44 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] |
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] |
23110 | Human injustice is not a permanent feature of communities [Rawls] |
15676 | Rawls defends the priority of right over good [Rawls, by Finlayson] |
4123 | A fair arrangement is one that parties can agree to without knowing how it will benefit them personally [Rawls, by Williams,B] |
21051 | Check your rationality by thinking of your opinion pronounced by the supreme court [Rawls] |
3279 | Utilitarianism inappropriately scales up the individual willingness to make sacrifices [Rawls, by Nagel] |
22406 | The maximisation of happiness must be done fairly [Rawls, by Smart] |
21137 | Rawls rejected cosmopolitanism because it doesn't respect the autonomy of 'peoples' [Rawls, by Shorten] |
3280 | Why does the rational agreement of the 'Original Position' in Rawls make it right? [Nagel on Rawls] |
20552 | The original position models the idea that citizens start as free and equal [Rawls, by Swift] |
18636 | Choose justice principles in ignorance of your own social situation [Rawls] |
18631 | All desirable social features should be equal, unless inequality favours the disadvantaged [Rawls] |
21119 | Power is only legitimate if it is reasonable for free equal citizens to endorse the constitution [Rawls] |
20538 | Utilitarians lump persons together; Rawls somewhat separates them; Nozick wholly separates them [Swift on Rawls] |
9277 | Rawls's account of justice relies on conventional fairness, avoiding all moral controversy [Gray on Rawls] |
23420 | In a pluralist society we can't expect a community united around one conception of the good [Rawls] |
20527 | Liberty Principle: everyone has an equal right to liberties, if compatible with others' liberties [Rawls] |
21018 | The social contract has problems with future generations, national boundaries, disabilities and animals [Rawls, by Nussbaum] |
21041 | Justice concerns not natural distributions, or our born location, but what we do about them [Rawls] |
23583 | If an aggression is unjust, the constraints on how it is fought are much stricter [Rawls] |
7602 | In the Bible God changes his mind (repenting of creating humanity, in the Flood) [Armstrong,K] |
7605 | Monotheism introduced intolerance into religious thinking [Armstrong,K] |
7599 | Around 800 BCE teachers superseded gods in India [Armstrong,K] |
7597 | There is virtually no sign of monotheism in the Pentateuch [Armstrong,K] |
7606 | The idea that Jesus was God was only settled in the fourth century [Armstrong,K] |
7596 | Faith is not just belief in propositions, but also putting trust in them [Armstrong,K] |