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Ideas for 'Writing the Book of the World', 'Philosophy of Logic' and 'Science of Logic'

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2 ideas

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

9024	Excluded middle has three different definitions [Quine]
	Full Idea: The law of excluded middle, or 'tertium non datur', may be pictured variously as 1) Every closed sentence is true or false; or 2) Every closed sentence or its negation is true; or 3) Every closed sentence is true or not true.
	From: Willard Quine (Philosophy of Logic [1970], Ch.6)
	A reaction: Unlike many top philosophers, Quine thinks clearly about such things. 1) is the classical bivalent reading of excluded middle; 2) is the purely syntactic version; 3) leaves open how we interpret the 'not-true' option.

5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic

10012	Quantification theory can still be proved complete if we add identity [Quine]
	Full Idea: Complete proof procedures are available not only for quantification theory, but for quantification theory and identity together. Gödel showed that the theory is still complete if we add self-identity and the indiscernability of identicals.
	From: Willard Quine (Philosophy of Logic [1970], Ch.5)
	A reaction: Hence one talks of first-order logic 'with identity', even though, as Quine observes, it is unclear whether identity is actually a logical or a mathematical notion.