Ideas from 'Higher-Order Logic' by Stewart Shapiro [2001], by Theme Structure
[found in 'Blackwell Guide to Philosophical Logic' (ed/tr Goble,Lou) [Blackwell 2001,0-631-20693-0]].
green numbers give full details |
back to texts
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expand these ideas
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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10301
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The axiom of choice is controversial, but it could be replaced
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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10588
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First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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10298
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Some say that second-order logic is mathematics, not logic
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10299
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If the aim of logic is to codify inferences, second-order logic is useless
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5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
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10300
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Logical consequence can be defined in terms of the logical terminology
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5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
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10290
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Second-order variables also range over properties, sets, relations or functions
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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10297
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The Löwenheim-Skolem theorem seems to be a defect of first-order logic
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10590
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Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
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10292
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Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model
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10296
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The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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10294
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Second-order logic has the expressive power for mathematics, but an unworkable model theory
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8. Modes of Existence / B. Properties / 11. Properties as Sets
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10591
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Logicians use 'property' and 'set' interchangeably, with little hanging on it
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