Ideas of David Bostock, by Theme

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

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2. Reason / D. Definition / 8. Impredicative Definition
 18137 Impredicative definitions are wrong, because they change the set that is being defined?
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
 13439 Venn Diagrams map three predicates into eight compartments, then look for the conclusion
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
 13422 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope
 13421 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
 13350 'Assumptions' says that a formula entails itself (φ|=φ)
 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
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
 13610 A logic with ¬ and → needs three axiom-schemas and one rule as foundation
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
 18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
 13846 A 'free' logic can have empty names, and a 'universally free' logic can have empty domains
4. Formal Logic / F. Set Theory ST / 1. Set Theory
 18114 There is no single agreed structure for set theory
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
 18107 A 'proper class' cannot be a member of anything
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
 18115 We could add axioms to make sets either as small or as large as possible
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
 18139 The Axiom of Choice relies on reference to sets that we are unable to describe
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
 18105 Replacement enforces a 'limitation of size' test for the existence of sets
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
 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
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
 13346 Truth is the basic notion in classical logic
 13545 Elementary logic cannot distinguish clearly between the finite and the infinite
 13822 Fictional characters wreck elementary logic, as they have contradictions and no excluded middle
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
 13623 The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem'
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
 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'
 13349 Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid'
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
 13617 MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ
 13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment)
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
 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
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
 13357 Truth-functors are usually held to be defined by their truth-tables
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
 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
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
 13360 In logic, a name is just any expression which refers to a particular single object
5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
 13361 An expression is only a name if it succeeds in referring to a real object
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
 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
 13848 We are only obliged to treat definite descriptions as non-names if only the former have scope
 13817 Definite descriptions are usually treated like names, and are just like them if they uniquely refer
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / c. Theory of definite descriptions
 13815 Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem
5. Theory of Logic / G. Quantification / 1. Quantification
 13438 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
 13818 If we allow empty domains, we must allow empty names
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
 18123 Substitutional quantification is just standard if all objects in the domain have a name
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
 13801 An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
 13619 Quantification adds two axiom-schemas and a new rule
 13622 Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine...
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
 13615 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ
 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
 13616 The Deduction Theorem greatly simplifies the search for proof
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
 13753 Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
 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
 13758 In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
 18120 The Deduction Theorem is what licenses a system of natural deduction
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
 13611 Tableau proofs use reduction - seeking an impossible consequence from an assumption
 13613 A completed open branch gives an interpretation which verifies those formulae
 13756 A tree proof becomes too broad if its only rule is Modus Ponens
 13762 Tableau rules are all elimination rules, gradually shortening 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
 13757 Unlike natural deduction, semantic tableaux have recipes for proving things
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
 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
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
 13364 Interpretation by assigning objects to names, or assigning them to variables first [PG]
5. Theory of Logic / I. Semantics of Logic / 5. Extensionalism
 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
5. Theory of Logic / K. Features of Logics / 2. Consistency
 13540 A set of formulae is 'inconsistent' when there is no interpretation which can make them all true
 13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
 13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ
5. Theory of Logic / K. Features of Logics / 6. Compactness
 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
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
 18125 Berry's Paradox considers the meaning of 'The least number not named by this name'
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
 18100 ω + 1 is a new ordinal, but its cardinality is unchanged
 18101 Each addition changes the ordinality but not the cardinality, prior to aleph-1
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
 18102 A cardinal is the earliest ordinal that has that number of predecessors
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
 18106 Aleph-1 is the first ordinal that exceeds aleph-0
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 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
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
 18093 For Eudoxus cuts in rationals are unique, but not every cut makes a real number
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
 18110 Infinitesimals are not actually contradictory, because they can be non-standard real numbers
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
 18156 Modern axioms of geometry do not need the real numbers
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
 18097 The Peano Axioms describe a unique structure
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
 13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all
 13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
 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
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
 18143 Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set!
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
 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
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
 18141 Nominalism about mathematics is either reductionist, or fictionalist
 18157 Nominalism as based on application of numbers is no good, because there are too many applications
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
 18150 Actual measurement could never require the precision of the real numbers
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
 18158 Ordinals are mainly used adjectively, as in 'the first', 'the second'...
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
 18127 Simple type theory has 'levels', but ramified type theory has 'orders'
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
 18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job
 18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
 18146 If Hume's Principle is the whole story, that implies structuralism
 18111 Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
 18129 Many crucial logicist definitions are in fact impredicative
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
 18159 Higher cardinalities in sets are just fairy stories
 18155 A fairy tale may give predictions, but only a true theory can give explanations
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
 18140 The best version of conceptualism is predicativism
 18138 Conceptualism fails to grasp mathematical properties, infinity, and objective truth values
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
 18131 If abstracta only exist if they are expressible, there can only be denumerably many of them
 18132 The predicativity restriction makes a difference with the real numbers
 18134 Predicativism makes theories of huge cardinals impossible
 18135 If mathematics rests on science, predicativism may be the best approach
 18136 If we can only think of what we can describe, predicativism may be implied
 18133 The usual definitions of identity and of natural numbers are impredicative
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
 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)
9. Objects / F. Identity among Objects / 5. Self-Identity
 13847 If non-existent things are self-identical, they are just one thing - so call it the 'null object'
10. Modality / A. Necessity / 6. Logical Necessity
 13820 The idea that anything which can be proved is necessary has a problem with empty names
19. Language / C. Assigning Meanings / 3. Predicates
 13363 A (modern) predicate is the result of leaving a gap for the name in a sentence
19. Language / F. Communication / 2. Assertion
 18121 In logic a proposition means the same when it is and when it is not asserted