Combining Philosophers

Ideas for Eubulides, Stewart Shapiro and Hannah Arendt

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57 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]
10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [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]
10298 Some say that second-order logic is mathematics, not logic [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]
10299 If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
10300 Logical consequence can be defined in terms of the logical terminology [Shapiro]
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
10259 The two standard explanations of consequence are semantic (in models) and deductive [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 / B. Logical Consequence / 5. Modus Ponens
10257 Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
10253 Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
10251 The law of excluded middle might be seen as a principle of omniscience [Shapiro]
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [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 / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
10209 A function is just an arbitrary correspondence between collections [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 / G. Quantification / 5. Second-Order Quantification
10290 Second-order variables also range over properties, sets, relations or functions [Shapiro]
5. Theory of Logic / G. Quantification / 6. Plural Quantification
10268 Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
10235 A sentence is 'satisfiable' if it has a model [Shapiro]
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
10240 Model theory deals with relations, reference and extensions [Shapiro]
13644 Semantics for models uses set-theory [Shapiro]
10239 The central notion of model theory is the relation of 'satisfaction' [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]
10214 Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
10238 The set-theoretical hierarchy contains as many isomorphism types as possible [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]
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [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]
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [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]
5. Theory of Logic / L. Paradox / 1. Paradox
6007 If you know your father, but don't recognise your father veiled, you know and don't know the same person [Eubulides, by Dancy,R]
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
6006 If you say truly that you are lying, you are lying [Eubulides, by Dancy,R]
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / b. The Heap paradox ('Sorites')
6008 Removing one grain doesn't destroy a heap, so a heap can't be destroyed [Eubulides, by Dancy,R]