1997 | Intermediate Logic |
p.356 | 13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects |
1.1 | p.3 | 13346 | Truth is the basic notion in classical logic |
1.2 | p.5 | 13347 | Validity is a conclusion following for premises, even if there is no proof |
1.3 | p.10 | 13348 | It seems more natural to express |= as 'therefore', rather than 'entails' |
1.3 | p.13 | 13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' |
2.5.A | p.30 | 13350 | 'Assumptions' says that a formula entails itself (φ|=φ) |
2.5.B | p.30 | 13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference |
2.5.C | p.31 | 13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z |
2.5.E | p.33 | 13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ |
2.5.F | p.33 | 13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ |
2.5.G | p.33 | 13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= |
2.5.H | p.33 | 13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ |
2.6 | p.39 | 13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope |
2.6 | p.39 | 13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope |
2.7 | p.46 | 13357 | Truth-functors are usually held to be defined by their truth-tables |
2.8 | p.48 | 13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers |
2.8 | p.48 | 13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all |
3.1 | p.71 | 13360 | In logic, a name is just any expression which refers to a particular single object |
3.1 | p.71 | 13361 | An expression is only a name if it succeeds in referring to a real object |
3.1 | p.72 | 13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality |
3.2 | p.74 | 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence |
3.4 | p.82 | 13364 | Interpretation by assigning objects to names, or assigning them to variables first [PG] |
3.7 | p.109 | 13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors |
3.8 | p.124 | 13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion |
4.1 | p.141 | 13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption |
4.1 | p.146 | 13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' |
4.5 | p.165 | 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true |
4.5 | p.167 | 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula |
4.5 | p.167 | 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ |
4.7 | p.176 | 13543 | A relation is not reflexive, just because it is transitive and symmetrical |
4.7 | p.177 | 13613 | A completed open branch gives an interpretation which verifies those formulae |
4.8 | p.183 | 13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact |
4.8 | p.184 | 13545 | Elementary logic cannot distinguish clearly between the finite and the infinite |
5.1 | p.191 | 13608 | Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) |
5.1 | p.192 | 13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' |
5.2 | p.194 | 13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation |
5.3 | p.202 | 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) |
5.3 | p.203 | 13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ |
5.3 | p.206 | 13616 | The Deduction Theorem greatly simplifies the search for proof |
5.3 | p.207 | 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ |
5.5 | p.217 | 13618 | Compactness means an infinity of sequents on the left will add nothing new |
5.6 | p.221 | 13619 | Quantification adds two axiom-schemas and a new rule |
5.6 | p.223 | 13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem |
5.7 | p.227 | 13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth |
5.8 | p.232 | 13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... |
6.1 | p.240 | 13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part |
6.2 | p.248 | 13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) |
6.2 | p.251 | 13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it |
6.4 | p.263 | 13756 | A tree proof becomes too broad if its only rule is Modus Ponens |
6.5 | p.269 | 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things |
6.5 | p.270 | 13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle |
7.1 | p.274 | 13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded |
7.2 | p.281 | 13760 | A sequent calculus is good for comparing proof systems |
7.3 | p.283 | 13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored |
7.3 | p.285 | 13762 | Tableau rules are all elimination rules, gradually shortening formulae |
8.1 | p.323 | 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) |
8.1 | p.324 | 13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity |
8.1 | p.327 | 13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English |
8.1 | p.328 | 13803 | If we are to express that there at least two things, we need identity |
8.1 | p.328 | 13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) |
8.2 | p.333 | 13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs |
8.2 | p.334 | 13812 | A 'zero-place' function just has a single value, so it is a name |
8.3 | p.342 | 13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors |
8.3 | p.342 | 13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer |
8.3 | p.343 | 13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem |
8.3 | p.344 | 13816 | Because of scope problems, definite descriptions are best treated as quantifiers |
8.3 | p.347 | 13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer |
8.4 | p.351 | 13818 | If we allow empty domains, we must allow empty names |
