Re: [isabelle] Representation of meta-mathematics in natural deduction (including Isabelle) and Hilbert-style systems (Church and Andrews)


Adhering to good military protocol, I state that I don't fraternize with the troops, of whom I command none, so I fraternize with no one, and prefer monologue to dialog.

On 12/7/2015 10:35 AM, Ken Kubota wrote:
Dear Members of the Research Community,

I am a community-of-one, a mere observer who can type.

If Professor Paulson decides to respond to this critique (possibly including
those in section 3.8), I would be interested in the answer, too, as my
communication with him was dominated by misunderstandings obviously caused by
different interpretations of the term "meta-language", as outlined above, such
that the main question remained unanswered: "3. Is the object language strictly
separated from the meta-language [...], and how is this done?" [Kubota, first
e-mail to Lawrence C. Paulson, May 22, 2015.] I had already previously written
It seems to me at the first glance that in Paulson's system Isabelle, in which
a new kind of variable (schematic variables) is introduced, the separation of
the object language and the meta-language is not as strict as in the works of
Alonzo Church and Peter B. Andrews. There are three reasons for this impression.

Central to your email is your emphasis on the traditional phrases of logic, "object language" and "meta-language". The problem I see with that is that though you use the phrases to try and frame the discussion about Isabelle, the phrases aren't used in the distribution documentation, which is a major representation of how the primaries want to frame the discussion.

Definitions and axioms are everything. You shouldn't be arguing subtleties if there's no common agreement on the meaning of central words and phrases.

In Logic and Computation, by Paulson, published 1987, Paulson was still sticking with tradition, and wrote:

To clear up the confusion we must distinguish the meta language from the object language. The formal language of terms and formulae is the object language. We
  make statements about the object language in the meta language.

In 1988, only a year later, he switched to "meta-logic" and "object-logic", as shown in the document you quote from, which I show again below.

I've always been under the impression that Paulson did that to give himself new or relatively unused terminology, to define specifically what he was doing, to separate his stuff from the crowd's stuff. In fact, if you search on "meta-logic" and "object-logic", the two compound words don't show up much, unlike "object language" and "metalanguage".

I'm actually a bit confused. I'm inclined to believe that Larry Paulson is who introduced their use into the language of logic, or at least, he's who made their use "popular". I could ask him about the history of the words, but he's a member of the set of everyone, and not to be fraternized with.

Back to the documentation. If the authors, in the documentation, don't use the words "object language" and "meta-language", and they do use "meta-logic" and "object-logic", doesn't that tell us there's a distinction they're trying to make?

That your points may be valid for some kind of general critique, that may be true, but you can't attack a claim that a person hasn't made. You want to talk about "object language" and "meta-language", when I only hear them talking about "meta-logic" and "object-logic".

I think they are at fault at something. It's that they describe Isabelle/HOL in terms of older logics, when, in my opinion, though it draws from far and wide, it is its own standard; it defines itself, in a very concrete way. There is something very specific to talk about.

Because they describe it in terms of other logics, to give people an idea of what it is, people then expect it to be what they want it to be, which usually is what the people are familiar with.

I did a search for "language", in PDF and DJVU verions of this book, "Isabelle - A Generic Theorem Prover":

That book is a big part of what's in the current documentation. It was basically intact, though split up, until at least 2008, in the docs ref.pdf, intro.pdf, logics.pdf, and logic-ZF.pdf.

I didn't see one use of "object language" and "meta-language" in those documents. I could have missed something.

Third, the strict distinction of the object language and the meta-language does
not seem to be a criterion for the development of Isabelle, as the term
"meta-logic" in the context of Isabelle is used for the _technical_
meta-language only, in which specific logics can be specified and implemented,
but not in the sense of the _mathematical_ meta-language: "A calculus of logics
is often called a logical framework; I prefer to speak of a meta-logic and its
object-logics. Isabelle-86 required a precise meta-logic suited to its aims and
methods. A fragment of higher-order logic (called M here for 'meta') now serves
this purpose." [Paulson, 1988, p. 3]

The lack of a strict distinction between object language and meta-language
seems inherent to natural deduction (and not only to Isabelle). Alonzo Church

You say, "Third, the strict distinction of the object language and the meta-language does not seem to be a criterion for the development of Isabelle..."

