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Re: Why is model theory needed?

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On Sunday, April 9, 2023 at 12:15:07 PM UTC-5, Mostowski Collapse wrote:

Jan, 

If I open Chang and Keisler, there is much more than first-order logic. In fact, they describe model theory as universal algebra combined with logic. One cannot only look at the logic component. Skolem's contribution pushes foundational studies back to the algebraic perspective.

I would have to do some serious digging to try to answer your specific question.

A modern model theory class would probably speak of Tarski. It might mention something like "East coast" and "West coast" approaches to the subject. If I recall correctly, the East coast approach is associated with Abraham Robinson (Tarski taught on the West coast of the United States).

But, whether spoken of in a class on model theory or not, first-order model theory rests on portraying a dichotomy between syntax and semantics.

Ewald's sourcebook attributes the first notions of a different approach to algebra to Peacock and Gregory. Moreover, this is a development within British mathematics. To Peacock, he attributes the idea of a purely symbolic algebra. To Gregory, he attributes the idea that algebraic symbols could equally well represent operations in addition to numbers. Ewald then says of de Morgan:

"He was the first mathematician to appreciate the importance of the new algebra for the analysis of logic, and the first to provide a reasonably complete and explicit description of a formal calculus."

In his paper, "On the foundations of algebra," de Morgan writes:

"Algebra now consists of two parts, the technical and the logical. Technical algebra is the art of using symbols under regulations which, when this part of the subject is considered independently of the other, are prescribed as the definition of the symbols. Logical algebra is the science which investigates the method of giving meaning to the primary symbols, and of interpreting all subsequent symbolic results."

Our modern language uses different words, but what de Morgan is describing is our distinction between syntax and semantics.

This distinction is more dramatically emphasized in his paper, "Trigonometry and double algebra" where he describes the scenario:

"The proficient in a symbolic calculus would naturally demand a supply of meaning. Suppose him left without the power of obtaining it from without: his teacher is dead, and he must invent meanings for himself. His problem is: Given symbols and laws of combination, required meanings for the symbols of which the right to make those combinations shall be a logical consequence. He tries, and succeeds;he invents a set of meanings which satisfy the conditions. Has he then supplied what his teacher would have given, if he had lived? In one particular, certainly: he has turned his symbolic calculus into a significant one. But, it does not follow that he has done it in a way which his teacher would have taught him had he lived."

So, imagine that Zermelo says, "This means that" while Skolem says "Not so fast." Augustus de Morgan had already identified the situation.

That this is associated with British mathematics is important because Russell had distinguished between material implication and formal implication in "Principles of Mathematics." Tarski had famously abandoned Lesniewski because he saw Russell's work as more fruitful. If I recall correctly, Lesniewski is attributed with identifying the importance of syntactic categories. So, Tarski's work on the semantic conception of truth combines the many facets needed to use the dichotomy between syntax and semantics for the notion of "truth in a model" and the sense of "logical consequence" as a model-theoretic construct.

There is another aspect to de Morgan's scenario given above. Meaningfulness conveyed through instruction may be distinguished from abstract interpretation. So, what is apparently involved is a trichotomy between syntax, semantics, and pragmatics. This trichotomy had been introduced/popularized by Carnap, where pragmatics is intended to refer to meaningfulness as it relates to a language user. This is why I now choose to understand semantics only with respect to the use of truth in derivations. One does not really find this aspect mentioned in typical courses on "mathematical logic." And, analytic philosophers now tend to portray pragmatics so that it no longer is actually bound to human agency. But, at some point an agent must interpret, eh?

There is something else from the second paper mentioned above:

"It is most important that the student should bear in mind that, with one exception, no word nor sign of arithmetic has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct symbolic algebras. [...]

"The one exception above noted, which has some share of meaning, is the sign = placed between two symbols, as in A=B. It indicates that the two symbols have the same resulting meaning, by whatever different steps attained."

Relative to Tarski's semantic conception of truth, the arguments to a sign of equality *must* denote. The necessary truth of reflexive equality is not "given." Both logicism and the first-order paradigm introduce this on the basis of principles different from the claimed semantic theory. Under Tarski's semantic theory of truth, it is possible for reflexive equality statements to be false. That is something else one will not find in a modern class on model theory. And, it is why "foundational" claims reducing mathematics to algebra fail to encompass analysis.

