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https://en.wikipedia.org/wiki/Model_theory
Why can't semantics be directly expressed in the formal language?

This is the key part of model theory that I don't understand:

https://www.lesswrong.com/posts/MG8Yhsxqu9JY4xRPr/mental-context-for-model-theory

At its core, model theory is the study of what you said, as opposed to what you meant. [...]

This ability to disconnect the intended interpretation from the available interpretations is the bedrock of model theory. [...]

It's about discovering how well (or poorly) a given theory constrains its interpretations. It's a toolset used to discuss interpretations in general.

At its core, model theory is the study of what a mathematical theory actually says, when you strip the intent from the symbols.

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    I know who this is. How many sockpuppet accounts are allowed per person? – user4894 May 30 at 6:04
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    ps -- Moderators, this handle is the same one used on Reddit by You Know Who. There's objective evidence for the identity of this sockpuppet, it's not just my opinion. – user4894 May 30 at 6:28
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    Model theory is the semantics: it is the mathematical way to formalize the semantics concepts needed to explain the way formal languages and theories work. – Mauro ALLEGRANZA May 30 at 6:59
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    No one ever said we need model theory to express semantics. It just so happens that model theory has a lot of useful applications in fields as diverse as philosophy of mathematics, philosophy of science, philosophy of language, philosophical logic, and metaphysics.The philosophical significance of that utility value is still being assessed, and to my mind we're still in a relatively nascent stage of understanding the relationship between philosophy and model theory in any systematic way. – transitionsynthesis May 30 at 15:28
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    Who are these people you're showing it to? It's perfectly open to you to reject model theory and try a Carnapian approach, but you should anticipate that it's not going to be widely considered to be plausible. Since you're going against orthodoxy, you need a strong idea of why your approach is superior to model theory. People may not agree with you but if you can show them you actually know your stuff, they'll let you do your thing. And, hey, stick to it if that's your thing. People like Chalmers have made an entire career out of reviving unfashionable Carnapian and Fregean ideas. – transitionsynthesis May 30 at 15:37
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I'm hesitant to answer this question, since I believe that the OP is someone I've already given up discussing this topic with, but on the off-chance since I think this is actually a reasonably well-posed question I'll take a stab at it.


I could talk about how this makes things more confusing - e.g. from the perspective of PA + "PA is inconsistent" it would be the case that 0=1 is false and not true but every sentence of the form "X is true" would be true - and this would already justify a strong negative response from the mathematical community, but all this pales next to the real point:

We already have words for what you're talking about.

Namely, "(T-)provable," "(T-)disprovable," and so forth. So this isn't a situation where you're identifying a new concept and giving it a name, but rather a situation where you want to make it harder for us to talk about a given kind of thing. Just because you don't want to consider something doesn't mean that the rest of us will accept not being able to consider that thing.

Does this mean the community is unwilling to talk about the meaning of truth? Not at all, but you have to actually say something. Mathematicians (myself included for that matter) will be much more sympathetic to contentful arguments than proposed language changes.

For example, maybe you want to argue that the community use of the word "true" has underlying problematic philosophical assumptions (I'd argue it does), and that a different notion captures many basic ideas about truth better (I'd argue that all I've seen don't, including yours). Depending on precisely what you have to say about it people may be interested.

You may even want to, in the course of that argument, avoid using the word "truth" in the model-theoretic way. That's also fine. The only thing that's problematic is when you insist on a different usage as being obviously correct. If you really don't want to use existing language, create your own (say, "ModelTrue" versus "TheoryTrue," etc.) - although people may roll their eyes, this is totally unobjectionable. But you have to precisely define what you're talking about, and not use existing technical language differently.


Now, what about model theory?

To "get the gist" of model theory, you need to look at more than just philosophy: you need to look at the actual mathematics (which it is, after all, a part of). Model theory isn't just the study of models, it's also the study of theories (and underlying formal logical systems) - really, it's the study of how the two are connected. It's only if we ignore the vast majority of actual mathematics that it becomes anti-formal-systems-y: given that the "semantic" side is far more entrenched in mathematics (formalism came way late to the party), you should actually think of model theory as a huge force in favor of the consideration of formal systems.

Even ignoring that synthesis as interesting in its own right, it has real applications - indeed, which make syntax look good! Model theory demonstrated that bringing "syntactic" ideas into "semantic" contexts we can prove purely "semantic" results, especially using the compactness theorem (e.g. proving that every injective polynomial map $\mathbb{C}^n\rightarrow\mathbb{C}^n$ is surjective, or the Ax-Kochen theorem). Crucial to its success is the recognition of this difference: that provability and truth-in-a-model-or-class-of-models are different notions (although in certain situations they coincide).


