18137 | Impredicative definitions are wrong, because they change the set that is being defined? |

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 |

18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism |

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

18114 | There is no single agreed structure for set theory |

18098 | Cantor proved that all sets have more subsets than they have members |

18107 | A 'proper class' cannot be a member of anything |

18115 | We could add axioms to make sets either as small or as large as possible |

18104 | Frege, unlike Russell, has infinite individuals because numbers are individuals |

18139 | The Axiom of Choice relies on reference to sets that we are unable to describe |

18130 | Axiom of Reducibility: there is always a function of the lowest possible order in a given level |

18105 | Replacement enforces a 'limitation of size' test for the existence of sets |

18108 | First-order logic is not decidable: there is no test of whether any formula is valid |

18109 | The completeness of first-order logic implies its compactness |

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 |

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 |

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

18123 | Substitutional quantification is just standard if all objects in the domain have a name |

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) |

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

18120 | The Deduction Theorem is what licenses a system of natural deduction |

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

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

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

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

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

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 |

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

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

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 |

18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' |

18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 |

18100 | ω + 1 is a new ordinal, but its cardinality is unchanged |

18102 | A cardinal is the earliest ordinal that has that number of predecessors |

18106 | Aleph-1 is the first ordinal that exceeds aleph-0 |

18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms |

18099 | The number of reals is the number of subsets of the natural numbers |

18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number |

18094 | Dedekind says each cut matches a real; logicists say the cuts are the reals |

18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers |

18156 | Modern axioms of geometry do not need the real numbers |

18096 | Zero is a member, and all successors; numbers are the intersection of sets satisfying this |

18097 | The Peano Axioms describe a unique structure |

18113 | PA concerns any entities which satisfy the axioms |

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 |

18149 | There are many criteria for the identity of numbers |

18148 | Hume's Principle is a definition with existential claims, and won't explain numbers |

18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it |

18142 | One-one correlations imply normal arithmetic, but don't explain our concept of a number |

18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! |

18116 | Numbers can't be positions, if nothing decides what position a given number has |

18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties |

18157 | Nominalism as based on application of numbers is no good, because there are too many applications |

18141 | Nominalism about mathematics is either reductionist, or fictionalist |

18150 | Actual measurement could never require the precision of the real numbers |

18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... |

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

18127 | Simple type theory has 'levels', but ramified type theory has 'orders' |

18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number |

18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job |

18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality |

18146 | If Hume's Principle is the whole story, that implies structuralism |

18129 | Many crucial logicist definitions are in fact impredicative |

18159 | Higher cardinalities in sets are just fairy stories |

18155 | A fairy tale may give predictions, but only a true theory can give explanations |

18140 | The best version of conceptualism is predicativism |

18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values |

18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them |

18134 | Predicativism makes theories of huge cardinals impossible |

18136 | If we can only think of what we can describe, predicativism may be implied |

18135 | If mathematics rests on science, predicativism may be the best approach |

18133 | The usual definitions of identity and of natural numbers are impredicative |

18132 | The predicativity restriction makes a difference with the real numbers |

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

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

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 |

18121 | In logic a proposition means the same when it is and when it is not asserted |