Combining Texts
Ideas for
'Monadology', 'Structures and Structuralism in Phil of Maths' and 'Foundations without Foundationalism'
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33 ideas
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
13627
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There is no 'correct' logic for natural languages [Shapiro]
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13642
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Logic is the ideal for learning new propositions on the basis of others [Shapiro]
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5. Theory of Logic / A. Overview of Logic / 2. History of Logic
13668
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Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
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13669
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Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
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13667
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Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13662
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First-order logic was an afterthought in the development of modern logic [Shapiro]
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13624
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The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
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13660
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Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
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13673
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The notion of finitude is actually built into first-order languages [Shapiro]
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
15944
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Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
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13629
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Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
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13650
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Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
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13645
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In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
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13649
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Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
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5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
13626
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Semantic consequence is ineffective in second-order logic [Shapiro]
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13637
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If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
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5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
2111
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Falsehood involves a contradiction, and truth is contradictory of falsehood [Leibniz]
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5. Theory of Logic / E. Structures of Logic / 1. Logical Form
13632
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Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
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5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
13674
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We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
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5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
10175
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Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price]
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5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
13633
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'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13644
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Semantics for models uses set-theory [Shapiro]
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5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13636
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An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
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13670
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Categoricity can't be reached in a first-order language [Shapiro]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13648
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The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
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13675
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Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
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13658
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Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
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13659
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Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
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5. Theory of Logic / K. Features of Logics / 3. Soundness
13635
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'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
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5. Theory of Logic / K. Features of Logics / 4. Completeness
13628
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We can live well without completeness in logic [Shapiro]
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5. Theory of Logic / K. Features of Logics / 6. Compactness
13630
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Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
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13646
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Compactness is derived from soundness and completeness [Shapiro]
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5. Theory of Logic / K. Features of Logics / 9. Expressibility
13661
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A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]
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