Combining Texts
Ideas for
'What is Logic?st1=Ian Hacking', 'On Formally Undecidable Propositions' and 'Mr Strawson on Logical Theory'
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14 ideas
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
13845
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The various logics are abstractions made from terms like 'if...then' in English [Hacking]
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13840
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First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking]
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13844
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A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking]
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
13842
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Second-order completeness seems to need intensional entities and possible worlds [Hacking]
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5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
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Logical languages are rooted in ordinary language, and that connection must be kept [Quine]
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5. Theory of Logic / E. Structures of Logic / 1. Logical Form
22434
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Reduction to logical forms first simplifies idioms and grammar, then finds a single reading of it [Quine]
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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13837
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With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking]
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5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
13839
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Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13843
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If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking]
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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17886
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The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
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5. Theory of Logic / K. Features of Logics / 2. Consistency
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Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P]
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5. Theory of Logic / K. Features of Logics / 3. Soundness
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If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh]
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5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
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10621
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Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
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