10476 | 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. |
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. |
10478 | 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! |
10477 | |= 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. |
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) |
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). |
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. |
10474 | |= 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. |
10473 | 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. |
10481 | 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, |
10475 | 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'. |
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) |
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) |
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. |
10480 | 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. |
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) |