Email Gateway (Aug 22 2022 at 12:03): Email Gateway (Aug 22 2022 at 12:03):From: Ken Kubota <mail@kenkubota.de>
Dear Gottfried Barrow and List Members,
Thank you for your comment at
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2015-December/msg00023.html
concerning my note available at
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2015-December/msg00018.html
Let me respond to the last three paragraphs of your comment quoted below which
summarize the most important questions.
"It is the logical reliability due to reduction that builds the core of the
LCF-approach to theorem proving (or LCF-style full expansiveness [cf. Gordon,
2000, p. 178]): 'All proofs are ultimately performed in terms of a small set of
primitive inferences, so provided this small logical kernel is correct the
results should be reliable.' [Harrison, 2009, p. 60]" [Kubota, 2015a, pp. 12 f.]
The idea behind the LCF-approach appealing also to those who do not share
metatheoretical concepts such as the semantic approach (model theory) is to
reduce the set of rules and elements, such that the logical core is minimized
without loss of expressiveness. I called it the "The Principle of Reductionism"
[Kubota, 2015a, p. 11].
In Peter B. Andrews' logic Q0 this principle is - by the use of syntactical
means only - consequently carried out, such that the basic means of the
language are reduced without loss of expressiveness to
In other words, with this simple minimum core logic of Q0 all of formal logic
and large parts of mathematics, and with type variables (polymorphic type
theory) and the binding of type variables with lambda (dependent type theory),
presumably all of mathematics can be construed, expressed and proved purely
syntactically. Due to its simplicity, the logical kernel becomes extremely
small and logically robust, i.e., mathematically safe.
In the R0 implementation, the propositional connectives therefore are not part
of the logical core, but their definitions are "outsourced" into a file and,
for example, the rule of modus ponens into a separate proof template file. In
fact, the expressiveness of R0 as a logistic system is stronger than that of
Isabelle/HOL. Isabelle/HOL allows for type variables and therefore has the
expressiveness of polymorphic type theory, but only R0 allows, in addition to
that, the binding of type variables with lambda as abstraction operator and
therefore enables the construction of types required for dependent type theory.
Moreover, axiomatic set theory as a whole is subject to critique: "Axiomatic
set theory uses axioms to justify the existence of certain sets. Unless the
proof of existence from these axioms is provided, any construction cannot be
introduced into the domain of discourse. This approach, however, is not
systematic and subject to arbitrary conditions. For example, the most common
formulation, the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC),
comprises axioms for the empty set, union, one-element sets, power sets etc.,
trying to cover the standard sets used in mathematics. But there always remain
legitimate sets not contained by these axioms; for example it is well known
that large cardinal numbers and certain universes cannot be treated within ZFC
(without assuming additional axioms), although from a purely logical point of
view there is no difference between the mathematical objects covered by ZFC and
those not covered.
This problem has its origin in the approach itself, as axiomatic set theory
makes use of non-logical axioms to establish sets, thus trying to avoid
paradoxes via content restrictions instead of the proper specification of the
formal language. Opting this method, axiomatic set theory including ZFC is
bound to remain a preliminary and auxiliary approach, but not the foundation of
mathematics." [Kubota, 2015a, p. 18].
The strict distinction of the object language and the meta-language was so
important to Church that he emphasized it a second time in his article: "We
must, of course, distinguish between formal theorems, or theorems of the
system, and syntactical theorems, or theorems about the system, this and
related distinctions being a necessary part of the process of using a known
language (English) to set up another (more exact) language." [Church, 1940, p.
61]
In natural deduction, as for example in Isabelle, metatheorems (in terms of
Hilbert-style systems) become part of the formal language (the object language)
itself. So from the point of view of axiomatic (Hilbert-style) deductive
systems, the expressiveness in natural deduction is shifted towards the
meta-level, possibly at the cost of expressiveness at the object level. This
caused me to introduce an own terminology with the notions of "object logic"
and "meta-logic" in my second publication on Goedel [cf. Kubota, 2015] until
realizing that the definitional line between "object logic" and "meta-logic" is
identical with that between axiomatic (Hilbert-style) deductive systems and
natural deduction, having found the number of allowed occurrences of the
deduction symbol (turnstile) in a theorem as a formal criterion. The
translation mechanism from a theorem to the existence of its proof (or vice
versa) in Goedel's First Incompleteness Theorem is in Paulson's presentation
his theorem 'proved_iff_proved_PfP' - called (P) in my article -, in Andrews'
presentation the implicit rule used for step 7101.4. Whereas in Andrews' system
this implicit rule immediately results in inconsistency [cf. Kubota, 2015, pp.
10 f.], I am not sure whether this applies to natural deduction, too. Because
of a limited expressiveness at the object level, the antinomy might be
construed without causing an openly visible inconsistency in the formal
language, since "the two occurrences of the deduction symbol [...] remain an
obstacle for obtaining a paradox [i.e., an inconsistency] in the metalogic
[i.e., in natural deduction]" [Kubota, 2015, p. 14]. Then, "Gödel's first
incompleteness theorem: If consistent, our theory is incomplete." (Paulson) is
rendered from a hypothetical judgement to a trivial tautology in the form of a
conclusio ex falso, since the theory under consideration is actually
inconsistent. However, an antinomy (paradox) should not be expressible at all
(in formal logic and mathematics).
