Ideas from 'FirstOrder Logic' by Wilfrid Hodges [2001], by Theme Structure
[found in 'Blackwell Guide to Philosophical Logic' (ed/tr Goble,Lou) [Blackwell 2001,0631206930]].
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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)

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

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There are three different standard presentations of semantics

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I = φ means that the formula φ is true in the interpretation I

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
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Down LöwenheimSkolem: if a countable language has a consistent theory, that has a countable model

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Up LöwenheimSkolem: if infinite models, then arbitrarily large models

5. Theory of Logic / K. Features of Logics / 6. Compactness
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If a firstorder theory entails a sentence, there is a finite subset of the theory which entails it

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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A 'set' is a mathematically wellbehaved class
