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In my reading of Alfred Tarski's model theory, a theory is a formal system whose sentences are without inherent meaning, but which becomes meaningful (e.g. having truth values) only after a model provides an interpretation of its sentences as referring to anything in which some notion of truth may obtain, such as the structure of the empirical world.

On the other hand a colloquial model, has an inherent structure all its own, such as a "model car" or some other structure which may be interpreted as referring to a truth-bearing structure, e.g. an actual car. Indeed, this quote taken from Julian Jaynes's "The Origin of Consciousness in the Breakdown of the Bicameral Mind" comports with this understanding of model, and goes further to refer to theory as the interpretation:

The Bohr model of the atom is that of a proton surrounded by orbiting electrons. It is something like the pattern of the solar system, and that is indeed one of its metaphoric sources. Bohr’s theory was that all atoms were similar to his model. The theory, with the more recent discovery of new particles and complicated interatomic relationships, has turned out not to be true. But the model remains. A model is neither true nor false only the theory of its similarity to what it represents.

The sincere investigator of truth might become befuddled by Tarski's seemingly reversed meanings from that in common use -- confused to the point that meta-understanding is subverted, as "understanding" is further described by Jaynes:

A theory is thus a metaphor between a model and data. And understanding in science is the feeling of similarity between complicated data and a familiar model.

Tarski was no lightweight so we are left with a rather urgent question regarding this befuddlement:

What is it about colloquial meta-understanding that is so wrong-headed regarding empirical reality that Tarski sought to reverse the meanings of these words OR what is it about my understanding of Tarski (and/or Jaynes and/or colloquial usage) that renders Tarski's meanings consilient with the colloquial?

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  • I don't know if Jaynes was correct, but at least if not he was interesting.
    – Scott Rowe
    Oct 29, 2023 at 17:12
  • The meanings are not reversed, what shifts is the target of discourse. In the scientific (Jaynes's) case, the target is reality and the model is "interpreted" by something which is real. In the mathematical (Tarski's) case, the target are models themselves (in something like the first sense), and it is the formal theory that is now "interpreted" by what's in them. This is particularly explicit in the semantic view of scientific theories, where models are taken to be set-theoretic structures, see SEP.
    – Conifold
    Oct 29, 2023 at 21:10
  • In how many ways can we de-befuddle this? Oct 30, 2023 at 2:54

3 Answers 3

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The word model has different meanings. According to the online Websters:

  1. a usually miniature representation of something a plastic model of the human heart
  2. a type or design of product (such as a car) offers eight new models for next year, including a completely restyled convertible
  3. a system of postulates, data, and inferences presented as a mathematical description of an entity or state of affairs
  4. ARCHETYPE
  5. an example for imitation or emulation
  6. one who is employed to display clothes or other merchandise has appeared as a model in ads for swimsuits
  7. a person or thing that serves as a pattern for an artist
  8. VERSION sense 2: an experimental model of a bionic arm
  9. a description or analogy used to help visualize something (such as an atom) that cannot be directly observed
  10. structural design
  11. an organism whose appearance a mimic imitates
  12. ANIMAL MODEL
  13. dialectal British : COPY, IMAGE
  14. obsolete : a set of plans for a building

By my reading, 2, 4, 6, 7, 10, 13, and 14 would be enough to justify Tarski's new usage (I assume 3 was added post-Tarski so I'm not including it).

The notion of a theory as an uninterpreted set of well-formed formulas (WFFs) predates Tarski's model theory. It was inspired by Hilbert's geometry and the discovery that the axioms of geometry could be changed. There was a movement toward removing all meaning from mathematics and treating the manipulation of symbols as all there was (largely among logicians, not mathematicians). The problem with this approach, however, it that it removes any usefulness from mathematics.

However, the meaning of the sentences was never really absent, what was going on was the the symbols were used with different meanings in different contexts. Non-Euclidean geometries were proven consistent by demonstrating other models (in Tarski's sense) of the axioms, different from the model Euclid had in mind. Tarski was just formalizing that relationship.

Jayne's view, on the other hand, seems to be specific to just one of the definitions: 9.

Amusing aside: it seems you can date when entry 8 was added within ten years or so by the example. That incorrect use of the word "bionic" for "super-prosthetic" goes back to the Six Million Dollar Man which came out in 1973. I speculate that the definition dates from 1973 to 1983 because by 1983 (according to my recollection) the term was no longer in use except in jokes about the show.

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  1. Tarski operates on the level of formal theories.
    He discriminates between an axiomatized theory on one hand, and different structures which satisfy the axioms on the other hand. Hence a model of an axiomatized theory is a map from the components of the theory to the elements of a set with certain properties, which is the model and its structure.

    Example: Mathematical group theory deal with groups. A group (G,°,e) is a set of elements together

    • with an associative composition °: G x G -> G,

    • and a distinguished element e from G, such that x°e=e°x for all elements x of G (neutral element),

    • and such that for each x from G exists an y from G satisfying x°y=y°x=e (existence of inverse).

    A model for group theory are the integer numbers (Z,+,zero) with Z the set of integers, + being the addition, and zero the distinguished element.

  2. The colloquial use of "model" is not a formalized concept. Nevertheless, a “model car” in the sense of a prototype is a concrete vehicle, which shows that a newly designed series of cars can be built and satisfies the requirements. In this sense the model car can be considered an image of the designed cars.

  3. I will not go into the exegesis of Jaynes's thoughts because already the above quote from his work did not convince me.

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You’re making the essential mistake everyone makes about Tarskian Truth, which is to assume that there’s only one model because there’s only one True.

Tarskian truth theories, and the model-theoretic semantics of interpretation, are a mathematical methodology for the treatment of formal languages. Models form an algebra, and one can quite happily build models of counterfactual theories.

But importantly, the models themselves are not things that are true or false in the sense described by Tarski’s truth predicate schemata, because to define Truth in a language builds on top of an existing interpretation of a language/theory using a model.

Rather, it’s exactly because of Tarski that it now makes to talk about the relative interpretation of models and theories in each others’ languages - that one can compare what each of them says about what is true in compositional terms.

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  • I don't see any indication the question makes that mistake. Oct 29, 2023 at 23:10
  • @DavidGudeman, thanks for the comment - I suppose I’ll need to amend my answer to more directly address the question’s use of the word “truth”. Importantly, this isn’t a mistake Jaynes is making.
    – Paul Ross
    Oct 30, 2023 at 7:47
  • In short, OP says Tarski’s theories follow from truthful models, but that’s not what Tarski does - Truth comes later. Models have semantic values, that we by convention call T and F.
    – Paul Ross
    Oct 30, 2023 at 7:50

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