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Ideas for 'Symposium', 'Foundations without Foundationalism' and 'Ordinatio'

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31 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]
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
13626 Semantic consequence is ineffective in second-order logic [Shapiro]
13637 If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
13632 Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
13674 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
13633 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13644 Semantics for models uses set-theory [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
13670 Categoricity can't be reached in a first-order language [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
5. Theory of Logic / K. Features of Logics / 3. Soundness
13635 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
5. Theory of Logic / K. Features of Logics / 4. Completeness
13628 We can live well without completeness in logic [Shapiro]
5. Theory of Logic / K. Features of Logics / 6. Compactness
13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
13646 Compactness is derived from soundness and completeness [Shapiro]
5. Theory of Logic / K. Features of Logics / 9. Expressibility
13661 A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]