8.4 | p.353 | 13819 | Aristotle's said some Fs are G or some Fs are not G, forgetting that there might be no Fs |
8.4 | p.354 | 13820 | The idea that anything which can be proved is necessary has a problem with empty names |
8.5 | p.357 | 13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle |
8.6 | p.360 | 13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains |
8.6 | p.362 | 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' |
8.8 | p.375 | 13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope |
2009 | Philosophy of Mathematics |
4.4 | p.98 | 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number |
4.4 | p.99 | 18094 | Dedekind says each cut matches a real; logicists say the cuts are the reals |
4.4 | p.100 | 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms |
4.4 | p.101 | 18096 | Zero is a member, and all successors; numbers are the intersection of sets satisfying this |
4.4 n20 | p.103 | 18097 | The Peano Axioms describe a unique structure |
4.5 | p.106 | 18098 | Cantor proved that all sets have more subsets than they have members |
4.5 | p.107 | 18099 | The number of reals is the number of subsets of the natural numbers |
4.5 | p.110 | 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 |
4.5 | p.110 | 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged |
4.5 | p.111 | 18102 | A cardinal is the earliest ordinal that has that number of predecessors |
5.2 | p.135 | 18104 | Frege, unlike Russell, has infinite individuals because numbers are individuals |
5.4 | p.144 | 18105 | Replacement enforces a 'limitation of size' test for the existence of sets |
5.4 | p.149 | 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 |
5.4 | p.151 | 18107 | A 'proper class' cannot be a member of anything |
5.5 | p.152 | 18108 | First-order logic is not decidable: there is no test of whether any formula is valid |
5.5 | p.153 | 18109 | The completeness of first-order logic implies its compactness |
5.5 | p.155 | 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers |
5.5 | p.159 | 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality |
6.3 | p.183 | 18113 | PA concerns any entities which satisfy the axioms |
6.4 | p.188 | 18115 | We could add axioms to make sets either as small or as large as possible |
6.4 | p.188 | 18114 | There is no single agreed structure for set theory |
6.4 | p.190 | 18116 | Numbers can't be positions, if nothing decides what position a given number has |
6.5 | p.192 | 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties |
7.2 | p.202 | 18120 | The Deduction Theorem is what licenses a system of natural deduction |
7.2 | p.207 | 18121 | In logic a proposition means the same when it is and when it is not asserted |
7.2 | p.214 | 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism |
7.3 n23 | p.217 | 18123 | Substitutional quantification is just standard if all objects in the domain have a name |
8.1 | p.227 | 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' |
8.1 | p.231 | 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' |
8.2 | p.237 | 18129 | Many crucial logicist definitions are in fact impredicative |
8.2 | p.238 | 18130 | Axiom of Reducibility: there is always a function of the lowest possible order in a given level |
8.2 | p.242 | 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them |
8.3 | p.244 | 18132 | The predicativity restriction makes a difference with the real numbers |
8.3 | p.246 | 18133 | The usual definitions of identity and of natural numbers are impredicative |
8.3 | p.251 | 18134 | Predicativism makes theories of huge cardinals impossible |
8.3 | p.252 | 18136 | If we can only think of what we can describe, predicativism may be implied |
8.3 | p.252 | 18135 | If mathematics rests on science, predicativism may be the best approach |
8.3 | p.252 | 18137 | Impredicative definitions are wrong, because they change the set that is being defined? |
8.4 | p.253 | 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values |
8.4 | p.259 | 18140 | The best version of conceptualism is predicativism |
8.4 n36 | p.256 | 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe |
9 | p.262 | 18141 | Nominalism about mathematics is either reductionist, or fictionalist |
9.5.iii | p.301 | 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... |
9.5.iii | p.303 | 18159 | Higher cardinalities in sets are just fairy stories |
9.A.2 | p.267 | 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! |
9.A.2 | p.267 | 18142 | One-one correlations imply normal arithmetic, but don't explain our concept of a number |
9.A.2 | p.268 | 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job |
9.A.2 | p.271 | 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it |
9.A.2 | p.272 | 18146 | If Hume's Principle is the whole story, that implies structuralism |
9.A.2 | p.272 | 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number |
9.A.2 | p.274 | 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers |
9.A.2 | p.275 | 18149 | There are many criteria for the identity of numbers |
9.A.3 | p.280 | 18150 | Actual measurement could never require the precision of the real numbers |
9.B.5 | p.290 | 18155 | A fairy tale may give predictions, but only a true theory can give explanations |
9.B.5.ii | p.295 | 18156 | Modern axioms of geometry do not need the real numbers |
9.B.5.iii | p.301 | 18157 | Nominalism as based on application of numbers is no good, because there are too many applications |