Dear Jon, All ...
Peirce's explorations in logic and the theory of signs opened several
directions of generalization from logics of complete information (LOCI)
to theories of partial information (TOPI). Naturally we hope all these
avenues of approach will eventually converge on a unified base camp from
which greater heights of understanding may be reached, but that is still
a work in progress, at least for me.
Any passage from logic as a critical, formal, or normative theory
of controlled sign transactions to the descriptive study of signs
“in the wild” involves relaxing logical norms to statistical norms.
One of the headings under which Peirce expands the scope of logic to
something more general, whether keeping or losing the name of “logic”
is a secondary issue, is found in his study of Generality and Vagueness
as affecting signs not fully primed for logical use. There's a bit about
that at the following places:
Peirce (CP 5.448)
https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS#Excerpt_6._Peirce_.28CP_5.448.29
FOM List • C.S. Peirce on “General” and “Vague”
• https://cs.nyu.edu/pipermail/fom/2009-March/thread.html#13437
1. https://cs.nyu.edu/pipermail/fom/2009-March/013437.html
2. https://cs.nyu.edu/pipermail/fom/2009-March/013446.html
3. https://cs.nyu.edu/pipermail/fom/2009-March/013448.html
As you can see, in this direction of generalization Peirce
considers relaxing both the principle of contradiction and
the principle of excluded middle.
Regards,
Jon
On 12/20/2020 4:09 PM, Jon Alan Schmidt wrote:
Jon A., List:
Indeed, Peirce's Law is one of several possible formalizations of excluded
middle within classical logic, although it seems likely that Peirce himself
would have objected to referring to it as such since he denied the
underlying assumption.
CSP: Logic requires us, with reference to each question we have in hand, to
hope some definite answer to it may be true. That *hope *with reference to
each case as it comes up is, by a *saltus* [leap], stated by logicians
as a *law*concerning all cases, namely, the law of excluded middle.
This law amounts to saying that the universe has a perfect reality.
(NEM 4:xiii, no date)
In a 2013 paper (in Spanish,
https://www.academia.edu/36792040/Cuadernos_de_Sistem%C3%A1tica_Peirceana_5,
pp. 5-24) Arnold Oostra discusses how Peirce's 1885 article, "On the
Algebra of Logic: A Contribution to the Philosophy of Notation" (CP
3.359-403), furnishes an axiomatization of classical propositional logic
(CPL); and the last of the five axioms is what today we call Peirce's Law.
As Oostra notes, "With this axiomatization of CPL, Peirce anticipated the
development of mathematical logic by about 40 years" (p. 21).
Oostra goes on to show that (ironically) by omitting the "law" that now
bears his name and adding several axioms defining conjunction and
disjunction without excluded middle accordingly, Peirce also could have
provided an axiomatization of intuitionistic propositional logic (IPL) well
in advance of Brouwer. In Oostra's words, "his axiomatization of 1885,
omitting Peirce's Law that he included as a last resort to prove the
completeness of CPL, hides the nucleus of an axiomatization of the IPL" (p.
22).
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
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