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10044
|
Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro]
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Full Idea:
Russell adduces two reasons against the extensional view of classes, namely the existence of the null class (which cannot very well be a collection), and the unit classes (which would have to be identical with their single elements).
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Structure and Ontology p.459
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A reaction:
Gödel believes in the reality of classes. I have great sympathy with Russell, when people start to claim that sets are not just conveniences to help us think about things, but actual abstract entities. Is the singleton of my pencil is on this table?
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21707
|
Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Russell/Whitehead, by Linsky,B]
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Full Idea:
Russell did not view logic as an uninterpreted calculus awaiting interpretations [the modern view]. Rather, logic is a single 'interpreted' body of a priori truths, of propositions rather than sentence forms - but maximally general and topic neutral.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 1
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A reaction:
This is the view which Wittgenstein challenged, saying logic is just conventional. Linsky claims that Russell's logicism is much more plausible, once you understand his view of logic.
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8683
|
Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend]
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Full Idea:
Unlike Frege, Russell and Whitehead were not realists about mathematical objects, and whereas Frege thought that only arithmetic and analysis are branches of logic, they think the vast majority of mathematics (including geometry) is essentially logical.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1
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A reaction:
If, in essence, Descartes reduced geometry to algebra (by inventing co-ordinates), then geometry ought to be included. It is characteristic of Russell's hubris to want to embrace everything.
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10093
|
The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
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Full Idea:
Russell and Whitehead's ramified theory of types worked not with sets, but with propositional functions (similar to Frege's concepts), with a more restrictive assignment of variables, insisting that bound, as well as free, variables be of lower type.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.3
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A reaction:
I don't fully understand this (and no one seems much interested any more), but I think variables are a key notion, and there is something interesting going on here. I am intrigued by ordinary language which behaves like variables.
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8691
|
The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
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Full Idea:
The Russell/Whitehead type theory reduces mathematics to a consistent founding discipline, but is criticised for not really being logic. They could not prove the existence of infinite sets, and introduced a non-logical 'axiom of reducibility'.
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From:
comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.6
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A reaction:
To have reduced most of mathematics to a founding discipline sounds like quite an achievement, and its failure to be based in pure logic doesn't sound too bad. However, it seems to reduce some maths to just other maths.
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10305
|
In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead]
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Full Idea:
In the system of 'Principia Mathematica', it is not only the axioms of infinity and reducibility which go beyond pure logic, but also the initial conception of a universal domain of individuals and of a domain of predicates.
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From:
comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.267) by Paul Bernays - On Platonism in Mathematics p.267
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A reaction:
This sort of criticism seems to be the real collapse of the logicist programme, rather than Russell's paradox, or Gödel's Incompleteness Theorems. It just became impossible to stick strictly to logic in the reduction of arithmetic.
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8746
|
To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro]
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Full Idea:
Russell insisted on the vicious circle principle, and thus rejected impredicative definitions, which resulted in an unwieldy ramified type theory, with the ad hoc axiom of reducibility. Ramsey's simpler theory was impredicative and avoided the axiom.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2
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A reaction:
Nowadays the theory of types seems to have been given up, possibly because it has no real attraction if it lacks the strict character which Russell aspired to.
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14664
|
Necessary beings (numbers, properties, sets, propositions, states of affairs, God) exist in all possible worlds [Plantinga]
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Full Idea:
A 'necessary being' is one that exists in every possible world; and only some objects - numbers, properties, pure sets, propositions, states of affairs, God - have this distinction.
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From:
Alvin Plantinga (Actualism and Possible Worlds [1976], 2)
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A reaction:
This a very odd list, though it is fairly orthodox among philosophers trained in modern modal logic. At the very least it looks rather parochial to me.
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14666
|
Socrates is a contingent being, but his essence is not; without Socrates, his essence is unexemplified [Plantinga]
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Full Idea:
Socrates is a contingent being; his essence, however, is not. Properties, like propositions and possible worlds, are necessary beings. If Socrates had not existed, his essence would have been unexemplified, but not non-existent.
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From:
Alvin Plantinga (Actualism and Possible Worlds [1976], 4)
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A reaction:
This is a distinctive Plantinga view, of which I can make little sense. I take it that Socrates used to have an essence. Being dead, the essence no longer exists, but when we talk about Socrates it is largely this essence to which we refer. OK?
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12033
|
An object is identical with itself, and no different indiscernible object can share that [Russell/Whitehead, by Adams,RM]
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Full Idea:
Trivially, the Identity of Indiscernibles says that two individuals, Castor and Pollux, cannot have all properties in common. For Castor must have the properties of being identical with Castor and not being identical with Pollux, which Pollux can't share.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913], I p.57) by Robert Merrihew Adams - Primitive Thisness and Primitive Identity 2
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A reaction:
I suspect that either the property of being identical with itself is quite vacuous, or it is parasytic on primitive identity, or it is the criterion which is actually used to define identity. Either way, I don't find this claim very illuminating.