I know this is not the answer for which you had hoped. But, unless you think God has told your teachers a secret, what we have to work from is the documented literature.

mitch



> But when did modern model theory start? What one would 
> nowadays hear in an university course about model theory? 
> It surely not Skolem paradox and non-categoricity of set theory. 
> 
> In first-order logic, only theories with a finite model can 
> be categorical. So what is usually meant by categoricity in 
> FOL is something weaker, and it can be defined without 
> 
> model theory, namely. A theory T is complete: 
> 
> For every formula A either T |- A or T |- ~A. 
> 
> And incomplete theory is automatically non-categorical. 
> How would you show that an incomplete theory is 
> non-categorical, with what peopel would learn today 
> 
> in an university course about model theory?
> Mitchell Smith schrieb am Sonntag, 9. April 2023 um 17:22:47 UTC+2: 
> > On Sunday, April 9, 2023 at 9:56:50 AM UTC-5, Mostowski Collapse wrote: 
> > 
> > The best answer, Jan, is probably with de Morgan and Peacock. I have not read Peacock, but, I can provide quotes from de Morgan's work on double algebra. Among other things, he explicitly discusses how an untrained intellect discovering a formal algebra is likely to apply an interpretation of the symbols independent from the intended interpretations of the original authors. 
> > 
> > One can probably relate this to the inscrutability of reference argument attributed to Quine. 
> > 
> > De Morgan's problem had been that signs for operations appeared to have the same functions across different types of numbers. Prior to complex numbers and quaternions, no one had actually noticed the problem. Thus, the meaning of operation symbols became relegated to the specification of a domain of discourse. 
> > 
> > The algebraic picture is re-introduced into "foundations" when Skolem points out that Zermelo's set theory cannot be categorical. Skolem also recognizes that the same is true of arighmetic and is credited with the first recognitikn of non-standard models. 
> > 
> > Gotta go.... can say more later. 
> > 
> > mitch 
> > > Quizz: And when did Model Theory start? In its modern form? 
> > > I mean mentioning Kant and Newton is Ok, but does it help to 
> > > give a picture of **modern** Model Theory. When did it start? 
> > > Mitchell Smith schrieb am Sonntag, 9. April 2023 um 16:01:54 UTC+2: 
> > > > On Saturday, April 8, 2023 at 2:17:04 PM UTC-5, olcott wrote: 
> > > > > Well-formed formulas have meaning only when an interpretation is given 
> > > > > for the symbols. Mendelson 
> > > > > 
> > > > > No one seems to know why model theory is needed. 
> > > > > A ∧ B → A is known to be true on the basis of the meaning of the 
> > > > > symbols, no model theory needed. 
> > > > > 
> > > > > -- 
> > > > > Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius 
> > > > > hits a target no one else can see." Arthur Schopenhauer 
> > > > Peter, 
> > > > 
> > > > A similar question had been asked on MSE two years ago. I have no reason to join the auspicious minds of MSE (who practice censorship), but, I have occasionally answered as a guest. 
> > > > 
> > > > I gave a two-part answer to the question at the link, 
> > > > 
> > > > https://math.stackexchange.com/questions/3970711/what-is-the-point-of-model-theory 
> > > > 
> > > > under 'mls'. 
> > > > 
> > > > You may find it informative. The original poster did accept my answer.

