Combining Texts

All the ideas for 'fragments/reports', 'On 'Insolubilia' and their solution' and 'What Numbers Are'

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6 ideas

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets

21563	The 'no classes' theory says the propositions just refer to the members [Russell]
	Full Idea: The contention of the 'no classes' theory is that all significant propositions concerning classes can be regarded as propositions about all or some of their members.
	From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.200)
	A reaction: Apparently this theory has not found favour with later generations of theorists. I see it in terms of Russell trying to get ontology down to the minimum, in the spirit of Goodman and Quine.

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

17813	Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers [White,NP]
	Full Idea: The Löwenheim-Skolem theorem tells us that any theory with a true interpretation has a model in the natural numbers.
	From: Nicholas P. White (What Numbers Are [1974], V)

5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / d. Richard's paradox

21565	Richard's puzzle uses the notion of 'definition' - but that cannot be defined [Russell]
	Full Idea: In Richard's puzzle, we use the notion of 'definition', and this, oddly enough, is not definable, and is indeed not a definite notion at all.
	From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.209)
	A reaction: The background for this claim is his type theory, which renders certain forms of circular reference meaningless.

5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox

21564	Vicious Circle: what involves ALL must not be one of those ALL [Russell]
	Full Idea: The 'vicious-circle principle' says 'whatever involves an apparent variable must not be among the possible values of that variable', or (less exactly) 'whatever involves ALL must not be one of ALL which it involves.
	From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.204)
	A reaction: He offers this as a parallel to his 'no classes' principle. That referred to classes, but this refers to propositions, and specifically the Liar Paradox (which he calls the 'Epimenedes').

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure

17812	Finite cardinalities don't need numbers as objects; numerical quantifiers will do [White,NP]
	Full Idea: Statements involving finite cardinalities can be made without treating numbers as objects at all, simply by using quantification and identity to define numerically definite quantifiers in the manner of Frege.
	From: Nicholas P. White (What Numbers Are [1974], IV)
	A reaction: [He adds Quine 1960:268 as a reference]

13. Knowledge Criteria / E. Relativism / 2. Knowledge as Convention

1556	By nature people are close to one another, but culture drives them apart [Hippias]
	Full Idea: I regard you all as relatives - by nature, not by convention. By nature like is akin to like, but convention is a tyrant over humankind and often constrains people to act contrary to nature.
	From: Hippias (fragments/reports [c.430 BCE]), quoted by Plato - Protagoras 337c8