119 ideas
| 17240 | Definitions are the first step in philosophy [Hobbes] |
| 17237 | Definitions of things that are caused must express their manner of generation [Hobbes] |
| 17239 | Definition is resolution of names into successive genera, and finally the difference [Hobbes] |
| 17241 | A defined name should not appear in the definition [Hobbes] |
| 9331 | How do we determine which of the sentences containing a term comprise its definition? [Horwich] |
| 17242 | 'Petitio principii' is reusing the idea to be defined, in disguised words [Hobbes] |
| 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] |
| 17245 | A part of a part is a part of a whole [Hobbes] |
| 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] |
| 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] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [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] |
| 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] |
| 17258 | If we just say one, one, one, one, we don't know where we have got to [Hobbes] |
| 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] |
| 17253 | Change is nothing but movement [Hobbes] |
| 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] |
| 16670 | Accidents are just modes of thinking about bodies [Hobbes] |
| 16621 | Accidents are not parts of bodies (like blood in a cloth); they have accidents as things have a size [Hobbes] |
| 16734 | The complete power of an event is just the aggregate of the qualities that produced it [Hobbes] |
| 17247 | The only generalities or universals are names or signs [Hobbes] |
| 14960 | Bodies are independent of thought, and coincide with part of space [Hobbes] |
| 17250 | If you separate the two places of one thing, you will also separate the thing [Hobbes] |
| 17249 | If you separated two things in the same place, you would also separate the places [Hobbes] |
| 17248 | If a whole body is moved, its parts must move with it [Hobbes] |
| 16790 | A body is always the same, whether the parts are together or dispersed [Hobbes] |
| 17244 | To make a whole, parts needn't be put together, but can be united in the mind [Hobbes] |
| 17233 | Particulars contain universal things [Hobbes] |
| 17246 | Some accidental features are permanent, unless the object perishes [Hobbes] |
| 17251 | The feature which picks out or names a thing is usually called its 'essence' [Hobbes] |
| 17257 | It is the same river if it has the same source, no matter what flows in it [Hobbes] |
| 12853 | Some individuate the ship by unity of matter, and others by unity of form [Hobbes] |
| 17256 | If a new ship were made of the discarded planks, would two ships be numerically the same? [Hobbes] |
| 16794 | As an infant, Socrates was not the same body, but he was the same human being [Hobbes] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 17255 | Two bodies differ when (at some time) you can say something of one you can't say of the other [Hobbes] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 16582 | We can imagine a point swelling and contracting - but not how this could be done [Hobbes] |
| 9333 | A priori belief is not necessarily a priori justification, or a priori knowledge [Horwich] |
| 9342 | Understanding needs a priori commitment [Horwich] |
| 9332 | Meaning is generated by a priori commitment to truth, not the other way around [Horwich] |
| 9341 | Meanings and concepts cannot give a priori knowledge, because they may be unacceptable [Horwich] |
| 9334 | If we stipulate the meaning of 'number' to make Hume's Principle true, we first need Hume's Principle [Horwich] |
| 9339 | A priori knowledge (e.g. classical logic) may derive from the innate structure of our minds [Horwich] |
| 17238 | Science aims to show causes and generation of things [Hobbes] |
| 17260 | Imagination is just weakened sensation [Hobbes] |
| 19373 | A 'conatus' is an initial motion, experienced by us as desire or aversion [Hobbes, by Arthur,R] |
| 2948 | Sensation is merely internal motion of the sentient being [Hobbes] |
| 17261 | Apart from pleasure and pain, the only emotions are appetite and aversion [Hobbes] |
| 17236 | Words are not for communication, but as marks for remembering what we have learned [Hobbes] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 16600 | Prime matter is body considered with mere size and extension, and potential [Hobbes] |
| 17252 | Acting on a body is either creating or destroying a property in it [Hobbes] |
| 17254 | An effect needs a sufficient and necessary cause [Hobbes] |
| 17235 | A cause is the complete sum of the features which necessitate the effect [Hobbes] |
| 17234 | Motion is losing one place and acquiring another [Hobbes] |
| 17259 | 'Force' is the quantity of movement imposed on something [Hobbes] |
| 17243 | Past times can't exist anywhere, apart from in our memories [Hobbes] |