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Why is model theory needed? olcott <polcott2@gmail.com> - 2023-04-08 14:16 -0500
  Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-08 12:36 -0700
    Re: Why is model theory needed? olcott <polcott2@gmail.com> - 2023-04-08 14:40 -0500
      Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-08 12:51 -0700
  Re: Why is model theory needed? André G. Isaak <agisaak@gm.invalid> - 2023-04-08 14:27 -0600
    Re: Why is model theory needed? olcott <polcott2@gmail.com> - 2023-04-08 15:35 -0500
    Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 16:04 -0500
      Re: Why is model theory needed? [superb answer] André G. Isaak <agisaak@gm.invalid> - 2023-04-08 15:15 -0600
        Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 16:23 -0500
          Re: Why is model theory needed? [superb answer] André G. Isaak <agisaak@gm.invalid> - 2023-04-08 15:34 -0600
            Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 14:43 -0700
            Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 17:51 -0500
              Re: Why is model theory needed? [superb answer] André G. Isaak <agisaak@gm.invalid> - 2023-04-08 17:05 -0600
                Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 18:20 -0500
                Re: Why is model theory needed? [superb answer] André G. Isaak <agisaak@gm.invalid> - 2023-04-08 17:30 -0600
                Re: Why is model theory needed? [superb answer] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-08 16:35 -0700
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 16:45 -0700
                Re: Why is model theory needed? [superb answer] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-08 17:12 -0700
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 17:17 -0700
                Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 19:25 -0500
                Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 19:21 -0500
                Re: Why is model theory needed? [superb answer] André G. Isaak <agisaak@gm.invalid> - 2023-04-08 18:46 -0600
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 17:52 -0700
                Re: Why is model theory needed? [superb answer] André G. Isaak <agisaak@gm.invalid> - 2023-04-08 18:57 -0600
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 18:00 -0700
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 18:27 -0700
                Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 20:30 -0500
                Re: Why is model theory needed? [superb answer] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-08 19:12 -0700
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-09 02:41 -0700
                Re: Why is model theory needed? [superb answer] olcott <polcott2@gmail.com> - 2023-04-08 20:21 -0500
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 16:23 -0700
                Re: Why is model theory needed? [superb answer] Julio Di Egidio <julio@diegidio.name> - 2023-04-08 16:25 -0700
    Re: Why is model theory needed? [superb answer][model] olcott <polcott2@gmail.com> - 2023-04-11 08:54 -0500
  Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-09 03:30 -0700
  Re: Why is model theory needed? Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-09 07:01 -0700
    Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-09 07:56 -0700
      Re: Why is model theory needed? Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-09 08:22 -0700
        Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-09 10:15 -0700
          Re: Why is model theory needed? Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-09 19:16 -0700
            Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-09 23:40 -0700
              Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-10 00:02 -0700
                Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-10 01:05 -0700
                Re: Why is model theory needed? Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-10 04:59 -0700
                Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-10 08:59 -0700
                Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-10 09:11 -0700
                Re: Why is model theory needed? [true] olcott <polcott2@gmail.com> - 2023-04-10 11:50 -0500
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-11 00:20 -0700
                Re: Why is model theory needed? [true] olcott <polcott2@gmail.com> - 2023-04-11 08:00 -0500
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-11 10:21 -0700
                Re: Why is model theory needed? [true] olcott <polcott2@gmail.com> - 2023-04-11 12:36 -0500
                Re: Why is model theory needed? [true] olcott <polcott2@gmail.com> - 2023-04-11 12:39 -0500
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-11 11:26 -0700
                Re: Why is model theory needed? [true] olcott <polcott2@gmail.com> - 2023-04-11 13:34 -0500
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-11 11:37 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-11 21:19 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-11 22:49 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 08:36 +0200
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-11 23:47 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 00:29 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 16:15 +0200
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-12 07:49 -0700
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-12 07:51 -0700
                Re: Why is model theory needed? [true] [Montague Grammar] olcott <polcott2@gmail.com> - 2023-04-12 10:22 -0500
                Re: Why is model theory needed? [true] [Montague Grammar] Mostowski Collapse <bursejan@gmail.com> - 2023-04-12 10:02 -0700
                Re: Why is model theory needed? [true] [Montague Grammar] Mostowski Collapse <bursejan@gmail.com> - 2023-04-12 10:10 -0700
                Re: Why is model theory needed? [true] [Montague Grammar] olcott <polcott2@gmail.com> - 2023-04-12 13:21 -0500
                Re: Why is model theory needed? [true] [Montague Grammar] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-12 11:56 -0700
                Re: Why is model theory needed? [true] [Montague Grammar] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 11:59 -0700
                Re: Why is model theory needed? [true] [Montague Grammar] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-12 12:13 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 19:48 +0200
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-12 11:11 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 11:53 -0700
                Re: Why is model theory needed? [true] olcott <polcott2@gmail.com> - 2023-04-12 14:12 -0500
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 12:17 -0700
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-17 07:53 -0700
                Re: Why is model theory needed? [true] olcott <polcott2@gmail.com> - 2023-04-17 10:13 -0500
                Re: Why is model theory needed? [true] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-11 22:35 -0700
                Re: Why is model theory needed? [true] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-12 12:28 -0700
                Re: Why is model theory needed? [true] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 12:33 -0700
                Re: Why is model theory needed? [true] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-12 13:29 -0700
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-12 14:04 -0700
                Re: Why is model theory needed? [true] Mostowski Collapse <bursejan@gmail.com> - 2023-04-12 14:06 -0700
                Re: Why is model theory needed? [true] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-12 14:14 -0700
                Re: Why is model theory needed? [true] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-12 14:16 -0700
              Re: Why is model theory needed? Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-10 04:16 -0700
    Re: Why is model theory needed? olcott <polcott2@gmail.com> - 2023-04-09 10:01 -0500
      Re: Why is model theory needed? Mostowski Collapse <bursejan@gmail.com> - 2023-04-09 08:15 -0700
        Re: Why is model theory needed? olcott <polcott2@gmail.com> - 2023-04-09 10:18 -0500
      Re: Why is model theory needed? Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-09 17:51 -0700
        Re: Why is model theory needed? [truth v provability] olcott <polcott2@gmail.com> - 2023-04-10 13:12 -0500
          Re: Why is model theory needed? [truth v provability] Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-12 19:15 -0700
            Re: Why is model theory needed? [truth v provability] Julio Di Egidio <julio@diegidio.name> - 2023-04-12 19:34 -0700
            Re: Why is model theory needed? [truth v provability] olcott <polcott2@gmail.com> - 2023-04-12 22:40 -0500
              Re: Why is model theory needed? [truth v provability] Mitchell Smith <mitchellsmith0860@gmail.com> - 2023-04-13 19:44 -0700
                Re: Why is model theory needed? [truth v provability] olcott <polcott2@gmail.com> - 2023-04-13 22:29 -0500
                Re: Why is model theory needed? [truth v provability] Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-04-14 11:17 -0700
                Re: Why is model theory needed? [truth v provability] olcott <polcott2@gmail.com> - 2023-04-15 13:43 -0500
            Re: Why is model theory needed? [truth v provability] olcott <polcott2@gmail.com> - 2023-04-15 23:22 -0500

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