Combining Texts

All the ideas for 'Philosophical Explanations', 'The Foundations of Mathematics' and 'The Runabout Inference Ticket'

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7 ideas

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

18844	You would cripple mathematics if you denied Excluded Middle [Hilbert]
	Full Idea: Taking the principle of Excluded Middle away from the mathematician would be the same, say, as prohibiting the astronomer from using the telescope or the boxer from using his fists.
	From: David Hilbert (The Foundations of Mathematics [1927], p.476), quoted by Ian Rumfitt - The Boundary Stones of Thought 9.4
	A reaction: [p.476 in Van Heijenoort]

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives

17896	We need to know the meaning of 'and', prior to its role in reasoning [Prior,AN, by Belnap]
	Full Idea: For Prior, so the moral goes, we must first have a notion of what 'and' means, independently of the role it plays as premise and as conclusion.
	From: report of Arthur N. Prior (The Runabout Inference Ticket [1960]) by Nuel D. Belnap - Tonk, Plonk and Plink p.132
	A reaction: The meaning would be given by the truth tables (the truth-conditions), whereas the role would be given by the natural deduction introduction and elimination rules. This seems to be the basic debate about logical connectives.

17898	Prior's 'tonk' is inconsistent, since it allows the non-conservative inference A |- B [Belnap on Prior,AN]
	Full Idea: Prior's definition of 'tonk' is inconsistent. It gives us an extension of our original characterisation of deducibility which is not conservative, since in the extension (but not the original) we have, for arbitrary A and B, A |- B.
	From: comment on Arthur N. Prior (The Runabout Inference Ticket [1960]) by Nuel D. Belnap - Tonk, Plonk and Plink p.135
	A reaction: Belnap's idea is that connectives don't just rest on their rules, but also on the going concern of normal deduction.

11021	Prior rejected accounts of logical connectives by inference pattern, with 'tonk' his absurd example [Prior,AN, by Read]
	Full Idea: Prior dislike the holism inherent in the claim that the meaning of a logical connective was determined by the inference patterns into which it validly fitted. ...His notorious example of 'tonk' (A → A-tonk-B → B) was a reductio of the view.
	From: report of Arthur N. Prior (The Runabout Inference Ticket [1960]) by Stephen Read - Thinking About Logic Ch.8
	A reaction: [The view being attacked was attributed to Gentzen]

13836	Maybe introducing or defining logical connectives by rules of inference leads to absurdity [Prior,AN, by Hacking]
	Full Idea: Prior intended 'tonk' (a connective which leads to absurdity) as a criticism of the very idea of introducing or defining logical connectives by rules of inference.
	From: report of Arthur N. Prior (The Runabout Inference Ticket [1960], §09) by Ian Hacking - What is Logic?

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.