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Logic consists of proofs, not bold assertions. Semantics means assigning truth-values, it's kind of unavoidable, syntax alone is just string concatenation. You can't do away with axioms either.

Also, is it correct to say that a model could suffer from infinite regress? I am under the impression that self-reference is common with diagonalization constructions, it results in a regress, but one that terminates. That's how Gödel's incompleteness results work, the model provides the semantics. A lot of models are incomplete or even inconsistent, but incompleteness isn't a terrible outcome. Models can be extended indefinitely.

I don't see where semantics coming from meta-semantics comes into play, and the regression, or how value-judgments are applicable.

I read the Tarski part ”The rules of logic have been given to us by Tarski, which in turn got them from Mr. Metatarski” from Girard's On the meaning of logical rules I : syntax vs. semantics, and he seems to be more criticizing Fregean Sinn or Sense, meaning natural language applications of logic, not talking about mathematical logic.

Someone told me that the point of ”we presuppose the existence of a meta-world, in which logical operations already make sense” is one related to formal logic, where I think it's referring to natural language, or formalizations of natural language and not mathematical models. He seems to be saying something similar to post-Wittgensteinian inferentialism, I guess. Inferentialist semantics. It seems to be more a theory of natural language, and not a mathematical model

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    I'm sorry, user59567, I don't understand the question.
    – Scott Rowe
    Commented Jul 13, 2022 at 1:55
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    Sounds many thoughts and questions here... Re your "A lot of models are incomplete or even inconsistent", it's a basic knowledge in logic that only deductively closed theory may be incomplete or inconsistent which doesn't apply to models. Re your ”we presuppose the existence of a meta-world, in which logical operations already make sense... where I think it's referring to natural language, or formalizations of natural language and not mathematical models", not necessarily as reflected in Tarsk's infinite hierarchical of formal languages as opposed to semantically close natural language... Commented Jul 13, 2022 at 3:22
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    Re your "Inferentialist semantics. It seems to be more a theory of natural language, and not a mathematical model", Inferentialist semantics is also commonly known as proof-theoretic semantics and equally applies to any logic/math system as Tarsk's, in addition to Wittgensteinian meaning-is-use doctrine applied in informal natural language... Commented Jul 13, 2022 at 3:51
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    But the regress' ghost is there: if we do not know how to use logical rules, how can we explain them with logical semantics, that itself uses rules? But this is the way language works. Commented Jul 13, 2022 at 6:52
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    What is the regress exactly, that any semantic interpretation needs another interpretation to be interpreted? If so, this is known as Wittgenstein's rule-following paradox, as is his solution:"there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" and "going against it"". Ryle's regress and epistemic regress are similar.
    – Conifold
    Commented Jul 13, 2022 at 11:22

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I think you're asking about whether the abstract idea of a "model" is well-founded. For a given axiomatic theory of logic, a model is intended to be some set of objects that satisfies all the formal axioms.

In theory, a model allows us to assign a truth value to every logical proposition, whether or not that proposition can be proved from the axioms.

When you say "model-theoretic regress" I believe you are talking about the regress from talking about propositions and axioms as purely textual things, to talking about objects in the model, to which the textual propositions supposedly refer.

I agree with you that this is problematic, because of the Pragmatic Maxim.

Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.

The notion of a model is unjustified, because it is disconnected from any physical object that the mathematician works with. The objects a human mathematician directly interacts with includes the text of the proof, diagrams and images on paper or in their imagination, and other activity of their own neurons. The mathematician performs calculations involving these physical objects. The mathematician is probably limited to what a Turing machine can do, if we suppose that the mathematician's brain can be simulated on a Turing machine, which seems likely, or at least has not been ruled out.

The notion of a model - that yields a true or false answer to every question we ask in the logical theory, whether or not the question can be answered according to the formal axioms - goes beyond anything a Turing machine could do, and goes beyond anything the mathematician could do. The model "decides" everything. What practical connection does the mathematician have to the model? If the model says a certain proposition is true - though this proposition can be neither proved nor disproved by any axiom (or meta-axiom, or informal rule of thought) the mathematician knows - then in what practical way can this possibly matter to the mathematician?

To take a specific example, consider the continuum hypothesis. The continuum hypothesis is independent of ZFC; we may take the axioms of ZFC and assume additionally the continuum hypothesis, or we may take the axioms of ZFC and assume additionally the negation of the continuum hypothesis. Usually mathematicians do not declare whether they are taking the continuum hypothesis to be true or to be false. It does not matter to their work.

So, if they haven't even considered the continuum hypothesis, what model are they working in? No model of ZFC + (continuum hypothesis) is also a model of ZFC + (negation of continuum hypothesis). By declining to take a position on the continuum hypothesis, a mathematician cannot be working in any specific model, and thus their propositions cannot "refer" to any specific model.

Even if the mathematician does declare their stance on the continuum hypothesis for the purposes of a particular paper, there are an infinite set of other propositions, that, like the continuum hypothesis, are also independent of ZFC, and which the mathematician has not declared a stance on. So, again, the propositions they derive cannot refer to any specific model.

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    "The notion of a model is unjustified, because..." Then you have to discard almost all of mathematics from ancient Mesopotamia to the invention of mathematical logic, which was almost all done on the basis of a model. Commented Jul 13, 2022 at 6:05
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    "So, if they haven't even considered the continuum hypothesis, what model are they working in?" It doesn't matter. The proofs apply to any model that satisfies the axioms that they use. Commented Jul 13, 2022 at 6:07
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    @DavidGudeman Yes, the proofs apply to any model that satisfies the axioms - which means that nothing in the proof refers to any specific object in any specific model! I don't think the Mesopotamians were using anything like the modern notion of a model of a logical system; the closest thing they had were physical pictures of a few circles and other geometric objects, or similar pictures in their mind's eye. They needed no notion of the set of all possible arrangements of geometric objects.
    – causative
    Commented Jul 13, 2022 at 6:10
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    OK, you're right that they didn't deal with models, but they did deal with model objects, things like natural numbers and ratios and geometric lines which are not physical objects. Up to the nineteenth century, all mathematics was done by thinking about these sorts of objects, not by thinking about textual axioms. Models are just a formalization of the connection between notation and traditional mathematical thought. Commented Jul 13, 2022 at 8:13
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    It certainly is a problem when we start talking about things we can't think about! Tattoo that on your arms, everyone!
    – Scott Rowe
    Commented Jul 13, 2022 at 15:36

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