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
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 (FirstOrder 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 truthvalue, a firstorder 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 (FirstOrder 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 interpretationplusvaluation is reduced to the same question for simpler formulas, or the 'HenkinHintikka style' in terms of games, or the 'BarwiseEtchemendy style' for computers.




From:
Wilfrid Hodges (FirstOrder 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 (FirstOrder 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 prooftheoretic symbol, whereas = is a semantic symbol (involving truth).

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
10288

Down LöwenheimSkolem: if a countable language has a consistent theory, that has a countable model




Full Idea:
Downward LöwenheimSkolem (the weakest form): If L is a firstorder 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 (FirstOrder Logic [2001], 1.10)

10289

Up LöwenheimSkolem: if infinite models, then arbitrarily large models




Full Idea:
Upward LöwenheimSkolem: every firstorder theory with infinite models has arbitrarily large models.




From:
Wilfrid Hodges (FirstOrder Logic [2001], 1.10)

5. Theory of Logic / K. Features of Logics / 6. Compactness
10287

If a firstorder theory entails a sentence, there is a finite subset of the theory which entails it




Full Idea:
Compactness Theorem: suppose T is a firstorder theory, ψ is a firstorder sentence, and T entails ψ. Then there is a finite subset U of T such that U entails ψ.




From:
Wilfrid Hodges (FirstOrder 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 wellbehaved class




Full Idea:
A 'set' is a mathematically wellbehaved class.




From:
Wilfrid Hodges (FirstOrder Logic [2001], 1.6)
