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'', 'Concepts' and 'First-Order Logic'
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10 ideas
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
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Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W]
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If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt]
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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
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Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt]
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The sense of a connective comes from primitively obvious rules of inference [Rumfitt]
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5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
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A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W]
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There are three different standard presentations of semantics [Hodges,W]
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I |= φ means that the formula φ is true in the interpretation I [Hodges,W]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
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Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
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5. Theory of Logic / K. Features of Logics / 6. Compactness
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If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W]
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