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Ideas of David Bostock, by Text

[British, fl. 1980, Of Merton College, Oxford.]

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