Click on the Idea Number for the full details | back to texts | expand these ideas

13819 | Aristotle's said some Fs are G or some Fs are not G, forgetting that there might be no Fs |

13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion |

13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope |

13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope |

13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference |

13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z |

13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ |

13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ |

13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= |

13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ |

13350 | 'Assumptions' says that a formula entails itself (φ|=φ) |

13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation |

13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains |

13545 | Elementary logic cannot distinguish clearly between the finite and the infinite |

13346 | Truth is the basic notion in classical logic |

13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle |

13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' |

13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' |

13347 | Validity is a conclusion following for premises, even if there is no proof |

13348 | It seems more natural to express |= as 'therefore', rather than 'entails' |

13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) |

13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ |

13803 | If we are to express that there at least two things, we need identity |

13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) |

13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity |

13357 | Truth-functors are usually held to be defined by their truth-tables |

13812 | A 'zero-place' function just has a single value, so it is a name |

13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs |

13360 | In logic, a name is just any expression which refers to a particular single object |

13361 | An expression is only a name if it succeeds in referring to a real object |

13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors |

13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer |

13816 | Because of scope problems, definite descriptions are best treated as quantifiers |

13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer |

13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope |

13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem |

13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors |

13818 | If we allow empty domains, we must allow empty names |

13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English |

13619 | Quantification adds two axiom-schemas and a new rule |

13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... |

13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ |

13616 | The Deduction Theorem greatly simplifies the search for proof |

13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem |

13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth |

13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) |

13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle |

13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it |

13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part |

13756 | A tree proof becomes too broad if its only rule is Modus Ponens |

13762 | Tableau rules are all elimination rules, gradually shortening formulae |

13757 | Unlike natural deduction, semantic tableaux have recipes for proving things |

13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption |

13613 | A completed open branch gives an interpretation which verifies those formulae |

13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' |

13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored |

13760 | A sequent calculus is good for comparing proof systems |

13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded |

13364 | Interpretation by assigning objects to names, or assigning them to variables first |

13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects |

13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality |

13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true |

13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ |

13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula |

13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact |

13618 | Compactness means an infinity of sequents on the left will add nothing new |

13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers |

13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all |

13608 | Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) |

13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) |

13543 | A relation is not reflexive, just because it is transitive and symmetrical |

13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' |

13820 | The idea that anything which can be proved is necessary has a problem with empty names |

13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence |