Thanks John, I would be interested in seeing you and Pieter argue over the contended points
in a public discourse arena, and I am hopeful that you could come to some brief conclusion
(oh noh, I did not want to start another logical war)
could you interpret at a glance the transition from the square to the dodecadron (that would save me having to study the paper for the mo)
On Fri, Oct 16, 2009 at 7:30 PM, John F. Sowa <sowa@xxxxxxxxxxx>
Thanks for citing that paper by Pieter Seuren:
It's a good summary of a lot of issues, but I prefer a different
way of explaining many of the same observations:
1. Logic is not the foundation for natural language semantics.
2. Instead, language comes first, and as Wittgenstein pointed
out in great detail, there is an open-ended number of
language games that people can play with the same syntax
3. The different varieties of logic are abstractions from
different kinds of language games that people play.
4. Different language games may be more popular in different
cultures and for different applications.
5. But all those language games are "natural" in the sense
that many people have found them to be useful expressions
for the way they think about and talk about situations
and activities that they are familiar with.
6. Hence, all the logics that have been abstracted from
those language games can be considered candidates for
a "natural logic".
7. Some logics might be considered "more natural" for
certain cultures or kinds of applications, but people
in any culture can learn to use any of those ways of
thinking, talking, and reasoning if they find a need
However, I do take issue with the following paragraph in
Seuren's paper, which repeats some frequently repeated
errors and misunderstandings about the history of logic:
PS> Standard modern predicate calculus (SMPC) was developed mainly
> by Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970),
> who collaborated with Alfred Whitehead (1861–1947) in writing
> _Principia Mathematica_ (Whitehead and Russell 1910–1913), the
> standard and highly revered reference work for modern logic.
> SMPC was developed in order to sever all links between logic
> and psychology and to do so by turning logic into a branch of
> mathematics, or vice versa (which of the two was correct was
> the object of much debate during the first half of the twentieth
> century). The mathematics used was the new set theory, based
> on the work by George Boole (1815–1864).
The first point is that the algebraic notation for logic was
invented by C. S. Peirce (1880, 1883, 1885) independently of
Frege's Begriffsschrift (1879), which nobody but Frege ever
used. Ernst Schröder adopted the algebraic notation from
Peirce, and he criticized Frege's notation as unreadable.
Peano adopted the algebraic notation from both Peirce and
Schröder (both of whom he cited in detail), and he also
criticized Frege's notation as unreadable. When Frege wrote
to Peano about those criticisms, Peano insisted that Frege
translate his notation to the algebraic form before he would
Russell learned the notation from Peano, and he did not add
anything to it -- all the symbols that Russell used in his
_Principles of Mathematics_ in 1903 were in Peano's book of
1895, including Peano's use of dots to show precedence.
The term 'Peano-Russell notation' is a crime against history,
because Russell added nothing to it.
Furthermore, Whitehead had written his book _Universal Algebra_
in 1898, and he had cited both Peirce and Schröder, not Frege.
The style of Whitehead's 1898 book is very similar to the
style of the _Principia_, which does not resemble the very
talky-talky style that Russell used in his book of 1903.
In Germany, Schröder had written his three-volume _Vorlesungen
über die Algebra der Logik_, which was the standard textbook
on logic from 1895 to 1910. In Germany, Hilbert, Löwenheim,
Zermelo, and others used Peirce-Schröder notation for many
years. When the _Principia_ came out in 1910, they were
underwhelmed by it, since they had gone quite a bit further.
The following point about Boole is also wrong:
> The mathematics used was the new set theory, based
> on the work by George Boole (1815–1864).
Boole had applied his algebra to propositions, monadic
predicates, and sets in exactly the same way. The
_expression_ 'P x Q', for example, represented 'P and Q'
for propositions or monadic predicates, and it also
represented 'P intersection Q' for sets.
Boole did not distinguish a set consisting of one element
from the element itself. In that sense, his "set theory"
was closer to the later theories of mereology. Modern
set theory was invented by Georg Cantor, who made a
sharp distinction between subset and elementOf.
In summary, Seuren made some good points, but his history
is wrong, and he should pay more attention to Wittgenstein.
Paola Di Maio
Networked Research Lab, UK
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