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I have never been able to understand any need or even any benefit of model theory. Both Rudolf Caranp and Richard Montague showed how to encode semantics directly in the syntax. Can you help me understand why model theory is not a useless appendage?

The above original question may have been phrased on the basis of false assumptions.
What I really need to know is what the gist (barest possible essence) of model theory is.

The terminology of model theory seems to be like the terminology of object oriented programming. Very common concepts are given entirely different names that retain all of their original meaning.

A method of an object is not some new newfangled thing that is very slightly different in totally inexplicable ways from a procedure of procedural programming. It is exactly a procedure from procedural programming.

The property of an object is not some new newfangled thing that is very slightly different in totally inexplicable ways from a variable of procedural programming. It is exactly a variable of procedural programming.

Model theory seems to be the same sort of thing as OOP where specific data is tied to a specific system as if it was a single packaged thing.

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    This question could be improved by answering some of the following questions: What does a good answer to this question look like (what use is an answer to this question to you)? What is the philosophical context in which you encountered a reference to model theory? What hypotheses have you formed? What has your research uncovered so far?
    – Joseph Weissman
    Sep 14, 2019 at 18:09
  • Some editing error @Josephweissman
    – Rushi
    Sep 14, 2019 at 18:26

2 Answers 2

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Model theory, as it is today understood, is a formal way to study how bits of language manage to represent the world. The fundamental idea of model theory is that you have a structure that assigns interpretations to bits of language in such a way that the structure makes each sentence in the language either true or false. You know this. The metalanguage thus formalizes important metatheoretical philosophical concepts like categoricity, completeness, and many more, as algebraic operations on the structure. Notions that can't be captured from within the structure are then seen to be external to the demands of interpretation arising solely from the statements in theory. It also easily captures metatheoretic notions like domain variation, though such notions are not widely well-understood in Carnap-style type theory. In short, it is a promising way to study theories and the philosophical properties of theories that are of interest to philosophers.

Model theory formalizes the metalanguage. Ok, and you say we don't need the metalanguage so we don't need the formalization of the metalanguage. But the elegance of the model theory makes it so that you can recover important philosophical notions, e.g. categoricity and semantic completeness (and there are many others). From the perspective of complexity engineering, it's possible the metalanguage and additional structures appear superfluous. But we can work freely in the metalanguage in a way understandable to everyone trained in elementary first-order logic, without having to simulate those notions artificially to save a pre-existing commitment to some kind of Carnapian utopia. (EDIT: Notice also the rise of model theory coincides with the more universal adoption of first-order logic over type theory in the middle of the 20th century).

It might behoove you to study less model theory and more philosophy and model theory and philosophy of model theory. Try reading the book Philosophy and Model Theory by Timothy Button and Sean Walsh. There, they walk you through the philosophical significance of model theory through detailed examples in a technically sophisticated way. Getting a grip on the philosophical perspective that motivates the use of model theory may help you see why it is something that many philosophers are prepared to accept without much hesitation. Moreover, model theory is simply now an autonomous branch of mathematics with its own fundamental philosophical issues. It may also help to point out that the philosophical concerns surrounding model theory are very different than those that were in the air during Carnap's time and to which Carnap was responding. Things like logicism and the foundation of mathematics were still fresh topics at that time, and projects like Carnap's would have had a much larger horse in the race.

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  • What benefit does model theory bring to formalizing conceptual knowledge (defined above) that could not be specified as relations between finite strings?
    – polcott
    Sep 16, 2019 at 20:03
  • Why can we bypass the need for model theory in the minimal case of specifying that the semantic property of Boolean true is construed simply as the satisfaction of specified relations between finite strings such as: "2 + 3 = 5" ↔ ⊤ and "2 + 3 = 17" ↔ ⊥
    – polcott
    Sep 16, 2019 at 20:19
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I will respond to your embedded answers from the POV of modern math.

1) You can get the gist of model theory by looking at problems that were solved by just this sort of model:

Model theory is instantiated in modern abstract algebra. Beyond syntax and embedding in a known domain, it provides a third way of looking at patterns, by giving them a range of different semantic interpretations that can be compared and contrasted.

2) Why does anyone need it?

Why does anyone need the instance-based models of algebra? By design, they always address things that can be described perfectly well by symbol manipulation: that is why the whole field is called algebra, after all.

But this approach derived because less intricate and 'fussy' ways of looking at things did not give us the right way to classify the solutions for polynomials. We could not see the reason why fifth-degree polynomials do not have a closed form solution like all simpler ones.

