Combining Philosophers

All the ideas for Michael Burke, Yale Kamisar and Peter Smith

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54 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
There cannot be a set theory which is complete [Smith,P]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order arithmetic can prove new sentences of first-order [Smith,P]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'partial function' maps only some elements to another set [Smith,P]
An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P]
A 'total function' maps every element to one element in another set [Smith,P]
The 'range' of a function is the set of elements in the output set created by the function [Smith,P]
Two functions are the same if they have the same extension [Smith,P]
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P]
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
A 'natural deduction system' has no axioms but many rules [Smith,P]
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
No nice theory can define truth for its own language [Smith,P]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P]
A 'bijective' function has one-to-one correspondence in both directions [Smith,P]
A 'surjective' ('onto') function creates every element of the output set [Smith,P]
5. Theory of Logic / K. Features of Logics / 3. Soundness
If everything that a theory proves is true, then it is 'sound' [Smith,P]
Soundness is true axioms and a truth-preserving proof system [Smith,P]
A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P]
5. Theory of Logic / K. Features of Logics / 4. Completeness
A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P]
A theory is 'negation complete' if it proves all sentences or their negation [Smith,P]
'Complete' applies both to whole logics, and to theories within them [Smith,P]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P]
5. Theory of Logic / K. Features of Logics / 7. Decidability
'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P]
A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P]
Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P]
A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P]
A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P]
The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P]
The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P]
5. Theory of Logic / K. Features of Logics / 9. Expressibility
Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
The truths of arithmetic are just true equations and their universally quantified versions [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
All numbers are related to zero by the ancestral of the successor relation [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic
Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P]
Baby Arithmetic is complete, but not very expressive [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / c. Robinson arithmetic
Robinson Arithmetic (Q) is not negation complete [Smith,P]
Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P]
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
Maybe the clay becomes a different lump when it becomes a statue [Burke,M, by Koslicki]
Burke says when two object coincide, one of them is destroyed in the process [Burke,M, by Hawley]
Sculpting a lump of clay destroys one object, and replaces it with another one [Burke,M, by Wasserman]
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]
24. Applied Ethics / C. Death Issues / 5. Euthanasia
We only allow voluntary euthanasia to someone who is both sane and crazed by pain [Kamisar]
People will volunteer for euthanasia because they think other people want them dead [Kamisar]