Combining Texts

Ideas for 'Thinking About Mathematics', 'Letter to Shumacher' and 'Introduction to 'Absolute Generality''

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4 ideas

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

8729	Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
	Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
	A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle.

5. Theory of Logic / G. Quantification / 2. Domain of Quantification

13449	We could have unrestricted quantification without having an all-inclusive domain [Rayo/Uzquiano]
	Full Idea: The possibility of unrestricted quantification does not immediately presuppose the existence of an all-inclusive domain. One could deny an all-inclusive domain but grant that some quantifications are sometimes unrestricted.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.1)
	A reaction: Thus you can quantify over anything you like, but only from what is available. Eat what you like (in this restaurant).

13450	Absolute generality is impossible, if there are indefinitely extensible concepts like sets and ordinals [Rayo/Uzquiano]
	Full Idea: There are doubts about whether absolute generality is possible, if there are certain concepts which are indefinitely extensible, lacking definite extensions, and yielding an ever more inclusive hierarchy. Sets and ordinals are paradigm cases.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.1)

5. Theory of Logic / G. Quantification / 5. Second-Order Quantification

13453	Perhaps second-order quantifications cover concepts of objects, rather than plain objects [Rayo/Uzquiano]
	Full Idea: If one thought of second-order quantification as quantification over first-level Fregean concepts [note: one under which only objects fall], talk of domains might be regimented as talk of first-level concepts, which are not objects.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2)
	A reaction: That is (I take it), don't quantify over objects, but quantify over concepts, but only those under which known objects fall. One might thus achieve naïve comprehension without paradoxes. Sound like fun.