John, (01)
On Tue, 2010-11-09 at 23:40 -0500, John F. Sowa wrote:
> Patrick,
>
> That paper isn't available on the WWW:
> (02)
Actually it is but only behind a number of annoying firewalls, etc. (03)
But it is also available in any number of good university libraries. (04)
Despite the presence of the WWW they continue in operation. (05)
When the flirtation with full-text access as meaningful access ends, I
expect libraries and librarians will take on ever increasingly important
roles with regard to meaningful access to ever increasing stores of
information. (06)
> > Well, but for a non-burning bush origin for FOL, see: Ferreirós, José
> > (2001), "The Road to Modern Logic-An Interpretation", Bulletin of
> > Symbolic Logic 7 (4): 441–484, doi:10.2307/2687794,
> > http://jstor.org/stable/2687794 .
>
> But as Kronecker said in 1886, "Die ganzen Zahlen hat der liebe Gott
> gemacht, alles andere ist Menschenwerk". [The dear God made the
> whole numbers, everything else is the work of man.]
>
> I believe that one can make a very strong case that FOL can be
> considered the equivalent for logic as the natural numbers for
> mathematics. It is a subset of every natural language, and it
> is simpler to specify than any of its subsets.
>
> In any case, the paper you cited is not available on the WWW. But the
> abstract seems to discuss the work that Peirce did in the years from
> 1897 to 1906, but everybody else ignored:
> (07)
The article runs some 45 pages and examines a number of the major
players in a fair amount of detail. (08)
For example, the author mentions that it was David Hilbert and Wilhelm
Ackermann, Grundzüge de theoretischen Logik, Springer, Berlin, 1928, (09)
***
"The first really modern treatise of formal logic is not Principia
Mathematica but Hilbert & Ackermann's Grundzüge de theoretischen Logik
[37]. The book is noteworthy because one can find, for the first time in
a treatise on logic, a study of FOL as a separate system (under the name
of "restricted functional calculus", posing the question of its
metatheoretical properties, e.g., completeness. But FOL appears only as
an interesting subsystem and the work culminates by presenting the
so-called "extended functional calculus", a peculiar version of the
theory of types."
*** (010)
ON Frege and FOL, the author comments in passing (Frege is treated at
length, along with other) (011)
***
Incidentally, it is worth reminding the reader that, while it is true
that Frege's system [24] is quite close to a modern formal one, some of
his notational simplifications gives rise to a *deceitful appearance
that the system is essentially a modern first-order one.* Upon more
careful reading it becomes clear that Frege's system is higher-order
throughout, and that he actually deployed higher-order logical tools
(this is explicit in the theory of series in the last part of
Begriffsschrift).
***
(emphasis added) (012)
The first section of the article is a quick historical survey before
turning to the details, where the author concludes: (013)
***
Presenting in a nutshell the results of our quick historical overview,
we can say that around 1900 logic was conceived as a theory of
sentences, set and relations; after World War I and as late as 1930 the
exemplar for modern logic was a higher-order system, simple type theory;
and only around 1940-1950 did the community of logicians as a whole come
to agree that the paradigm logical system is FOL.
*** (014)
None of which is to take anything away from the usefulness of FOL for a
variety of purposes but to point out that its *usefulness* for some
given task is the question that needs to be asked. (015)
No one doubts the utility of Euclidean geometry but mathematics would be
a good deal poorer in the absence of non-Euclidean geometries. (016)
> > Among the latter, one may emphasize the spirit of modern axiomatics,
> > the situation of foundational insecurity in the 1920s, the resulting
> > desire to find systems well-behaved from a proof-theoretical point
> > of view, and the metatheoretical results of the 1930s. Not surprisingly,
> > the mathematical and, more specifically, the foundational context in
> > which Firs-Order-Logic matured will be seen to have played a primary
> > role in its shaping.
>
> First of all, FOL was born mature in the versions by Frege and Peirce.
> It is true that Frege had a better formalized axiomatization in 1879
> than Peirce had in 1885. But their versions are equivalent to every
> version invented since then.
>
> I certainly agree that the axiomatization by Frege, a version of
> which Whitehead and Russell adopted for PM, was not bad for 1879,
> but it was horribly cumbersome. And the version presented in PM
> probably made more people hate logic than any other system ever
> inflicted on poor, innocent students.
>
> But in 1897, Peirce developed the existential graphs, which have
> the simplest axiomatization, proof procedure, and model theory
> ever discovered:
>
> 1. The only axiom for EGs is a blank sheet of paper, and Peirce's
> rules of inference are a streamlined version of natural deduction,
> which Gentzen invented over 30 years later.
>
> 2. Peirce also invented a very simple and elegant model theory for
> EGs, which he called _endoporeutic_ (for outside-in evaluation).
> Nobody understood Peirce's endorporeutic until Hintikka's
> student Pietarinen showed that it was a version of Game
> Theoretical Semantics.
>
> 3. Every other common axiomatization and proof procedure invented
> for FOL can be shown to be derived theorems or derived rules
> of inference from Peirce's system.
>
> 4. But as Frithjof Dau showed, Peirce's rules are strictly more
> powerful than the other common proof procedures: certain proofs
> that take exponential time with those methods can be proved
> in polynomial time with Peirce's rules. (The case he showed
> was for cut-free proofs with Gentzen's clauses, but there
> would be similar kinds of proofs for other cases as well.)
>
> For a brief summary of Peirce's system, see his own tutorial:
>
> http://www.jfsowa.com/peirce/ms514.htm
>
> Bottom line: God made FOL, and Peirce was his prophet.
> (017)
Actually not. FOL was born out of concerns in the mathematical community
that could have well resolved themselves differently. (018)
As to Peirce being a prophet, despite the fact I like Peirce a lot,
well..., let's say opinions vary and leave it at that. ;-) (019)
Hope you are having a great day! (020)
Patrick (021)
> John
>
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