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’s own words were: “Il est surprenant que tant de grands algébristes n’aient pas trouvé moyen d’essayer à porter leur méthode dans les autres sciences … On est dans l’habitude de regarder ces autres sciences comme d’une autre nature; il semble qu’il y a une autre sorte de vérité, une autre sorte d’évidence, un autre ciel, un autre soleil tout différent pour celles-ci”.

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.