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Ideas for 'Defending the Axioms', 'Elucidation of some points in E.Schrder' and 'Foundations without Foundationalism'

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14 ideas

5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
13627 There is no 'correct' logic for natural languages [Shapiro]
13642 Logic is the ideal for learning new propositions on the basis of others [Shapiro]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
13668 Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
13669 Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
13667 Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13662 First-order logic was an afterthought in the development of modern logic [Shapiro]
13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
13673 The notion of finitude is actually built into first-order languages [Shapiro]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]