Combining Philosophers

All the ideas for Michael Burke, Melvin Fitting and David Bostock

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136 ideas

2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions are wrong, because they change the set that is being defined? [Bostock]
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock]
'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]
'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
'Negation' says that Γ,φ|= iff Γ|=φ [Bostock]
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
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 [Bostock]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
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 [Bostock]
4. Formal Logic / E. Nonclassical Logics / 8. Intensional Logic
If terms change their designations in different states, they are functions from states to objects [Fitting]
Intensional logic adds a second type of quantification, over intensional objects, or individual concepts [Fitting]
4. Formal Logic / E. Nonclassical Logics / 9. Awareness Logic
Awareness logic adds the restriction of an awareness function to epistemic logic [Fitting]
4. Formal Logic / E. Nonclassical Logics / 10. Justification Logics
Justication logics make explicit the reasons for mathematical truth in proofs [Fitting]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
There is no single agreed structure for set theory [Bostock]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
A 'proper class' cannot be a member of anything [Bostock]
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 [Bostock]
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 [Bostock]
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 [Bostock]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
The completeness of first-order logic implies its compactness [Bostock]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Truth is the basic notion in classical logic [Bostock]
Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock]
Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock]
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
Classical logic is deliberately extensional, in order to model mathematics [Fitting]
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' [Bostock]
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
Validity is a conclusion following for premises, even if there is no proof [Bostock]
It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
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) [Bostock]
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
If we are to express that there at least two things, we need identity [Bostock]
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 [Bostock]
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 [Bostock]
A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
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 [Bostock]
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 [Bostock]
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 [Bostock]
Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock]
Because of scope problems, definite descriptions are best treated as quantifiers [Bostock]
Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock]
We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock]
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 [Bostock]
5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
λ-abstraction disambiguates the scope of modal operators [Fitting]
5. Theory of Logic / G. Quantification / 1. Quantification
'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
If we allow empty domains, we must allow empty names [Bostock]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Substitutional quantification is just standard if all objects in the domain have a name [Bostock]
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 [Bostock]
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Quantification adds two axiom-schemas and a new rule [Bostock]
Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock]
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock]
The Deduction Theorem greatly simplifies the search for proof [Bostock]
Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock]
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock]
Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock]
In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
The Deduction Theorem is what licenses a system of natural deduction [Bostock]
Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock]
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
A tree proof becomes too broad if its only rule is Modus Ponens [Bostock]
Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock]
Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock]
A completed open branch gives an interpretation which verifies those formulae [Bostock]
Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock]
In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock]
Tableau rules are all elimination rules, gradually shortening formulae [Bostock]
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 [Bostock]
A sequent calculus is good for comparing proof systems [Bostock]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG]
5. Theory of Logic / I. Semantics of Logic / 5. Extensionalism
Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock]
If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
5. Theory of Logic / K. Features of Logics / 2. Consistency
For 'negation-consistent', there is never |-(S)φ and |-(S)φ [Bostock]
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock]
5. Theory of Logic / K. Features of Logics / 6. Compactness
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
Compactness means an infinity of sequents on the left will add nothing new [Bostock]
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' [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
The number of reals is the number of subsets of the natural numbers [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Modern axioms of geometry do not need the real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
The Peano Axioms describe a unique structure [Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
There are many criteria for the identity of numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Numbers can't be positions, if nothing decides what position a given number has [Bostock]
Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Actual measurement could never require the precision of the real numbers [Bostock]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Many crucial logicist definitions are in fact impredicative [Bostock]
If Hume's Principle is the whole story, that implies structuralism [Bostock]
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Higher cardinalities in sets are just fairy stories [Bostock]
A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
The best version of conceptualism is predicativism [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
The usual definitions of identity and of natural numbers are impredicative [Bostock]
If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
The predicativity restriction makes a difference with the real numbers [Bostock]
Predicativism makes theories of huge cardinals impossible [Bostock]
If mathematics rests on science, predicativism may be the best approach [Bostock]
If we can only think of what we can describe, predicativism may be implied [Bostock]
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock]
9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
Persistence conditions cannot contradict, so there must be a 'dominant sortal' [Burke,M, by Hawley]
The 'dominant' of two coinciding sortals is the one that entails the widest range of properties [Burke,M, by Sider]
9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
'The rock' either refers to an object, or to a collection of parts, or to some stuff [Burke,M, by Wasserman]
9. Objects / B. Unity of Objects / 3. Unity Problems / b. Cat and its tail
Tib goes out of existence when the tail is lost, because Tib was never the 'cat' [Burke,M, by Sider]
9. Objects / B. Unity of Objects / 3. Unity Problems / c. Statue and clay
Sculpting a lump of clay destroys one object, and replaces it with another one [Burke,M, by Wasserman]
Burke says when two object coincide, one of them is destroyed in the process [Burke,M, by Hawley]
Maybe the clay becomes a different lump when it becomes a statue [Burke,M, by Koslicki]
9. Objects / B. Unity of Objects / 3. Unity Problems / d. Coincident objects
Two entities can coincide as one, but only one of them (the dominant sortal) fixes persistence conditions [Burke,M, by Sider]
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' [Bostock]
10. Modality / A. Necessity / 6. Logical Necessity
The idea that anything which can be proved is necessary has a problem with empty names [Bostock]
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
Definite descriptions pick out different objects in different possible worlds [Fitting]
19. Language / C. Assigning Meanings / 3. Predicates
A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock]
19. Language / F. Communication / 2. Assertion
In logic a proposition means the same when it is and when it is not asserted [Bostock]