53 ideas
| 19917 | Without reason and human help, human life is misery [Spinoza] |
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| 19922 | People are only free if they are guided entirely by reason [Spinoza] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| 19935 | Peoples are created by individuals, not by nature, and only distinguished by language and law [Spinoza] |
| 19914 | In nature everything has an absolute right to do anything it is capable of doing [Spinoza] |
| 19915 | Natural rights are determined by desire and power, not by reason [Spinoza] |
| 7487 | Society exists to extend human awareness [Spinoza, by Watson] |
| 19943 | The state aims to allow personal development, so its main purpose is freedom [Spinoza] |
| 19930 | Sovereignty must include the power to make people submit to it [Spinoza] |
| 19936 | Kings tend to fight wars for glory, rather than for peace and liberty [Spinoza] |
| 19937 | Monarchs are always proud, and can't back down [Spinoza] |
| 19940 | Deposing a monarch is dangerous, because the people are used to royal authority [Spinoza] |
| 19931 | Every state is more frightened of its own citizens than of external enemies [Spinoza] |
| 19920 | Democracy is a legitimate gathering of people who do whatever they can do [Spinoza] |
| 19933 | If religion is law, then piety is justice, impiety is crime, and non-believers must leave [Spinoza] |
| 19938 | Allowing religious ministers any control of the state is bad for both parties [Spinoza] |
| 19923 | Slavery is not just obedience, but acting only in the interests of the master [Spinoza] |
| 19939 | Government is oppressive if opinions can be crimes, because people can't give them up [Spinoza] |
| 19944 | Without liberty of thought there is no trust in the state, and corruption follows [Spinoza] |
| 19942 | Treason may be committed as much by words as by deeds [Spinoza] |
| 19924 | The freest state is a rational one, where people can submit themselves to reason [Spinoza] |
| 7827 | Spinoza wanted democracy based on individual rights, and is thus the first modern political philosopher [Stewart,M on Spinoza] |
| 19926 | The sovereignty has absolute power over citizens [Spinoza] |
| 19918 | Forming a society meant following reason, and giving up dangerous appetites and mutual harm [Spinoza] |
| 19919 | People only give up their rights, and keep promises, if they hope for some greater good [Spinoza] |
| 19921 | Once you have given up your rights, there is no going back [Spinoza] |
| 19925 | In democracy we don't abandon our rights, but transfer them to the majority of us [Spinoza] |
| 19928 | No one, in giving up their power and right, ceases to be a human being [Spinoza] |
| 19929 | Everyone who gives up their rights must fear the recipients of them [Spinoza] |
| 19932 | The early Hebrews, following Moses, gave up their rights to God alone [Spinoza] |
| 19916 | The order of nature does not prohibit anything, and allows whatever appetite produces [Spinoza] |
| 19927 | State and religious law can clash, so the state must make decisions about religion [Spinoza] |
| 19934 | Hebrews were very hostile to other states, who had not given up their rights to God [Spinoza] |
| 4300 | The Bible has nothing in common with reasoning and philosophy [Spinoza] |