56 ideas
| 10888 | Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo] |
| 10889 | The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo] |
| 10890 | A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo] |
| 10886 | Determinacy: an object is either in a set, or it isn't [Zalabardo] |
| 10887 | Specification: Determinate totals of objects always make a set [Zalabardo] |
| 10897 | A first-order 'sentence' is a formula with no free variables [Zalabardo] |
| 10893 | Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo] |
| 10899 | Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 10896 | Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo] |
| 10898 | The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo] |
| 10902 | We can do semantics by looking at given propositions, or by building new ones [Zalabardo] |
| 10892 | We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo] |
| 10895 | 'Logically true' (|= φ) is true for every truth-assignment [Zalabardo] |
| 10900 | Logically true sentences are true in all structures [Zalabardo] |
| 10894 | A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true [Zalabardo] |
| 10901 | Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo] |
| 10903 | A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 10891 | If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 23616 | Legal excuses are duress, ignorance, and diminished responsibility [McMahan] |
| 23606 | Liberty Rights are permissions, and Claim Rights are freedom from intervention [McMahan] |
| 23620 | A person or state may be attacked if they are responsible for an unjustified threat [McMahan] |
| 23598 | You (e.g. a police officer) are not liable to attack just because you pose a threat [McMahan] |
| 23594 | Wars can be unjust, despite a just cause, if they are unnecessary or excessive or of mixed cause [McMahan] |
| 23597 | Just war theory says all and only persons posing a threat are liable to attack [McMahan] |
| 23595 | The worst unjustified wars have no aim at all [McMahan] |
| 23619 | A defensive war is unjust, if it is responding to a just war [McMahan] |
| 23600 | Proportionality in fighting can't be judged independently of the justice of each side [McMahan] |
| 23603 | Can an army start an unjust war, and then fight justly to defend their own civilians? [McMahan] |
| 23611 | Soldiers cannot freely fight in unjust wars, just because they behave well when fighting [McMahan] |
| 23612 | The law of war differs from criminal law; attacking just combatants is immoral, but legal [McMahan] |
| 23617 | If the unjust combatants are morally excused they are innocent, so how can they be killed? [McMahan] |
| 23599 | You don't become a legitimate target, just because you violently resist an unjust attack [McMahan] |
| 23596 | If all combatants are seen as morally equal, that facilitates starting unjust wars [McMahan] |
| 23604 | Volunteer soldiers accept the risk of attack, but they don't agree to it, or to their deaths [McMahan] |
| 23608 | If being part of a big collective relieves soldiers of moral responsibility, why not the leaders too? [McMahan] |
| 23610 | If soldiers can't refuse to fight in unjust wars, can they choose to fight in just wars? [McMahan] |
| 23613 | Equality is both sides have permission, or both sides are justified, or one justified the other permitted [McMahan] |
| 23615 | Fighting unjustly under duress does not justify it, or permit it, but it may excuse it [McMahan] |
| 23605 | Soldiers cannot know enough facts to evaluate the justice of their war [McMahan] |
| 23602 | Innocence implies not being morally responsible, rather than merely being guiltless [McMahan] |
| 23618 | Unconditional surrender can't be demanded, since evil losers still have legitimate conditions [McMahan] |