Combining Philosophers

All the ideas for Peter Smith, Keith Hossack and Friend/Kimpton-Nye

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88 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]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice is a non-logical principle of set-theory [Hossack]
The Axiom of Choice guarantees a one-one correspondence from sets to ordinals [Hossack]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
Predicativism says only predicated sets exist [Hossack]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Maybe we reduce sets to ordinals, rather than the other way round [Hossack]
4. Formal Logic / G. Formal Mereology / 3. Axioms of Mereology
Extensional mereology needs two definitions and two axioms [Hossack]
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 / 2. Logical Connectives / d. and
The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'partial function' maps only some elements to another set [Smith,P]
A 'total function' maps every element to one element in another set [Smith,P]
An argument is a 'fixed point' for a function if it is mapped back to itself [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 / 6. Relations in Logic
'Before' and 'after' are not two relations, but one relation with two orders [Hossack]
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 / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
Plural definite descriptions pick out the largest class of things that fit the description [Hossack]
5. Theory of Logic / G. Quantification / 6. Plural Quantification
Plural reference will refer to complex facts without postulating complex things [Hossack]
Plural reference is just an abbreviation when properties are distributive, but not otherwise [Hossack]
A plural comprehension principle says there are some things one of which meets some condition [Hossack]
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 it proves all sentences or their negation [Smith,P]
'Complete' applies both to whole logics, and to theories within them [Smith,P]
A theory is 'negation complete' if one of its sentences or its negation can always be proved [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]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / d. Russell's paradox
Plural language can discuss without inconsistency things that are not members of themselves [Hossack]
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 / e. Ordinal numbers
The theory of the transfinite needs the ordinal numbers [Hossack]
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]
I take the real numbers to be just lengths [Hossack]
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 / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
All numbers are related to zero by the ancestral of the successor relation [Smith,P]
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [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 / e. Peano arithmetic 2nd-order
A plural language gives a single comprehensive induction axiom for arithmetic [Hossack]
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
Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
In arithmetic singularists need sets as the instantiator of numeric properties [Hossack]
Set theory is the science of infinity [Hossack]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
Numbers are properties, not sets (because numbers are magnitudes) [Hossack]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
We can only mentally construct potential infinities, but maths needs actual infinities [Hossack]
7. Existence / D. Theories of Reality / 11. Ontological Commitment / a. Ontological commitment
We are committed to a 'group' of children, if they are sitting in a circle [Hossack]
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]
8. Modes of Existence / B. Properties / 1. Nature of Properties
Humeans see properties as having no more essential features and relations than their distinctness [Friend/Kimpton-Nye, by PG]
Dispositions are what individuate properties, and they constitute their essence [Friend/Kimpton-Nye]
8. Modes of Existence / C. Powers and Dispositions / 1. Powers
Powers are properties which necessitate dispositions [Friend/Kimpton-Nye]
8. Modes of Existence / C. Powers and Dispositions / 2. Powers as Basic
Dispositional essentialism (unlike the grounding view) says only fundamental properties are powers [Friend/Kimpton-Nye]
8. Modes of Existence / C. Powers and Dispositions / 4. Powers as Essence
A power is a property which consists entirely of dispositions [Friend/Kimpton-Nye]
Powers are qualitative properties which fully ground dispositions [Friend/Kimpton-Nye]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / a. Dispositions
Dispositions have directed behaviour which occurs if triggered [Friend/Kimpton-Nye]
'Masked' dispositions fail to react because something intervenes [Friend/Kimpton-Nye]
A disposition is 'altered' when the stimulus reverses the disposition [Friend/Kimpton-Nye]
A disposition is 'mimicked' if a different cause produces that effect from that stimulus [Friend/Kimpton-Nye]
A 'trick' can look like a stimulus for a disposition which will happen without it [Friend/Kimpton-Nye]
Some dispositions manifest themselves without a stimulus [Friend/Kimpton-Nye]
We could analyse dispositions as 'possibilities', with no mention of a stimulus [Friend/Kimpton-Nye]
9. Objects / C. Structure of Objects / 5. Composition of an Object
Complex particulars are either masses, or composites, or sets [Hossack]
The relation of composition is indispensable to the part-whole relation for individuals [Hossack]
9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
Leibniz's Law argues against atomism - water is wet, unlike water molecules [Hossack]
The fusion of five rectangles can decompose into more than five parts that are rectangles [Hossack]
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
Dispositionalism says modality is in the powers of this world, not outsourced to possible worlds [Friend/Kimpton-Nye]
18. Thought / A. Modes of Thought / 1. Thought
A thought can refer to many things, but only predicate a universal and affirm a state of affairs [Hossack]
26. Natural Theory / D. Laws of Nature / 7. Strictness of Laws
Hume's Dictum says no connections are necessary - so mass and spacetime warping could separate [Friend/Kimpton-Nye]
27. Natural Reality / C. Space / 2. Space
We could ignore space, and just talk of the shape of matter [Hossack]