159 ideas
| 19807 | Both nature and reason require that everything has a cause [Rousseau] |
| 13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
| 13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
| 13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock] |
| 13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock] |
| 13350 | 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock] |
| 13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock] |
| 13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock] |
| 13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock] |
| 13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock] |
| 13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock] |
| 13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock] |
| 13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
| 13346 | Truth is the basic notion in classical logic [Bostock] |
| 13545 | Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock] |
| 13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock] |
| 13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
| 13347 | Validity is a conclusion following for premises, even if there is no proof [Bostock] |
| 13348 | It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock] |
| 13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock] |
| 13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock] |
| 13803 | If we are to express that there at least two things, we need identity [Bostock] |
| 13357 | Truth-functors are usually held to be defined by their truth-tables [Bostock] |
| 13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
| 13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
| 13360 | In logic, a name is just any expression which refers to a particular single object [Bostock] |
| 13361 | An expression is only a name if it succeeds in referring to a real object [Bostock] |
| 18946 | Unreflectively, we all assume there are nonexistents, and we can refer to them [Reimer] |
| 13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock] |
| 13816 | Because of scope problems, definite descriptions are best treated as quantifiers [Bostock] |
| 13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock] |
| 13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
| 13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
| 13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock] |
| 13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock] |
| 13818 | If we allow empty domains, we must allow empty names [Bostock] |
| 13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock] |
| 13619 | Quantification adds two axiom-schemas and a new rule [Bostock] |
| 13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock] |
| 13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock] |
| 13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
| 13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock] |
| 13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock] |
| 13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock] |
| 13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock] |
| 13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock] |
| 13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock] |
| 13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
| 13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
| 13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
| 13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
| 13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
| 13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
| 13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock] |
| 13760 | A sequent calculus is good for comparing proof systems [Bostock] |
| 13364 | Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG] |
| 13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock] |
| 13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
| 13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock] |
| 13618 | Compactness means an infinity of sequents on the left will add nothing new [Bostock] |
| 13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock] |
| 13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock] |
| 13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
| 13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 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] |