Combining Philosophers

Ideas for Melvin Fitting, Alfred R. Mele and Shaughan Lavine

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4 ideas

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

15926	Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
	Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.3)
	A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'.

5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics

11026	Classical logic is deliberately extensional, in order to model mathematics [Fitting]
	Full Idea: Mathematics is typically extensional throughout (we write 3+2=2+3 despite the two terms having different meanings). ..Classical first-order logic is extensional by design since it primarily evolved to model the reasoning of mathematics.
	From: Melvin Fitting (Intensional Logic [2007], §1)

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

15934	Mathematical proof by contradiction needs the law of excluded middle [Lavine]
	Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.1)
	A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku.

5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction

11028	λ-abstraction disambiguates the scope of modal operators [Fitting]
	Full Idea: λ-abstraction can be used to abstract and disambiguate a predicate. De re is [λx◊P(x)](f) - f has the possible-P property - and de dicto is ◊[λxP(x)](f) - possibly f has the P-property. Also applies to □.
	From: Melvin Fitting (Intensional Logic [2007], §3.3)
	A reaction: Compare the Barcan formula. Originated with Church in the 1930s, and Carnap 1947, but revived by Stalnaker and Thomason 1968. Because it refers to the predicate, it has a role in intensional versions of logic, especially modal logic.