Combining Philosophers

All the ideas for Hans Reichenbach, Peter Smith and Gabriel M.A. Segal

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66 ideas

1. Philosophy / G. Scientific Philosophy / 1. Aims of Science
Science is in the business of carving nature at the joints [Segal]
2. Reason / A. Nature of Reason / 5. Objectivity
Contextual values are acceptable in research, but not in its final evaluation [Reichenbach, by Reiss/Sprenger]
2. Reason / A. Nature of Reason / 8. Naturalising Reason
Psychology studies the way rationality links desires and beliefs to causality [Segal]
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 '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]
Two functions are the same if they have the same extension [Smith,P]
A 'partial function' maps only some elements to another set [Smith,P]
The 'range' of a function is the set of elements in the output set created by the function [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
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]
An 'injective' ('one-to-one') function creates a distinct output element from each original [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
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]
'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [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
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' has basic axioms, quantifiers and first-order logic [Smith,P]
Robinson Arithmetic (Q) is not negation complete [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]
10. Modality / A. Necessity / 5. Metaphysical Necessity
Is 'Hesperus = Phosphorus' metaphysically necessary, but not logically or epistemologically necessary? [Segal]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / b. Conceivable but impossible
If claims of metaphysical necessity are based on conceivability, we should be cautious [Segal]
12. Knowledge Sources / B. Perception / 5. Interpretation
Kant showed that our perceptions are partly constructed from our concepts [Reichenbach]
14. Science / D. Explanation / 3. Best Explanation / c. Against best explanation
The success and virtue of an explanation do not guarantee its truth [Segal]
18. Thought / A. Modes of Thought / 4. Folk Psychology
Folk psychology is ridiculously dualist in its assumptions [Segal]
18. Thought / C. Content / 5. Twin Earth
Humans are made of H2O, so 'twins' aren't actually feasible [Segal]
Externalists can't assume old words refer to modern natural kinds [Segal]
If 'water' has narrow content, it refers to both H2O and XYZ [Segal]
18. Thought / C. Content / 6. Broad Content
If content is external, so are beliefs and desires [Segal]
Concepts can survive a big change in extension [Segal]
Maybe experts fix content, not ordinary users [Segal]
Must we relate to some diamonds to understand them? [Segal]
Externalism can't explain concepts that have no reference [Segal]
Maybe content involves relations to a language community [Segal]
18. Thought / C. Content / 7. Narrow Content
If content is narrow, my perfect twin shares my concepts [Segal]
18. Thought / C. Content / 10. Causal Semantics
If thoughts ARE causal, we can't explain how they cause things [Segal]
Even 'mass' cannot be defined in causal terms [Segal]
26. Natural Theory / C. Causation / 5. Direction of causation
A theory of causal relations yields an asymmetry which defines the direction of time [Reichenbach, by Salmon]
27. Natural Reality / D. Time / 2. Passage of Time / g. Time's arrow
The direction of time is grounded in the direction of causation [Reichenbach, by Ladyman/Ross]