The key principle throughout all of this is:

More is better.

For example, model theory demonstrates that even if one isn't interested in formal theories, being able to talk about them is useful for proving things one is interested in. Conversely, we often demonstrate the consistency of a formal theory - which is a purely "syntactic" fact - via model-theoretic means (e.g. consider the proof that ZFC+CH is consistent if ZFC is), so model theory provides support for model-theoretic truth too (rather unsurprisingly).

I think this is the real sticking point. You ask in a comment "Wouldn't it be simpler to define axioms as true, and thus avoid the whole separate infrastructure?" but that's missing the point. You don't have to use anything you don't want to, although you do need to accept the way technical terms are (currently) defined. It's there for those who want it.

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The problem with your alternative, which was recognized by Carnap himself, is that you need an infinite hierarchy of (meta)languages.

Every arithmetical sentence G1 which is, for instance, irresolvable [neither provable nor refutable] in the language S1 is yet determinate [analytic or contradictory] in the language S1; in the first place there exists a richer syntax language S2, within which the proof that G1 is analytic or that G1 is contradictory can be stated; and secondly, there exists an object-language S3 of which S1 is a proper sub-language, such that G1 is resolvable in S3. But there exists neither a language in which all arithmetical terms can be defined nor one in which all arithmetic sentences are resolvable. [. . . ] In other words, everything mathematical can be formalized, but mathematics cannot be exhausted by one system; it requires an infinite series of ever richer languages. (Syntax §60d)

In some contexts, when expressing only a limited set of truths is okay, using a finite hierarchy is the solution. For example, in the so-called "axiomatic semantics" of programming languages there is a simple distinction between the underlying programming language and the FOL (first order logic) used to say "semantic" stuff about the program fragments. But there are limits (see link for details) for the kinds of truths you can express in that approach.

  • I just don't get the gist of model theory, see the second half of my post to see what I don't understand. – user39777 May 31 at 16:10
  • @exfalsoquodlibet: the article you link to (now) already tries to explain that "for dummies", which might not always work out as intended. What is your level of math and logic knowledge? Undergraduate? Which year? And in which major(s)? I'm asking because just telling you some facts you probably read somewhere already might not help improving your understanding. Unless you can further explain why you don't see model theory as useful... – Fizz Jun 2 at 0:34
  • Also, the person who that lesswrong piece is just learning model theory (his article also contained some errors, which were pointed out in the comments). So it might not be the best advocacy piece for model theory... Try reading the SEP article on model theory plato.stanford.edu/entries/model-theory It's still somewhat accessible to neophytes, at least in its introductory part. – Fizz Jun 2 at 0:44
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There’s a complaint once made by someone famous that once mathematicians get hold of a concept they write about it in such a way that not even the originators of the concept no longer understand it. Einstein made the same complaint about relativity and mathematicians and he was no slouch at mathematics. It’s still a complaint made today, including by mathematicians. For example, Alexandrov, a topologist said there are two kinds of mathematical books: those you can’t get beyond the first page, and those you cannot get past the first sentence.

Model theory is a formalisation of what semantics means and not semantics itself. Consider a word like ‘kick’. We all know what kick means. To kick a football along the ground. To kick a habit. The kick of a gun and so on. These are typically what is understood as part of the semantic field of the word and are the kind of examples you might find in a dictionary to help learn this word in use.

However, the formal role played by this word is a verb, this tells us how it relates to other words in a sentence in a formal way. Generally we think of this as syntax or grammar. In Kantian terms, it’s analytic. It’s meaning only comes from how the words relate to each other and not to the real word. Semantics then, in Kantian terms is synthetic. It tells us how a word corresponds to things or actions in the world. It can’t be deduced from a merely analytic analysis of the sentence. One has to go outside of the sentence to the world itself. This more broadly is part of what is philosophically known as the correspondence theory of truth: that propositions about the world are found to be true or false by referring facts of the real world and more generally, it’s meaning.

But this is nothing like how model theory works. It merely models this correspondence and is not the actual correspondence itself (most likely this is why it’s called model theory because it models, modelling!), that is it models the relation between syntax and semantics, but is neither the syntax, nor the grammar, nor the semantics of an actual real world language. To put it in another way, it’s not semantics-in-itself but mathematics-for-itself.

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