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### Ideas of David Bostock, by Text

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

 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 13421 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope
 2.6 p.39 13422 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction 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 13361 An expression is only a name if it succeeds in referring to a real object
 3.1 p.71 13360 In logic, a name is just any expression which refers to a particular single 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 13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ
 4.5 p.167 13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
 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.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.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.100 18095 Instead of by cuts or series convergence, real numbers could be defined by axioms
 4.4 n20 p.103 18097 The Peano Axioms describe a unique structure
 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
 5.4 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
 6.4 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.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.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