Combining Texts

All the ideas for 'Philosophical Explanations', 'What Numbers Are' and 'Paradox without Self-Reference'

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5 ideas

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 / 6. Paradoxes in Language / a. The Liar paradox

9138	An infinite series of sentences asserting falsehood produces the paradox without self-reference [Yablo, by Sorensen]
	Full Idea: Banning self-reference is too narrow to avoid the liar paradox. With 1) all the subsequent sentences are false, 2) all the subsequent sentences are false, 3) all the subsequent... the paradox still arises. Self-reference is a special case of this.
	From: report of Stephen Yablo (Paradox without Self-Reference [1993]) by Roy Sorensen - Vagueness and Contradiction 11.1
	A reaction: [Idea 9137 pointed out that the ban was too narrow. Sorensen p.168 explains why this one is paradoxical] This is a nice example of progress in philosophy, since the Greeks would have been thrilled with this idea (unless they knew it, but it was lost).

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]

11. Knowledge Aims / A. Knowledge / 4. Belief / c. Aim of beliefs

3570	Maybe knowledge is belief which 'tracks' the truth [Nozick, by Williams,M]
	Full Idea: Nozick suggests that knowledge is just belief which 'tracks the truth' (hence leaving out justification).
	From: report of Robert Nozick (Philosophical Explanations [1981]) by Michael Williams - Problems of Knowledge Ch. 2

13. Knowledge Criteria / C. External Justification / 4. Tracking the Facts

2748	A true belief isn't knowledge if it would be believed even if false. It should 'track the truth' [Nozick, by Dancy,J]
	Full Idea: Nozick says Gettier cases aren't knowledge because the proposition would be believed even if false. Proper justification must be more sensitive to the truth ("track the truth").
	From: report of Robert Nozick (Philosophical Explanations [1981], 3.1) by Jonathan Dancy - Intro to Contemporary Epistemology 3.1
	A reaction: This is a bad idea. I see a genuine tree in my garden and believe it is there, so I know it. That I might have believed it if I was in virtually reality, or observing a mirror, won't alter that.