45 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] |
7689 | The modal logic of C.I.Lewis was only interpreted by Kripke and Hintikka in the 1960s [Jacquette] |
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] |
7681 | Logic describes inferences between sentences expressing possible properties of objects [Jacquette] |
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] |
7682 | Logic is not just about signs, because it relates to states of affairs, objects, properties and truth-values [Jacquette] |
7697 | On Russell's analysis, the sentence "The winged horse has wings" comes out as false [Jacquette] |
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] |
7701 | Can a Barber shave all and only those persons who do not shave themselves? [Jacquette] |
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] |
7707 | To grasp being, we must say why something exists, and why there is one world [Jacquette] |
7692 | Being is maximal consistency [Jacquette] |
7687 | Existence is completeness and consistency [Jacquette] |
7679 | Ontology is the same as the conceptual foundations of logic [Jacquette] |
7678 | Ontology must include the minimum requirements for our semantics [Jacquette] |
7683 | Logic is based either on separate objects and properties, or objects as combinations of properties [Jacquette] |
7684 | Reduce states-of-affairs to object-property combinations, and possible worlds to states-of-affairs [Jacquette] |
7703 | If classes can't be eliminated, and they are property combinations, then properties (universals) can't be either [Jacquette] |
7685 | An object is a predication subject, distinguished by a distinctive combination of properties [Jacquette] |
7699 | Numbers, sets and propositions are abstract particulars; properties, qualities and relations are universals [Jacquette] |
7691 | The actual world is a consistent combination of states, made of consistent property combinations [Jacquette] |
7688 | The actual world is a maximally consistent combination of actual states of affairs [Jacquette] |
7695 | Do proposition-structures not associated with the actual world deserve to be called worlds? [Jacquette] |
7694 | We must experience the 'actual' world, which is defined by maximally consistent propositions [Jacquette] |
7706 | If qualia supervene on intentional states, then intentional states are explanatorily fundamental [Jacquette] |
7704 | Reduction of intentionality involving nonexistent objects is impossible, as reduction must be to what is actual [Jacquette] |
7702 | The extreme views on propositions are Frege's Platonism and Quine's extreme nominalism [Jacquette] |
23569 | The only just cause for a war is a wrong received [Vitoria] |
23601 | Leaders can only attack foreigners who have done wrong (as with their own subjects) [Vitoria] |
23566 | Princes should not justify a war to their subjects, and doing so would undermine the state [Vitoria] |
23576 | Sacking a city is lawful if it motivates the attacking troops [Vitoria] |