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Ideas from 'Introduction to 'Absolute Generality'' by Rayo,A/Uzquiasno,G [2006], by Theme Structure

[found in 'Absolute Generality' (ed/tr Rayo,A/Uzquiano,G) [OUP 2006,978-0-19-927643-1]].

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4. Formal Logic / F. Set Theory ST / 1. Set Theory
13451 The two best understood conceptions of set are the Iterative and the Limitation of Size
                        Full Idea: The two best understood conceptions of set are the Iterative Conception and the Limitation of Size Conception.
                        From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
13452 Some set theories give up Separation in exchange for a universal set
                        Full Idea: There are set theories that countenance exceptions to the Principle of Separation in exchange for a universal set.
                        From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2)
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
13449 We could have unrestricted quantification without having an all-inclusive domain
                        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
                        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
                        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.
19. Language / F. Communication / 5. Pragmatics / a. Contextual meaning
13448 The domain of an assertion is restricted by context, either semantically or pragmatically
                        Full Idea: We generally take an assertion's domain of discourse to be implicitly restricted by context. [Note: the standard approach is that this restriction is a semantic phenomenon, but Kent Bach (2000) argues that it is a pragmatic phenomenon]
                        From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.1)
                        A reaction: I think Kent Bach is very very right about this. Follow any conversation, and ask what the domain is at any moment. The reference of a word like 'they' can drift across things, with no semantics to guide us, but only clues from context and common sense.