37 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] |
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] |
16665 | There are entities, and then positive 'modes', modifying aspects outside the thing's essence [Suárez] |
16666 | A mode determines the state and character of a quantity, without adding to it [Suárez] |
16667 | Substances are incomplete unless they have modes [Suárez, by Pasnau] |
17007 | Forms must rule over faculties and accidents, and are the source of action and unity [Suárez] |
16780 | Partial forms of leaf and fruit are united in the whole form of the tree [Suárez] |
16758 | The best support for substantial forms is the co-ordinated unity of a natural being [Suárez] |
16743 | We can get at the essential nature of 'quantity' by knowing bulk and extension [Suárez] |
16742 | We only know essences through non-essential features, esp. those closest to the essence [Suárez] |
22143 | Identity does not exclude possible or imagined difference [Suárez, by Boulter] |
22146 | Minor Real distinction: B needs A, but A doesn't need B [Suárez, by Boulter] |
22145 | Major Real distinction: A and B have independent existences [Suárez, by Boulter] |
22144 | Real Essential distinction: A and B are of different natural kinds [Suárez, by Boulter] |
22147 | Conceptual/Mental distinction: one thing can be conceived of in two different ways [Suárez, by Boulter] |
22148 | Modal distinction: A isn't B or its property, but still needs B [Suárez, by Boulter] |
22149 | Scholastics assess possibility by what has actually happened in reality [Suárez, by Boulter] |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
16682 | Other things could occupy the same location as an angel [Suárez] |