Ideas of Wilfrid Hodges, by Theme

[British, b.1941, Of Bedford College, then Queen Mary and Westfield, London.]

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2. Reason / D. Definition / 7. Contextual Definition
The idea that groups of concepts could be 'implicitly defined' was abandoned
     Full Idea: Late nineteenth century mathematicians said that, although plus, minus and 0 could not be precisely defined, they could be partially 'implicitly defined' as a group. This nonsense was rejected by Frege and others, as expressed in Russell 1903.
     From: Wilfrid Hodges (Model Theory [2005], 2)
     A reaction: [compressed] This is helpful in understanding what is going on in Frege's 'Grundlagen'. I won't challenge Hodges's claim that such definitions are nonsense, but there is a case for understanding groups of concepts together.
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
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 / A. Overview of Logic / 5. First-Order Logic
Since first-order languages are complete, |= and |- have the same meaning
     Full Idea: In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!).
     From: Wilfrid Hodges (Model Theory [2005], 3)
     A reaction: [actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy!
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
|= in model-theory means 'logical consequence' - it holds in all models
     Full Idea: If every structure which is a model of a set of sentences T is also a model of one of its sentences φ, then this is known as the model-theoretic consequence relation, and is written T |= φ. Not to be confused with |= meaning 'satisfies'.
     From: Wilfrid Hodges (Model Theory [2005], 3)
     A reaction: See also Idea 10474, which gives the other meaning of |=, as 'satisfies'. The symbol is ALSO used in propositional logical, to mean 'tautologically implies'! Sort your act out, logicians.
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
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)
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).
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.
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
|= should be read as 'is a model for' or 'satisfies'
     Full Idea: The symbol in 'I |= S' reads that if the interpretation I (about word meaning) happens to make the sentence S state something true, then I 'is a model for' S, or I 'satisfies' S.
     From: Wilfrid Hodges (Model Theory [2005], 1)
     A reaction: Unfortunately this is not the only reading of the symbol |= [no space between | and =!], so care and familiarity are needed, but this is how to read it when dealing with models. See also Idea 10477.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory studies formal or natural language-interpretation using set-theory
     Full Idea: Model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Tarski's truth definition as a paradigm.
     From: Wilfrid Hodges (Model Theory [2005], Intro)
     A reaction: My attention is caught by the fact that natural languages are included. Might we say that science is model theory for English? That sounds like Quine's persistent message.
Models in model theory are structures, not sets of descriptions
     Full Idea: The models in model-theory are structures, but there is also a common use of 'model' to mean a formal theory which describes and explains a phenomenon, or plans to build it.
     From: Wilfrid Hodges (Model Theory [2005], 5)
     A reaction: Hodges is not at all clear here, but the idea seems to be that model-theory offers a set of objects and rules, where the common usage offers a set of descriptions. Model-theory needs homomorphisms to connect models to things,
A 'structure' is an interpretation specifying objects and classes of quantification
     Full Idea: A 'structure' in model theory is an interpretation which explains what objects some expressions refer to, and what classes some quantifiers range over.
     From: Wilfrid Hodges (Model Theory [2005], 1)
     A reaction: He cites as examples 'first-order structures' used in mathematical model theory, and 'Kripke structures' used in model theory for modal logic. A structure is also called a 'universe'.
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
Up L÷wenheim-Skolem: if infinite models, then arbitrarily large models
     Full Idea: Upward L÷wenheim-Skolem: every first-order theory with infinite models has arbitrarily large models.
     From: Wilfrid Hodges (First-Order Logic [2001], 1.10)
Down L÷wenheim-Skolem: if a countable language has a consistent theory, that has a countable model
     Full Idea: Downward L÷wenheim-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)
5. Theory of Logic / K. Features of Logics / 6. Compactness
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 / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
First-order logic can't discriminate between one infinite cardinal and another
     Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another.
     From: Wilfrid Hodges (Model Theory [2005], 4)
     A reaction: This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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)