39 ideas
| 21021 | Keep premises as weak as possible, to avoid controversial difficulties [Nussbaum] |
| 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] |
| 6356 | Maybe a reliable justification must come from a process working with its 'proper function' [Plantinga, by Pollock/Cruz] |
| 21007 | Storytelling is never neutral; some features of the world must be emphasised [Nussbaum] |
| 21025 | Particularism gives no guidance for the future [Nussbaum] |
| 21026 | Compassion is unreliable, because it favours people close to us [Nussbaum] |
| 21019 | Social contracts assume equal powers among the participants [Nussbaum] |
| 21011 | We shouldn't focus on actual preferences, which may be distorted by injustices [Nussbaum] |
| 21008 | Liberalism does not need a comprehensive account of value [Nussbaum] |
| 21012 | Women are often treated like children, and not respected for their choices [Nussbaum] |
| 21015 | Negative liberty is incoherent; all liberties, to do and to be, require the prevention of interference [Nussbaum] |
| 21017 | Political freedom is an incoherent project, because some freedoms limit other freedoms [Nussbaum] |
| 21016 | Political and civil rights are not separate from economic and social rights [Nussbaum] |
| 21009 | Capabilities: Life, Health, Safety, Mental life, Love, Planning, Joining in, Nature, Play, Control [Nussbaum, by PG] |
| 21010 | Justice requires that the ten main capabilities of people are reasonably enabled [Nussbaum] |
| 21013 | Capabilities are grounded in bare humanity and agency; qualifying as rational is not needed [Nussbaum] |
| 21014 | Rights are not just barriers against state interference; governments must affirm capabilities of citizens [Nussbaum] |
| 21020 | Any establishment belief system is incompatible with full respect for all citizens [Nussbaum] |
| 21023 | We should respect animals in the way that we respect the animal nature in humans [Nussbaum] |
| 21024 | It may be no harm to kill an animal which cannot plan for its future [Nussbaum] |
| 21022 | The Capabilities Approach sees animals as agents, not just as having feelings [Nussbaum] |