107 ideas
12274 | Begin examination with basics, and subdivide till you can go no further [Aristotle] |
12260 | Dialectic starts from generally accepted opinions [Aristotle] |
12291 | There can't be one definition of two things, or two definitions of the same thing [Aristotle] |
12292 | Definitions are easily destroyed, since they can contain very many assertions [Aristotle] |
12283 | In definitions the first term to be assigned ought to be the genus [Aristotle] |
12272 | We describe the essence of a particular thing by means of its differentiae [Aristotle] |
12279 | The differentia indicate the qualities, but not the essence [Aristotle] |
12289 | The genera and the differentiae are part of the essence [Aristotle] |
12261 | Differentia are generic, and belong with genus [Aristotle] |
12263 | 'Genus' is part of the essence shared among several things [Aristotle] |
12285 | The definition is peculiar to one thing, not common to many [Aristotle] |
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] |
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] |
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] |
13355 | 'Disjunction' 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] |
13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [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] |
13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [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] |
13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [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] |
13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
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] |
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] |
13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [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] |
13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [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] |
11261 | Puzzles arise when reasoning seems equal on both sides [Aristotle] |
12273 | Unit is the starting point of number [Aristotle] |
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] |
12267 | There are ten categories: essence, quantity, quality, relation, place, time, position, state, activity, passivity [Aristotle] |
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] |
12282 | An individual property has to exist (in past, present or future) [Aristotle] |
12264 | An 'accident' is something which may possibly either belong or not belong to a thing [Aristotle] |
12280 | Genus gives the essence better than the differentiae do [Aristotle] |
13269 | In the case of a house the parts can exist without the whole, so parts are not the whole [Aristotle] |
12284 | Everything that is has one single essence [Aristotle] |
12262 | An 'idion' belongs uniquely to a thing, but is not part of its essence [Aristotle] |
12290 | Destruction is dissolution of essence [Aristotle] |
12286 | If two things are the same, they must have the same source and origin [Aristotle] |
13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
12266 | 'Same' is mainly for names or definitions, but also for propria, and for accidents [Aristotle] |
12287 | Two identical things have the same accidents, they are the same; if the accidents differ, they're different [Aristotle] |
12288 | Numerical sameness and generic sameness are not the same [Aristotle] |
12259 | Reasoning is when some results follow necessarily from certain claims [Aristotle] |
13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
12271 | Induction is the progress from particulars to universals [Aristotle] |
12293 | We say 'so in cases of this kind', but how do you decide what is 'of this kind'? [Aristotle] |
13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
7453 | Galen's medicine followed the mean; each illness was balanced by opposite treatment [Galen, by Hacking] |
12276 | Justice and self-control are better than courage, because they are always useful [Aristotle] |
12277 | Friendship is preferable to money, since its excess is preferable [Aristotle] |
12275 | We value friendship just for its own sake [Aristotle] |
12281 | Man is intrinsically a civilized animal [Aristotle] |
12265 | All water is the same, because of a certain similarity [Aristotle] |
12278 | 'Being' and 'oneness' are predicated of everything which exists [Aristotle] |