Ideas of David Bostock, by Theme

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

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2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions are wrong, because they change the set that is being defined?
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
Aristotle's said some Fs are G or some Fs are not G, forgetting that there might be no Fs
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
'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope
'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Assumptions' says that a formula entails itself (φ|=φ)
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z
'Negation' says that Γ,φ|= iff Γ|=φ
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ
'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|=
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
A logic with and → needs three axiom-schemas and one rule as foundation
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
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
There is no single agreed structure for set theory
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
Cantor proved that all sets have more subsets than they have members
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
A 'proper class' cannot be a member of anything
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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 / f. Axiom of Infinity V
Frege, unlike Russell, has infinite individuals because numbers are individuals
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice relies on reference to sets that we are unable to describe
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
Axiom of Reducibility: there is always a function of the lowest possible order in a given level
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Replacement enforces a 'limitation of size' test for the existence of sets
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
The completeness of first-order logic implies its compactness
First-order logic is not decidable: there is no test of whether any formula is valid
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Truth is the basic notion in classical logic
Elementary logic cannot distinguish clearly between the finite and the infinite
Fictional characters wreck elementary logic, as they have contradictions and no excluded middle
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
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 |=
Validity is a conclusion following for premises, even if there is no proof
It seems more natural to express |= as 'therefore', rather than 'entails'
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid'
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment)
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity
If we are to express that there at least two things, we need identity
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b)
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Truth-functors are usually held to be defined by their truth-tables
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'zero-place' function just has a single value, so it is a name
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
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
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
Definite descriptions don't always pick out one thing, as in denials of existence, or errors
We are only obliged to treat definite descriptions as non-names if only the former have scope
Definite desciptions resemble names, but can't actually be names, if they don't always refer
Because of scope problems, definite descriptions are best treated as quantifiers
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
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
'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
If we allow empty domains, we must allow empty names
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Substitutional quantification is just standard if all objects in the domain have a name
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
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
Quantification adds two axiom-schemas and a new rule
Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine...
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
The Deduction Theorem greatly simplifies the search for proof
'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ
Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it
In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E)
The Deduction Theorem is what licenses a system of natural deduction
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
Tableau proofs use reduction - seeking an impossible consequence from an assumption
Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed'
A completed open branch gives an interpretation which verifies those formulae
In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored
Tableau rules are all elimination rules, gradually shortening formulae
A tree proof becomes too broad if its only rule is Modus Ponens
Unlike natural deduction, semantic tableaux have recipes for proving things
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded
A sequent calculus is good for comparing proof systems
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Interpretation by assigning objects to names, or assigning them to variables first
5. Theory of Logic / I. Semantics of Logic / 6. Extensionalism
Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects
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
A set of formulae is 'inconsistent' when there is no interpretation which can make them all true
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
For 'negation-consistent', there is never |-(S)φ and |-(S)φ
5. Theory of Logic / K. Features of Logics / 6. Compactness
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
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
Berry's Paradox considers the meaning of 'The least number not named by this name'
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
ω + 1 is a new ordinal, but its cardinality is unchanged
Each addition changes the ordinality but not the cardinality, prior to aleph-1
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
A cardinal is the earliest ordinal that has that number of predecessors
6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
Aleph-1 is the first ordinal that exceeds aleph-0
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Instead of by cuts or series convergence, real numbers could be defined by axioms
The number of reals is the number of subsets of the natural numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
For Eudoxus cuts in rationals are unique, but not every cut makes a real number
Dedekind says each cut matches a real; logicists say the cuts are the reals
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / l. Infinitesimals
Infinitesimals are not actually contradictory, because they can be non-standard real numbers
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Modern axioms of geometry do not need the real numbers
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
Zero is a member, and all successors; numbers are the intersection of sets satisfying this
The Peano Axioms describe a unique structure
PA concerns any entities which satisfy the axioms
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / f. Mathematical induction
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
Many things will satisfy Hume's Principle, so there are many interpretations of it
Hume's Principle is a definition with existential claims, and won't explain numbers
There are many criteria for the identity of numbers
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set!
One-one correlations imply normal arithmetic, but don't explain our concept of a number
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
Numbers can't be positions, if nothing decides what position a given number has
Structuralism falsely assumes relations to other numbers are numbers' only properties
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
Nominalism about mathematics is either reductionist, or fictionalist
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
Actual measurement could never require the precision of the real numbers
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Ordinals are mainly used adjectively, as in 'the first', 'the second'...
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Mathematics has no special axioms of its own, but follows from principles of logic (with definitions)
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Simple type theory has 'levels', but ramified type theory has 'orders'
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Neo-logicists agree that HP introduces number, but also claim that it suffices for the job
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
Many crucial logicist definitions are in fact impredicative
If Hume's Principle is the whole story, that implies structuralism
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Higher cardinalities in sets are just fairy stories
A fairy tale may give predictions, but only a true theory can give explanations
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
The best version of conceptualism is predicativism
Conceptualism fails to grasp mathematical properties, infinity, and objective truth values
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
If abstracta only exist if they are expressible, there can only be denumerably many of them
The predicativity restriction makes a difference with the real numbers
The usual definitions of identity and of natural numbers are impredicative
Predicativism makes theories of huge cardinals impossible
If we can only think of what we can describe, predicativism may be implied
If mathematics rests on science, predicativism may be the best approach
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
Relations can be one-many (at most one on the left) or many-one (at most one on the right)
A relation is not reflexive, just because it is transitive and symmetrical
9. Objects / F. Identity among Objects / 5. Self-Identity
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
The idea that anything which can be proved is necessary has a problem with empty names
19. Language / A. Language / 6. Predicates
A (modern) predicate is the result of leaving a gap for the name in a sentence
19. Language / H. Pragmatics / 1. Assertion
In logic a proposition means the same when it is and when it is not asserted