137 ideas
| 2676 | Didactic argument starts from the principles of the subject, not from the opinions of the learner [Aristotle] |
| 2675 | Reasoning is a way of making statements which makes them lead on to other statements [Aristotle] |
| 19807 | Both nature and reason require that everything has a cause [Rousseau] |
| 2677 | Dialectic aims to start from generally accepted opinions, and lead to a contradiction [Aristotle] |
| 2674 | Competitive argument aims at refutation, fallacy, paradox, solecism or repetition [Aristotle] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 16967 | 'Are Coriscus and Callias at home?' sounds like a single question, but it isn't [Aristotle] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 16149 | Generic terms like 'man' are not substances, but qualities, relations, modes or some such thing [Aristotle] |
| 11840 | Only if two things are identical do they have the same attributes [Aristotle] |
| 7235 | Without freedom of will actions lack moral significance [Rousseau] |
| 19791 | Natural mankind is too fragmented for states of peace, or of war and enmity [Rousseau] |
| 20501 | Rousseau assumes that laws need a people united by custom and tradition [Rousseau, by Wolff,J] |
| 7237 | The act of becoming 'a people' is the real foundation of society [Rousseau] |
| 19792 | To overcome obstacles, people must unite their forces into a single unified power [Rousseau] |
| 19812 | Human nature changes among a people, into a moral and partial existence [Rousseau] |
| 19814 | A state must be big enough to preserve itself, but small enough to be governable [Rousseau] |
| 19815 | Too much land is a struggle, producing defensive war; too little makes dependence, and offensive war [Rousseau] |
| 19822 | If the state enlarges, the creators of the general will become less individually powerful [Rousseau] |
| 19823 | If the population is larger, the government needs to be more powerful [Rousseau] |
| 7232 | Man is born free, and he is everywhere in chains [Rousseau] |
| 7234 | No man has any natural authority over his fellows [Rousseau] |
| 19816 | A state's purpose is liberty and equality - liberty for strength, and equality for liberty [Rousseau] |
| 7247 | The greatest social good comes down to freedom and equality [Rousseau] |
| 19838 | The measure of a successful state is increase in its population [Rousseau] |
| 19848 | The sovereignty does not appoint the leaders [Rousseau] |
| 20567 | Rousseau insists that popular sovereignty needs a means of expressing consent [Rousseau, by Oksala] |
| 19801 | Sovereignty is the exercise of the general will, which can never be delegated [Rousseau] |
| 19805 | Just as people control their limbs, the general-will state has total control of its members [Rousseau] |
| 19818 | Political laws are fundamental, as they firmly organise the state - but they could still be changed [Rousseau] |
| 19790 | Force can only dominate if it is seen as a right, and obedience as a duty [Rousseau] |
| 7233 | The social order is a sacred right, but based on covenants, not nature [Rousseau] |
| 19842 | The government is instituted by a law, not by a contract [Rousseau] |
| 7239 | The social pact is the total subjection of individuals to the general will [Rousseau] |
| 19793 | We need a protective association which unites forces, but retains individual freedom [Rousseau] |
| 7240 | To foreign powers a state is seen as a simple individual [Rousseau] |
| 19795 | The act of association commits citizens to the state, and the state to its citizens [Rousseau] |
| 19797 | Citizens must ultimately for forced to accept the general will (so freedom is compulsory!) [Rousseau] |
| 19796 | Individual citizens still retain a private will, which may be contrary to the general will [Rousseau] |
| 7244 | The general will is common interest; the will of all is the sum of individual desires [Rousseau] |
| 19802 | The general will is always right, but the will of all can err, because it includes private interests [Rousseau] |
| 19803 | If the state contains associations there are fewer opinions, undermining the general will [Rousseau] |
| 19804 | If a large knowledgeable population votes in isolation, their many choices will have good results [Rousseau] |
| 19808 | The general will changes its nature when it focuses on particulars [Rousseau] |
| 7246 | The general will is always good, but sometimes misunderstood [Rousseau] |
| 7250 | Laws are authentic acts of the general will [Rousseau] |
| 19844 | Assemblies must always confirm the form of government, and the current administration [Rousseau] |
| 19846 | The more unanimous the assembly, the stronger the general will becomes [Rousseau] |
