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'Mathematical Methods in Philosophy', '06: Romans' and 'Higher-Order Logic'
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12 ideas
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
10588
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First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
10298
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Some say that second-order logic is mathematics, not logic [Shapiro]
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10299
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If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
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5. Theory of Logic / A. Overview of Logic / 9. Philosophical Logic
18739
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Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-) [Horsten/Pettigrew]
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5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
10300
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Logical consequence can be defined in terms of the logical terminology [Shapiro]
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5. Theory of Logic / E. Structures of Logic / 1. Logical Form
18741
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Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew]
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5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
10290
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Second-order variables also range over properties, sets, relations or functions [Shapiro]
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
18744
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Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10292
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Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
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10590
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Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
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10296
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The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
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10297
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The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
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