Pierre-Stanislas Bédard : a precursor of
symbolic logic

Notes by Luigi
Logrippo, October 2013

w3.uqo.ca/**luigi**/**

https://www.site.uottawa.ca/~luigi/

Pierre-Stanislas Bédard
(Charlesbourg 1762 - Trois-Rivières 1829) was a noted
Québec politician, journalist, scholar and judge (he would have called himself
a ‘Canadien’). Ample information about him can be
found on the Web.

In 2012, in a CBC radio broadcast entitled “The art of reasoning”
dedicated to Bédard, the host, Paul Kennedy,
mentioned having studied a notebook attributed to him, including extensive use
of logical notation: http://www.cbc.ca/ideas/episodes/2012/03/08/the-art-of-reasoning/.
In addition, Bédard is believed to be the author of a
manuscript *« Traité du droit naturel démontré par des formules algébriques*». Unfortunately this manuscript
is lost but by its title Bédard appears to be a real
pioneer in this area, by more than a century!

Being interested in legal logic, I took a trip to Québec to examine the
notebook mentioned by Paul Kennedy. I had located it in La Bibliothèque
du Séminaire de Québec, Fonds
Ancien, see: http://www.cfqlmc.org/reseau-des-archives/921
l The manuscript number is M-241. For this visit,
I was very fortunate to be accompanied by Gilles Gallichan,
a Bédard expert, and assisted by Peter Gagné and Géneviève Vézina, librarians of the Bibliothèque.

The following notes are the result of some hours spent examining the
manuscript, on the 16^{th} and 17^{th} October 2013.
Unfortunately I haven’t been able to go again in order to double-check what I
write below.

M-241 is a leather-bound notebook of about 540 pages, measuring
39.5Hx17cmW. It is known by the library under the title *« Notes de philosophie, de mathématiques, de chimie, de musique, de
grammaire, de politique et notes de journal, 1798-1810* ». The only information about authorship is
the initials PB on the cover. However the manuscript has been attributed to
Pierre S. Bédard since its discovery and this view is
endorsed by Mr. Gallichan, who is transcribing Bédard’s correspondence. At the end of the notebook there
is an alphabetical index of contents.

M-241 is densely filled with neatly written notes and quotations on many
subjects: history, music theory, philosophy, simple mathematics, elementary physics and, most strikingly given the time and
place, what appear to be symbolic logic expressions and derivations with a
modern ‘look’. The symbols used seem to be Bédard’s
invention, and include some symbols used in modern logic, although surely with
different meanings. Unfortunately, Bédard does not
explain these derivations in natural language, nor does he seem to explain his
method and so some time should be spent in order to understand them. I had the
impression that they were exercises that Bédard did in order to experiment with
various formalisms. In some cases he uses as examples arithmetic properties. It
seems clear that Bédard was mimicking the algebraic
method in order to formalize logical reasoning.

Reproducing Bédard’s notation in this html
document would require some experimentation of including small drawings, and I
am not prepared to do this. For example,
he uses an equality symbol with a longer lower line, and he describes it as ‘a dépend de b’. Does he use this symbol to mean inclusion,
since the upper line is included in the lower? But if the upper line is longer,
this mean ‘a ne dépend pas
de b’, rather than ‘b dépend de a’ (and so there does
not seem to be a symbol for negation). Another similar symbol that he uses has
the lower line shifted to the right, does this mean nonempty intersection? Also
he uses a half-arrow symbol ___\ . There are many
expressions and apparent derivations using these and other symbols, and I had
the impression that his use of symbols evolved over time. However it is
difficult to figure out how this evolution occurred because the notebook does
not follow a time sequence: Bédard left empty spaces that
he filled later. At one point he seems to be trying to discover the concept of
logical implication, but by his brief explanations it seems that he did not
quite get it. His notion of ‘evident proposition’ is limited to identity; I
haven’t found mention of logical laws.

There is a section entitled: *“L’art de raisonner”*, which
includes a basic account of syllogism forms, without symbolic notation. In
another section entitled *“Un langage universel”* he
experiments with symbols to represent modalities in natural language. The
concept of universal language can be found in Locke and Leibniz,
however I suspect that Bédard got it from the former,
who was more familiar to him.

A systematic study could be made of Bédard’s
quotations, to determine his influences. I have seen quotations of Malebranche,
Locke, Descartes, others. Leibniz is also quoted but I could find no references
to his works on logic, or on legal logic (Leibniz had the concept of symbolic
logic, but he never published about it). Obviously Bédard
was limited to the books he could find around him, and we know that he
complained about this.

My first reaction after this consultation was of disappointment, because
I hadn’t found any information on what Bédard could
have written in his book on algebraic demonstration of natural law. The
notebook includes short attempts to define informally the concepts of obligation
and permission, but unfortunately there is no mention of the logical
relationships between them. Knowledge of Leibniz’s deontic logic would have
made a big difference. The end date of 1810 given in the title of the
manuscript may well be accurate and Bédard could have
developed later his ‘legal algebra’. After several vicissitudes, in late 1812 Bédard moved to a tranquil life to be a judge in Trois Rivières.

My second reaction to this reading was that this notebook reveals the
work of an isolated precursor of symbolic logic, “the method of representing
[and manipulating] logical expressions through the use of symbols and
variables, rather than in ordinary language”, as defined in http://www.philosophy-index.com/logic/symbolic/.
Bédard seems to have discovered the concept all by
himself, inspired by the power of algebraic formalism. He was discovering not
only the use of symbols to represent logical concepts, but also the idea of
algebraic manipulation of logical expressions. A philosopher who had published
on logical calculus before Bédard was Gottfried Ploucquet (1716-1790), but for what I have been able to
read, Ploucquet’s symbols and concepts were quite
different. The logical works of De Morgan and Boole were published after
Bédard’s time. Surely he would have found them a most engrossing reading: by
comparison, his notes were mere attempts. And he would have agreed with Peano’s words one century later (1913): “Symbolismo da alas ad mente de
homo sed suo usu exige studio et labore” (“Symbols give wings to
human mind but their use demands study and hard work” in Peano’s
own ‘Interlingua’).

Bédard is known to have bought many notebooks from his supplier, but perhaps
only two or three have arrived to us. I also quickly examined another notebook
in the same library, M-202 also attributed to him, but this one did not include
any mathematical symbols. It is possible that Bédard’s
best work in logic and algebra was elsewhere, especially in manuscripts that he
may have written at a more mature stage with publication in mind.

I trust that a technical study of this work will be done one day, and I
hope that other Bédard manuscripts will come to
light.