If natural deduction is ruled out for the foundation of mathematics, there
would be no other "fix" for Isabelle than switching to an axiomatic
(Hilbert-style) deductive system. Then, if one does not want to downgrade from
higher-order logic (which would artificially limit the expressiveness of the
language) and prefers to remain in the realm of Russell's invention, namely
type theory [cf. Russell, 1908], basically three options would be left:
a) Church's Hilbert-style system of 1940 [cf. Church, 1940] (simple type
theory, i.e., without type variables)
b) Andrews' logic Q0 [cf. Andrews, 2002, pp. 210-215], an improved formulation
with identity (equality) as the main notion (also simple type theory, i.e.,
without type variables), first published in 1963 as a simplification of
Henkin's variant of Church's formulation [cf. Andrews, 1963, pp. 345 f., 350
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Email Gateway (Aug 22 2022 at 12:04):From: Gottfried Barrow <igbi@gmx.com>
Ken,
I am addressing you some, but not soley. I think I've said enough to
make it clear that I'm not a representative, in any way, of the
professional community that surrounds Isabelle.
It is my goal to follow through and not have my name end up here much,
but everyone serves their purpose, and there are things of value I can
bring attention to, even if it's mixed in with a lot of noise. The
existence of a meta-logic and object-logic in Isabelle is an endless
source of confusion for newcomers, and a few basics can clear that up.
My last email served to bring attention to that once again, along with
the reference to "Logic and Computation", by Paulson.
Here is the thesis of this email: proof assistants are a game-changer. I
will try to bring my points back around to that.
Ken, please don't miss the part below, in my lecture on etiquette,where
I say this:
If, in a reply to me, you add emails to people who weren't party to the
discussion, you are a gross violator of decent etiquette.
2ND THING 1ST, PUT UP OR...
The last email, I thought that I bordered on being rude, relative to the
normal professionalism here, but it didn't get through to you.
With proof assistants, for formal logic it now becomes "deeds not
words". I say that as a buffer to the rude form: "put up or shut up". It
could be my "put up or shut up" theme doesn't deserve to get through to
the mailing list.
When we criticize someone's software, there's a simple comeback for the
author, which is some variation of "If you're so smart, then why don't
you go ahead and fix it?"
If you're so bold as to say that the logic of Isabelle may be
inconsistent, then why don't you "put up" by showing that it's
inconsistent. Otherwise, "shut up" about it. Deeds not words.
Meta-think is obviously needed to set up or analyze an object language
or an object-logic (and even a meta-logic). Consequently, it may appear
that the domain of the meta-think remains domain of the human mind.
Not at all. (I'm polluting my point somehow. Semantics anyone? But for
meta-thinkers only. Syntax is sufficient for object-think.) Any smart
meta-thinker who can program should be able to code up the meta-ideas,
and proof assistants move the expectations to "You're boring me with
your talk. Code it up, so we have something fixed to talk about".
If you're so bold as to claim that Isabelle is imperfect, then "put up"
and implement the perfect logic. Well, you appear to have coded up the
basics of something. And the world now is supposed to stop and jump on
your logic wagon?
The problem with "put up" is that it's subjective. I'll state my
opinion. You haven't. I'll leave the "shut up" as implied.
*
1ST THING 2ND, LEARN SOME DECENT ETIQUETTE*
What I say now is at least as important as what I just said.
I myself abuse etiquette, but I don't add email addresses in a reply to
people who haven't been party to a thread.
Your first email was to 6 people plus the user list. I responded back
only to the user list address. That was a very intentional act on my part.
You could have at least addressed it to "Dear List Members and Gottfried
Barrow".
I am not in the same class as "List Members". List members in general
most likely have a PhD and are qualified, or more qualified than me to
be meta-thinkers. I have a B.S. in math and am qualified to potentially
be a respectable object-thinker.
We're all a team, and we don't even have to like each other. It's
object-think-users who are needed here, not meta-thinkers.
But even object-thinkers can recognize basic mistakes in thinking, like
a person making a big deal about a theorem holding under one set of
axioms, but not under another set of axioms, where I'm assuming I don't
have a total misunderstanding of what you've said.
There is no need for me to be a meta-thinker. I want to do math, not
forever be bogged down in foundations. First-order logic comes for free
with HOL. With functional-prog-logic, I'm way ahead of the Z-Crowd
already, with their 5 millimeters. I need true, false, and the Law of
Excluded Middle. The Axiom of Choice? Of course. But your stuff? I need it?
Don't diss the team, dude, and don't spam the team leaders, or those
external, while also making me a part of it.But don't worry, I'm a
nobody. Nothing will be lost by not including me with all the other people.
