Combining Philosophers
Ideas for Lynch,MP/Glasgow,JM, Wilfrid Hodges and Ray Monk
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13 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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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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Since first-order languages are complete, |= and |- have the same meaning [Hodges,W]
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5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
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|= in model-theory means 'logical consequence' - it holds in all models [Hodges,W]
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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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10284
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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 / I. Semantics of Logic / 4. Satisfaction
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|= should be read as 'is a model for' or 'satisfies' [Hodges,W]
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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Model theory studies formal or natural language-interpretation using set-theory [Hodges,W]
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10475
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A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W]
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10481
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Models in model theory are structures, not sets of descriptions [Hodges,W]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
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10288
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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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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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