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This collage represents both great
logicians: Aristotle (384-322 BC), the creator of the Science of Logic,
and Gottfried Wilhelm von Leibniz (1646-1716), the first who transmuted
it into a mathematical discipline. They are pronouncing two winged phrases.
Magister
dixit ("The Master said it") is the scholastic conjuration reducing
any proof to Aristotle's authority (of course, he was not implicated in
the mediaeval speculations with his name). On the contrary, Calculemus!
("Let's calculate!") reflects Leibniz's conviction that all human reasoning
may be turned into an object of mathematical demonstration and in such
a way, any controversal truth can obtain the evidence of 2+2=4.
* * *
Arithmetizations
of Syllogistic à la Leibniz
A detailed history of Leibniz's ideas together with full proofs see in my paper published in the
Special Issue of the Journal of Applied Non-Classical
Logics
(v. 9, no. 2-3/1999, pp. 387-405) dedicated to the memory of my colleague
and friend George Gargov (1947-1996):
Leibniz's arithmetical translation was extended on the logic of properties in the
contributed paper presented
at the Logic
Colloquium '99 (Utrecht,
August 1-6, 1999):
Monadic
Predicate Calculus Arithmetized
à
la Leibniz
Further on, the arithmetical translation involved equality as well. The
text containing full demonstrations see in Comptes Rendues de l'Académie bulgare
des Sciences (v. 54, no. 1, 2001, pp. 9-10):
Monadic
Predicate Calculus with Equality
Arithmetized
à la Leibniz
Different syllogistics were uniformly considered in the contributed paper presented
at the 2nd
Panhellenic Logic
Symposium (Delphi, Greece, July 13 - 17,
1999: Proceedings, pp. 197-200):
Various
Syllogistics from the Algebraic
Point
of View
A popular exposé of Leibniz's programme together with my own results in
its realization (in Bilgarian) see in the report presented at the 29th
Spring Conference of the Union of the Bulgarian Mathematicians
(Lovetch,
April 3-6, 2000: Proceedings, pp. 36-45):
The
Leibniz Programme:
Calculation
in Lieu of Disputation
In 1999, the next volume of Leibniz's philosophical heritage appeared after
a 20-year wait. It was prepared by Leibniz-Forschungsstelle in Münster
and included the logical manuscripts of Leibniz written between 1677 and 1690. Fortunately they are accessible nowadays from the Internet: you can download the full Band VI-4 of his Philosophische Schriften and especially the PDF-file (2 MB) containing the most interesting for us manuscripts.
My Essay Review on this edition was published in History and Philosophy
of Logic (v. 22, 2001, n. 3, pp. 164-168). You can find there a
contemporary view on Leibniz's logical ideas, a critical exposition of
his philosophical dreams, and a reconstruction of his logical systems:
Essay Review
"Scholastic" arithmetical semantics: mode d'emploi. Namely, one of both Leibniz's models was tested on some simple examples in my paper announced on the 8th International Leibniz Congress (Hanover, July 2006). Note that the negation of a notion in this model is impossible when the universe of notions is infinite. Is any philosophical peculiarity of our thinking hidden in this phenomenon?
Leibniz's "Calculemus!" at Work
Leibnizian arithmetical semantics: mode d'emploi. Leibniz's preferable
model was tested and compared with the
"Scholastic" arithmetical semantics in the report presented on the 9th International Leibniz Congress (Hanover,
September 2011). Boolean algebra of the 8 fundamental colours is another elegant
illustration of the intensional semantics at work and this Easter egg
is its picturesque visualization.
Leibniz's Intensional Semantics of Syllogistics
(a Reconstruction)
* * *
Acknowledgement
Many
people helped my investigations in this field. I am grateful to the former
Director of Leibniz-Forschungsstelle in Münster Prof. Hans Schepers,
to Prof. Martin Schneider, and to Dr. Herma Kliege-Biller for their consultations,
for the opportunity to touch some Leibniz's manuscripts interesting for
me, and for assigning me the extremely valuable edition of Leibniz's writings.
Due to the hospitality of Prof. Wolfgang Lenzen from Osnabrück I obtained
an access to the literature lacking in Bulgaria. Prof. Hans Poser and his
colleagues from the Technical University in Berlin made possible my visit
to the 7th Leibniz-Kongress in 2001. The invitation from Prof. Hiroakura
Ono and the grant from the Japan Society for the Promotion of Science (JSPS) assured
my one-month stay in 1999 at the Japan Advanced Institute of Science and
Technology (JAIST) in Hokuriku near Kanazawa. In 2003 I visited JAIST for the second time thanks to a grant from JSPS as well as Kyoto University thanks
the the invitation from Prof. Soshichi Uchii. Prof. Herbert Breger and his colleagues from
the University of Hanover assured my participation in the 8th Leibniz-Kongress in
2006 and in the next Kongress in 2012 as well. My last results are a part of a joint
project with Prof. W. Lenzen and have been supported by grant LE 510/17-1
(2010-2011) from Deutsche Forschungsgemeinschaft.
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