If, in a reply to me, you add emails to people who weren't party to the
discussion, you are a gross violator of decent etiquette. (I heavily
watered that sentence down, to also try and respect the professionals here.)
QUOTE? LINKS INSTEAD, CONCERNING PURITY
I guess I had a point to make. I'll go ahead and pull a quote of yours,
after all:
"Since my philosophical background rejects any sort of metatheories... "
Hmm. Sounds like a heavy burden to carry. My philosophical background is
the extent to which I've spoken philosophically, and I only require that
to the extent I've said stupid things, I don't say them again.
So you're talking about purity. I guess I don't care here, so I don't
need to speak for Larry Paulson, and say whether he is or isn't a purist.
I do know he takes practical things into consideration, as demonstrated
by a recent comment he made on the FOM mailing list, which I link to below.
Where is there room here for me to speak of the
"all-or-nothing-requirement", for people more picky than me, about what
they need to get them to abandon their 5 millimeter mechanical pencils?
It all has to be good.
For mechanized proof, where math is literature, a great logic is useless
without the-rest-of-it, which you ain't got.
If you want purity, in the way you want purity, code it up, and suffer
the impracticalities of your own product. But syntactically, how can
anything be pure, when programming languages have to be used, with the
thousands of details involved? The absence of details is actually part
of the current illusion-of-elegance for traditional logics, as if
there's more than one in force.
Anyway,
http://www.cs.nyu.edu/pipermail/fom/2015-October/019312.html
http://www.cs.nyu.edu/pipermail/fom/2015-October/019282.html
http://www.cs.nyu.edu/pipermail/fom/2015-October/date.html
http://www.cs.nyu.edu/pipermail/fom/2015-September/date.html
Pay attention, or not, to Larry Paulson:
It looks like I need to clarify my remark about “imposing unreasonable costs”.
Under this I do not include the difficulty of implementing a proof assistant,
because our job is to build things and let the computer do the work. The costs
rather are those imposed on users who have to work within the free logic. I
often compare theorem proving in a formal logic to building a ship in a bottle,
and nobody wants to make the bottle any smaller, or the ship any bigger. To be
more precise, I have seen proof-theoretic work showing that very small changes
to a proof calculus can increase the length of a formal proof, in the worst case,
by a non-elementary factor. It isn’t obvious that the use of a free logic would
be as costly as that, but I get the impression that users who have tried such
systems have not been happy with the effort involved. I don’t believe that good
automation could entirely eliminate this effort.
*PAYING HOMAGE TO A MATHEMATICIAN WHO CARED * I forgot how I was supposed to use this, but it's about Robert M. Solovay.
Steven Obua published "Partizan Games in Isabelle/HOLZF":
https://mediatum.ub.tum.de/doc/1094432/1094432.pdf
In it, he references Robert Solovay as saying that axioms added to
Isabelle/HOL will be consistent. That gave me enough confidence to
consider it wouldn't be a waste of my time to mess around with adding
the axioms, similar to how Obua does it.
Initially, I couldn't find the reference, then one day, I searched
again, and it came up:
https://groups.google.com/forum/#!msg/fa.isabelle/fKOEdb7uqJk/qYY1-MNxwSsJ
Solovay writes:
This alternative is certainly possible, but [if I understand it
correctly]
is not what I'm interested in.
Your version: [HOL in ZFC] has the same consistency strength as ZFC.
My version [ZFC in HOL] would have consistency strength considerably
greater than ZFC [just a shade less than Zermolo + Choice + "There is an
inacessible cardinal].
How can I tie this into you, Ken?
You think you're the only one with ideas? Ideas are a dime a dozen.
Ideas aren't a problem. Time is a problem, with the upper limit of
mortality being the ultimate arbiter of time.
In overstating things, I said no one cares about a raw logic. I can also
say that I'm interested in everything, but have time for nothing.
THE NEW GAME
The new game for logic will be proof assistants. The game will not be
played in full force until people can get educated in the rules of the
game, in a reasonable amount of time.
When the game is in full force, cheap talk won't be tolerated much, but
then, the idea of wanting to engage in cheap talk will be foreign to the
vast number of people who want to play the game of logic.
These days, find a bunch of programmers, doing meta-talk. In general,
they're doers. They all write programs. They talk about what others have
done, or what they've done. Talk and do. They do them both.
That'll be the future of the game of mathematics. Whether low or high,
there will be no place for cheap and undeserved glory.
DISCO, TRUST ME ON THIS ONE
I hate giving away my potential secrets to success, but I'm a benevolent
person.
Logic? Meta-think? Definitely crowded games to play, since the last mods
for fixing naive sets was, when? Yea, not recent.
Hopeless situation? No, not with the trifecta-of-potential-success for
True Artistes: dance, trance, and disco.
That's not to say that some things don't stay the same.
For example, you never want show up on the dance floor and start dissin'
John Travolta, and not have, at least, disco
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Last updated: Nov 21 2024 at 12:39 UTC