We needed Galois theory to show the roots are bound into groups of permutations and basic facts about the possibilities there that were solved by looking at the internal relationships between instances of 'groupness'. Those are both model theories -- one is the model theory of fields and the other is the model theory of groups.

Humans simply have a real gift for comparative semantic projections which is stronger than their ability to process symbols more directly. It provided leverage to solve problems that were impervious to more direct approaches.

3) Formal proofs are not the target of model theory. They are not even the target for models of set theory.

Their purpose is to play the same role in logic that groups, rings, and fields play in modern algebra -- particularly (perhaps, unfortunately) in establishing negative results. The independence of Choice and the Continuum Hypothesis from the rest of set theory and Real Analysis was done by 'forcing' conflicting homomorphisms, a technique model theory developed through its application in the form of abstract algebra.

But like groups, rings, and fields, they have a beautiful internal structure of their own. Like those things, we expect that structure to have applications to unrelated problems.

Consider Woodin's work on ordinals. It is an abstract study of what kinds of orderings there can be and whether and how all those kinds can be reduced into something we can describe in a finite manner.

Yes, ultimately it is going to be a set of proofs. But they won't matter as much as the perspective that they provide on how humans think about orderings. A large part of the encoding for how they are related is built on which models of set theories provide structures into which multiple models of the overall set theory can be embedded.

It is just the right way to look at the problem.

4) There are syntactic models of semantics.

No, syntax is necessarily weaker than semantics, unless you throw out parts of semantics like Real Analysis.

For instance, the Real Numbers are uncountable, and any syntactic formulation that describes them will have countable models because there are only countable finite strings of symbols, and that is what syntax is made of. That is basic model theory, the Lowenheim-Skolem theorem.

So any structure you can actually nail down syntactically is not the shared semantic model on which Real Analysts actually do their work.

You can claim that the differences are necessarily irrelevant. But how could you possibly know? Delving into second-order logic give us some really strange results about how the countable interpretations of set theory "Henkin models" behave. They do not fit our intuition -- they are not true to our natural understanding of what we are actually meaning to analyze.

To the point of the big bold part

As a software engineer, you might also realize that problems can sometimes be best addressed by simulations and heuristics, especially NP-complete problems.

We optimize graph layouts by creating fake energy fields. We improve curve fitting estimates by creating colonies of competing parameter sets and 'breeding' them. We apply social models of preference or simulate little auctions to guide which combinations of alternatives to test for optimality. We do all kinds of metaphorical 'semantic' nonsense...

Every such simulation is always a completely unnecessary elaboration of details theoretically irrelevant to the problem at hand, added onto the complexity of the problem itself.

Simulations inject rules irrelevant to the real problem to force better convergence, etc. And those produce multiple different biased approaches that need to be compared and evaluated... All superfluous.

And yet, much in the spirit of point 2 above, it is often way easier to get things done this way. And even if you need an exact solution, which cannot come from estimation, these approaches get you close faster and provide you with tools you need.

Well, simulating the problem is generating irrelevant semantic models for something instead of analyzing it in sheerly syntactical manner.

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    It seems that human conceptual knowledge (defined above) is more of an acyclic directed graph inheritance hierarchy of connections between concepts that might be directly represented as connections between finite strings. How is the extra layer of model theory helpful with this?
    – polcott
    Sep 17, 2019 at 4:46
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    @polcott Stop moving the goalposts. I answered the question. If you want to ask another question, then ask another question. i do not want to embark on some kind of endless goose chase where every time anyone reaches the end of your question, you just add more.
    – user9166
    Sep 18, 2019 at 1:08
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    @polcott What use is bread? You can't build a bridge out of it. So then it is useless...
    – user9166
    Sep 18, 2019 at 1:19
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    @polcott Besides, "There is no closed form for solving fifth degree polynbomials", "The axiom of choice and the continuum hypothesis are independent of the rest of set theory." the classfication of finite simple groups and the other useful parts of abstract algebra are parts of "human conceptual knowledge". To date, they only have proofs by means of model theory. So if you read the answer, it addresses your new question. Strange that you have to ask, since that is kind of obvious.
    – user9166
    Sep 18, 2019 at 7:42
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    @polcott I am still not giving another answer to your question if you keep essentially changing it. It is not helpful. If you have a different question, ask a different question.
    – user9166
    Sep 18, 2019 at 18:55

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