| 19817 | Citizens should be independent of each other, and very dependent on the state [Rousseau] |
| 19840 | A citizen is a subject who is also sovereign [Rousseau] |
| 19839 | The flourishing of arts and letters is too much admired [Rousseau] |
| 19798 | Ancient monarchs were kings of peoples; modern monarchs more cleverly rule a land [Rousseau] |
| 19831 | The highest officers under a monarchy are normally useless; the public could choose much better [Rousseau] |
| 19833 | Hereditary monarchy is easier, but can lead to dreadful monarchs [Rousseau] |
| 19834 | Attempts to train future kings don't usually work, and the best have been unprepared [Rousseau] |
| 19829 | Natural aristocracy is primitive, and hereditary is dreadful, but elective aristocracy is best [Rousseau] |
| 7249 | Natural aristocracy is primitive, hereditary is bad, and elective aristocracy is the best [Rousseau] |
| 19830 | Large states need a nobility to fill the gap between a single prince and the people [Rousseau] |
| 19827 | Law makers and law implementers should be separate [Rousseau] |
| 19820 | The state has a legislature and an executive, just like the will and physical power in a person [Rousseau] |
| 19821 | I call the executive power the 'government', which is the 'prince' - a single person, or a group [Rousseau] |
| 19824 | Large populations needs stronger control, which means power should be concentrated [Rousseau] |
| 19826 | Democracy for small states, aristocracy for intermediate, monarchy for large [Rousseau] |
| 19837 | If inhabitants are widely dispersed, organising a revolt is much more difficult [Rousseau] |
| 19843 | The state is not bound to leave civil authority to its leaders [Rousseau] |
| 19825 | If the sovereign entrusts government to at least half the citizens, that is 'democracy' [Rousseau] |
| 19832 | Democratic elections are dangerous intervals in government [Rousseau] |
| 7243 | Silence of the people implies their consent [Rousseau] |
| 7251 | The English are actually slaves in between elections [Rousseau] |
| 7238 | Minorities only accept majority-voting because of a prior unanimous agreement [Rousseau] |
| 19828 | Democracy leads to internal strife, as people struggle to maintain or change ways of ruling [Rousseau] |
| 19835 | When ministers change the state changes, because they always reverse policies [Rousseau] |
| 19849 | In early theocracies the god was the king, and there were as many gods as nations [Rousseau] |
| 19841 | Sometimes full liberty is only possible at the expense of some complete enslavement [Rousseau] |
| 19847 | We can never assume that the son of a slave is a slave [Rousseau] |
| 7242 | Appetite alone is slavery, and self-prescribed laws are freedom [Rousseau] |
| 19800 | The social compact imposes conventional equality of rights on people who may start unequally [Rousseau] |
| 7248 | No citizen should be rich enough to buy another, and none so poor as forced to sell himself [Rousseau] |
| 19794 | If we all give up all of our rights together to the community, we will always support one another [Rousseau] |
| 7241 | In society man loses natural liberty, but gains a right to civil liberty and property [Rousseau] |
| 19806 | We alienate to society only what society needs - but society judges that, not us [Rousseau] |
| 19799 | Private property must always be subordinate to ownership by the whole community [Rousseau] |
| 19819 | The state ensures liberty, so civil law separates citizens, and binds them to the state [Rousseau] |
| 7245 | Natural justice, without sanctions, benefits the wicked, who exploit it [Rousseau] |
| 19809 | We accept the death penalty to prevent assassinations, so we must submit to it if necessary [Rousseau] |
| 19810 | A trial proves that a criminal has broken the social treaty, and is no longer a member of the state [Rousseau] |
| 19811 | Only people who are actually dangerous should be executed, even as an example [Rousseau] |
| 7236 | War gives no right to inflict more destruction than is necessary for victory [Rousseau] |
| 23607 | Wars are between States, not people, and the individuals are enemies by accident [Rousseau] |
| 19850 | By separating theological and political systems, Jesus caused divisions in the state [Rousseau] |
| 19852 | Civil religion needs one supreme god, an afterlife, justice, and the sanctity of the social contract [Rousseau] |
| 19853 | All religions should be tolerated, if they tolerate each other, and support citizenship [Rousseau] |
| 19851 | Every society has a religion as its base [Rousseau] |
| 19836 | The amount of taxation doesn't matter, if it quickly circulates back to the citizens [Rousseau] |
| 7252 | A tyrant exploits Christians because they don't value this life, and are made to be slaves [Rousseau] |