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14662
|
Possible worlds clarify possibility, propositions, properties, sets, counterfacts, time, determinism etc. [Plantinga]
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Full Idea:
The idea of possible worlds has delivered insights on logical possibility (de dicto and de re), propositions, properties and sets, counterfactuals, time and temporal relations, causal determinism, the ontological argument, and the problem of evil.
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|
From:
Alvin Plantinga (Actualism and Possible Worlds [1976], Intro)
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A reaction:
This date (1976) seems to be the high-water mark for enthusiasm about possible worlds. I suppose if we just stick to 'insights' rather than 'answers' then the big claim might still be acceptable. Which problems are created by possible worlds?
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16472
|
Plantinga's actualism is nominal, because he fills actuality with possibilia [Stalnaker on Plantinga]
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Full Idea:
Plantinga's critics worry that the metaphysics is actualist in name only, since it is achieved only by populating the actual world with entities whose nature is explained in terms of merely possible things that would exemplify them if anything did.
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From:
comment on Alvin Plantinga (Actualism and Possible Worlds [1976]) by Robert C. Stalnaker - Mere Possibilities 4.4
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A reaction:
Plantinga seems a long way from the usual motivation for actualism, which is probably sceptical empiricism, and building a system on what is smack in front of you. Possibilities have to be true, though. That's why you need dispositions in actuality.
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10040
|
Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory [Russell/Whitehead, by Gödel]
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Full Idea:
By analyzing the paradoxes to which Cantor's set theory had led, ..Russell brought to light the amazing fact that our logical intuitions (concerning such notions as truth, concept, being, class) are self-contradictory.
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|
From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.452
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|
A reaction:
The main intuition that failed was, I take it, that every concept has an extension, that is, there are always objects which will or could fall under the concept.
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21725
|
The multiple relations theory says assertions about propositions are about their ingredients [Russell/Whitehead, by Linsky,B]
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Full Idea:
The multiple relations theory of judgement proposes that assertions about propositions are dependent upon genuine facts involving belief and other attitude relations, subjects of those attitudes, and the constituents of the belief.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 7.2
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|
A reaction:
This seems to require a commitment to universals (especially relations) with which we can be directly acquainted. I prefer propositions, but as mental entities, not platonic entities.
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16469
|
Plantinga has domains of sets of essences, variables denoting essences, and predicates as functions [Plantinga, by Stalnaker]
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Full Idea:
The domains in Plantinga's interpretation of Kripke's semantics are sets of essences, and the values of variables are essences. The values of predicates have to be functions from possible worlds to essences.
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|
From:
report of Alvin Plantinga (Actualism and Possible Worlds [1976]) by Robert C. Stalnaker - Mere Possibilities 4.4
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|
A reaction:
I begin to think this is quite nice, as long as one doesn't take the commitment to the essences too seriously. For 'essence' read 'minimal identity'? But I take essences to be more than minimal, so use identities (which Kripke does?).
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16470
|
Plantinga's essences have their own properties - so will have essences, giving a hierarchy [Stalnaker on Plantinga]
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Full Idea:
For Plantinga, essences are entities in their own right and will have properties different from what instantiates them. Hence he will need individual essences of individual essences, distinct from the essences. I see no way to avoid a hierarchy of them.
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|
From:
comment on Alvin Plantinga (Actualism and Possible Worlds [1976]) by Robert C. Stalnaker - Mere Possibilities 4.4
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A reaction:
This sounds devastating for Plantinga, but it is a challenge for traditional Aristotelians. Only a logician suffers from a hierarchy, but a scientist might have to live with an essence, which contains a super-essence.
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14663
|
Are propositions and states of affairs two separate things, or only one? I incline to say one [Plantinga]
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|
Full Idea:
Are there two sorts of thing, propositions and states of affairs, or only one? I am inclined to the former view on the ground that propositions have a property, truth or falsehood, not had by states of affairs.
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|
From:
Alvin Plantinga (Actualism and Possible Worlds [1976], 1)
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|
A reaction:
Might a proposition be nothing more than an assertion that a state of affairs obtains? It would then pass his test. The idea that a proposition is a complex of facts in the external world ('Russellian' propositions?) quite baffles me.
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23453
|
Propositions as objects of judgement don't exist, because we judge several objects, not one [Russell/Whitehead]
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Full Idea:
A 'proposition', in the sense in which a proposition is supposed to be the object of a judgement, is a false abstraction, because a judgement has several objects, not one.
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|
From:
B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus 2E
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|
A reaction:
This is the rejection of the 'Russellian' theory of propositions, in favour of his multiple-relations theory of judgement. But why don't the related objects add up to a proposition about a state of affairs?
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