Ideas from 'First-Order Logic' by Wilfrid Hodges [2001], by Theme Structure

[found in 'Blackwell Guide to Philosophical Logic' (ed/tr Goble,Lou) [Blackwell 2001,0-631-20693-0]].

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5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic

10282

Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former)

			Full Idea: A logic is a collection of closely related artificial languages, and its older meaning is the study of the rules of sound argument. The languages can be used as a framework for studying rules of argument.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.1)
			A reaction: [Hodges then says he will stick to the languages] The suspicion is that one might confine the subject to the artificial languages simply because it is easier, and avoids the tricky philosophical questions. That approximates to computer programming.

5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic

10283

A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables

			Full Idea: To have a truth-value, a first-order formula needs an 'interpretation' (I) of its constants, and a 'valuation' (ν) of its variables. Something in the world is attached to the constants; objects are attached to variables.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.3)

10284

There are three different standard presentations of semantics

			Full Idea: Semantic rules can be presented in 'Tarski style', where the interpretation-plus-valuation is reduced to the same question for simpler formulas, or the 'Henkin-Hintikka style' in terms of games, or the 'Barwise-Etchemendy style' for computers.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.3)
			A reaction: I haven't yet got the hang of the latter two, but I note them to map the territory.

10285

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

			Full Idea: I |= φ means that the formula φ is true in the interpretation I.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.5)
			A reaction: [There should be no space between the vertical and the two horizontals!] This contrasts with |-, which means 'is proved in'. That is a syntactic or proof-theoretic symbol, whereas |= is a semantic symbol (involving truth).

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

10288

Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model

			Full Idea: Downward Löwenheim-Skolem (the weakest form): If L is a first-order language with at most countably many formulas, and T is a consistent theory in L. Then T has a model with at most countably many elements.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.10)

10289

Up Löwenheim-Skolem: if infinite models, then arbitrarily large models

			Full Idea: Upward Löwenheim-Skolem: every first-order theory with infinite models has arbitrarily large models.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.10)

5. Theory of Logic / K. Features of Logics / 6. Compactness

10287

If a first-order theory entails a sentence, there is a finite subset of the theory which entails it

			Full Idea: Compactness Theorem: suppose T is a first-order theory, ψ is a first-order sentence, and T entails ψ. Then there is a finite subset U of T such that U entails ψ.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.10)
			A reaction: If entailment is possible, it can be done finitely.

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

10286

A 'set' is a mathematically well-behaved class

			Full Idea: A 'set' is a mathematically well-behaved class.
			From: Wilfrid Hodges (First-Order Logic [2001], 1.6)