Maybe so, or maybe not, or maybe it's a lot easier to just talk about"meta-logic" and "object-logic", since they're precisely defined by Isabelle/Pure, along with the primary object-logic, Isabelle/HOL.

The beauty of proof assistants is that, unlike a logic like traditional ZFC, at build time, the heap for Isabelle/HOL is precisely and exactly defined. When I'm told that Isabelle/Pure is the meta-logic, I know exactly what "meta-logic" is. In fact, my understanding of "meta-logic" has no bearing on what the meta-logic of Isabelle/Pure is. Per binary file, called the heap, it's locked in, assuming no randomness in the physics of it all.

Compare that to abstract thoughts of the mind, where the outcome is dependent on understanding.

Maintaining the distinction between object language and meta-language, Goedel's
First Incompleteness Theorem (and hence, the Second) does not work in Peter B.
Andrews' logic Q0. Dealing with Goedel's First Incompleteness Theorem
seriously, one cannot simply ignore the fact that this proof fails in one of
the most important mathematical systems currently available to the public. As
mentioned earlier at

I'm not out to critique anything that I don't understand. I'll interpret this sentence broadly: "one cannot simply ignore the fact that this proof fails in one of the most important mathematical systems currently available to the public".

I think, actually, most everything gets ignored, in the grand scheme of things. Larry Paulson put a lot of work into the ZF logic, but then it's not used. Google introduced Dart to try and replace Javascript, but it hasn't. They tried to displace Facebook with something-or-another. They didn't. They're big, but not so big as to not get ignored in many of their new ventures.

What people care about with Isabelle is this: Isabelle/HOL. It's the result of a collective effort by many people, a product of 30 years or so, with millions of dollars having been poured into it. Tobias Nipkow, the wizard of mysterious low-level, under appreciated, under-the-hood magic, teamed up with Larry Paulson and switched the emphasis to Isabelle/HOL, which is based on the work of Mike Gordon. The large group at TUM gave it the huge boost it needed to get to the point it is now.

Nobody cares about a raw logic. Nobody cares about raw much-of-anything. If we work real hard, and give people a polished, or semi-polished product, then they might care about something we have to give away, or they may not.

A technical meta-logic as in the case of Isabelle or technical meta-languages
like ML have the advantage of offering the easy implementation and comparison
of several logics. But this advantage is relevant for the experimental stage
only. As for the final implementation of the particular preferred logic, I
would advise against using some kind of meta-logic or meta-language, as
additional features may weaken the rigor of (or even create an inconsistency
in) the logical kernel, but would suggest restricting oneself to the _direct_
encoding of the logic in order to preserve logical rigor or necessity. This
implies the use of a purely imperative computer programming language without an
implicit type system for the mathematical objects to be represented (i.e., C++).

You're looking at things wrong. If the rigor of the logic of Isabelle/HOL is weakened, and it's possibly inconsistent, then what is that? It's an opportunity for someone like you to get some bragging rights.

What did we see recently? A major accumulation of bragging rights by 3 people, though I'm not completely clear on how the bragging rights deserve to be divvied out. There's the Czech. Will he just get a Ph.D? No. He also gets the summa cum laude of bragging rights, for showing a logic is inconsistent.

Scary stuff, inconsistent logics. "Falso, you stay from me! You scary scary dude! You scare me, worse than a talking Barbie doll!!"

Really, what would be of more value? A generalized logic that stands the test of time, such as Isabelle/Pure and Isabelle/HOL, or a very narrow logic, such as what you advise?

How are people supposed to test the generalized logic, over many years, if the safer route is taken. And an inconsistent logic is not a crisis anyway. An inconsistent logic that can't be fixed with "don't do that" is a crisis. A loss of 30 years, that's a crisis.

ZFC is an inconsistent logic that was fixed with "don't do that". It's called the Axiom Schema of Separation. It's ugly, because it demands a new axiom for every subset of a set. Or is "ugly" just a matter of perception?


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