot' and 'cold' [Armstrong] |
| 8536 | We want to know what constituents of objects are grounds for the application of predicates [Armstrong] |
| 8531 | In most sets there is no property common to all the members [Armstrong] |
| 15753 | Essences might support Resemblance Nominalism, but they are too coarse and ill-defined [Armstrong] |
| 8533 | Predicates need ontological correlates to ensure that they apply [Armstrong] |
| 4035 | There must be some explanation of why certain predicates are applicable to certain objects [Armstrong] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| 8541 | Regularities theories are poor on causal connections, counterfactuals and probability [Armstrong] |
| 8540 | The introduction of sparse properties avoids the regularity theory's problem with 'grue' [Armstrong] |