Combining Philosophers

All the ideas for Anaxarchus, Barbara Vetter and Shaughan Lavine

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77 ideas

2. Reason / E. Argument / 1. Argument
Slippery slope arguments are challenges to show where a non-arbitrary boundary lies [Vetter]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
Deontic modalities are 'ought-to-be', for sentences, and 'ought-to-do' for predicates [Vetter]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
S5 is undesirable, as it prevents necessities from having contingent grounds [Vetter]
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
The Barcan formula endorses either merely possible things, or makes the unactualised impossible [Vetter]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
The Power Set is just the collection of functions from one collection to another [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Replacement was immediately accepted, despite having very few implications [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception of set wasn't suggested until 1947 [Lavine]
The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Mathematical proof by contradiction needs the law of excluded middle [Lavine]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Every rational number, unlike every natural number, is divisible by some other number [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
The infinite is extrapolation from the experience of indefinitely large size [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
The intuitionist endorses only the potential infinite [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory will found all of mathematics - except for the notion of proof [Lavine]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionism rejects set-theory to found mathematics [Lavine]
7. Existence / A. Nature of Existence / 1. Nature of Existence
The world is either a whole made of its parts, or a container which contains its parts [Vetter]
7. Existence / C. Structure of Existence / 1. Grounding / b. Relata of grounding
Grounding can be between objects ('relational'), or between sentences ('operational') [Vetter]
7. Existence / C. Structure of Existence / 5. Supervenience / d. Humean supervenience
The Humean supervenience base entirely excludes modality [Vetter]
8. Modes of Existence / B. Properties / 3. Types of Properties
A determinate property must be a unique instance of the determinable class [Vetter]
8. Modes of Existence / C. Powers and Dispositions / 4. Powers as Essence
Essence is a thing's necessities, but what about its possibilities (which may not be realised)? [Vetter]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / a. Dispositions
I have an 'iterated ability' to learn the violin - that is, the ability to acquire that ability [Vetter]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / c. Dispositions as conditional
We should think of dispositions as 'to do' something, not as 'to do something, if ....' [Vetter]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / d. Dispositions as occurrent
Nomological dispositions (unlike ordinary ones) have to be continually realised [Vetter]
8. Modes of Existence / C. Powers and Dispositions / 7. Against Powers
How can spatiotemporal relations be understood in dispositional terms? [Vetter]
9. Objects / D. Essence of Objects / 4. Essence as Definition
Real definition fits abstracta, but not individual concrete objects like Socrates [Vetter]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties
Modal accounts make essence less mysterious, by basing them on the clearer necessity [Vetter]
9. Objects / E. Objects over Time / 12. Origin as Essential
Why does origin matter more than development; why are some features of origin more important? [Vetter]
We take origin to be necessary because we see possibilities as branches from actuality [Vetter]
10. Modality / A. Necessity / 2. Nature of Necessity
The modern revival of necessity and possibility treated them as special cases of quantification [Vetter]
It is necessary that p means that nothing has the potentiality for not-p [Vetter]
10. Modality / A. Necessity / 5. Metaphysical Necessity
Metaphysical necessity is even more deeply empirical than Kripke has argued [Vetter]
10. Modality / B. Possibility / 1. Possibility
Maybe possibility is constituted by potentiality [Vetter]
Possible worlds allow us to talk about degrees of possibility [Vetter]
Possibilities are potentialities of actual things, but abstracted from their location [Vetter]
All possibility is anchored in the potentiality of individual objects [Vetter]
Possibility is a generalised abstraction from the potentiality of its bearer [Vetter]
10. Modality / B. Possibility / 4. Potentiality
Potentiality is the common genus of dispositions, abilities, and similar properties [Vetter]
Water has a potentiality to acquire a potentiality to break (by freezing) [Vetter]
A potentiality may not be a disposition, but dispositions are strong potentialities [Vetter, by Friend/Kimpton-Nye]
Potentiality does the explaining in metaphysics; we don't explain it away or reduce it [Vetter]
Potentiality logic is modal system T. Stronger systems collapse iterations, and necessitate potentials [Vetter]
Potentialities may be too weak to count as 'dispositions' [Vetter]
There are potentialities 'to ...', but possibilities are 'that ....'. [Vetter]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / c. Possible but inconceivable
The apparently metaphysically possible may only be epistemically possible [Vetter]
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Closeness of worlds should be determined by the intrinsic nature of relevant objects [Vetter]
10. Modality / E. Possible worlds / 2. Nature of Possible Worlds / c. Worlds as propositions
If worlds are sets of propositions, how do we know which propositions are genuinely possible? [Vetter]
10. Modality / E. Possible worlds / 3. Transworld Objects / e. Possible Objects
Are there possible objects which nothing has ever had the potentiality to produce? [Vetter]
13. Knowledge Criteria / D. Scepticism / 1. Scepticism
Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius]
14. Science / D. Explanation / 2. Types of Explanation / a. Types of explanation
Explanations by disposition are more stable and reliable than those be external circumstances [Vetter]
Grounding is a kind of explanation, suited to metaphysics [Vetter]
26. Natural Theory / D. Laws of Nature / 5. Laws from Universals
The view that laws are grounded in substance plus external necessity doesn't suit dispositionalism [Vetter]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / b. Scientific necessity
Dispositional essentialism allows laws to be different, but only if the supporting properties differ [Vetter]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / c. Essence and laws
Laws are relations of kinds, quantities and qualities, supervening on the essences of a domain [Vetter]
27. Natural Reality / D. Time / 1. Nature of Time / f. Eternalism
If time is symmetrical between past and future, why do they look so different? [Vetter]
27. Natural Reality / D. Time / 1. Nature of Time / h. Presentism
Presentists explain cross-temporal relations using surrogate descriptions [Vetter]