John, Jon, List,
JAS: Peirce anticipates aspects of the formal system that is now known as
intuitionistic logic--e.g., defining negation as the implication of
falsity...
JFS: Nobody knows what Peirce would have said about the less dogmatic
treatment of intuitionistic logic by Heyting and others, including Oostra.
But it's doubtful that he would have approved of the drastic reduction in
acceptable mathematical theories. He was always highly suspicious of any
attempt to block the way of inquiry.
"Dogmatic treatment of intuitionistic logic, etc."?? How in the world is
"intuitionistic logic--e.g., defining negation as the *implication* of
falsity..." (emphasis added) a "drastic reduction in acceptable
mathematical theories"? Acceptable to whom? You? Who else?
And, really, it seems fairly over the top to describe it as "an attempt to
block the way of inquiry?" Certainly your saying so does not make it so.
Please explain how this "blocks the way of inquiry" for folk like me who
are apparently radically deficient in mathematics and logic so simply can't
see it as such.
As JAS wrote:
JAS: . . . Peirce seeks to account for *objective *or *ontological
*indeterminacy--there
are some propositions that *are *neither true nor false because *reality *is
indeterminate. Years later, he briefly explores the logical implications of
this in his "Logic Notebook."
CSP: Triadic Logic is that logic which, though not rejecting entirely the
Principle of Excluded Middle, nevertheless recognizes that every
proposition, S is P, is either true, or false, or else S has a lower mode
of being such that it can neither be determinately P, nor determinately
not-P, but is at the limit between P and not P. (R 339:515[344r], 1909)
JAS: Intuitionistic logic also recognizes this limit but does not assign a
third truth value to it, which is what Peirce goes on to propose.
Accordingly, in today's terminology, *what he [Peirce] is primarily
rejecting in these passages is not the (so-called) law of excluded middle
("either a proposition or its negation is true"), but rather the principle
of bivalence ("every proposition is either true or false") *[emphasis
added].
In a word: Who is actually being 'dogmatic' here?
Best,
Gary R
“Let everything happen to you
Beauty and terror
Just keep going
No feeling is final”
― Rainer Maria Rilke
*Gary Richmond*
*Philosophy and Critical Thinking*
*Communication Studies*
*LaGuardia College of the City University of New York*
On Tue, May 18, 2021 at 10:59 PM John F. Sowa <s...@bestweb.net> wrote:
> Jon AS,
>
> I'm writing an article about Peirce's writings on logic in 1911, which
> I'll post to P-list soon. And I'm glad that we can agree on that point.
>
> JFS> However, Peirce and Brouwer were on opposite sides of fundamental
> issues about the nature of mathematics. ... In general, Brouwer's
> assumptions were diametrically opposed to Peirce's fundamental assumptions.
>
> JAS> I wholeheartedly agree.
>
> But in R670. Peirce definitively rejected the idea of defining negation in
> terms of implication and falsity. I show that in my article.
>
> JAS> Peirce anticipates aspects of the formal system that is now known as
> intuitionistic logic--e.g., defining negation as the implication of
> falsity...
>
> Nobody knows what Peirce would have said about the less dogmatic treatment
> of intuitionistic logic by Heyting and others, including Oostra. But it's
> doubtful that he would have approved of the drastic reduction in acceptable
> mathematical theories. He was always highly suspicious of any attempt to
> block the way of inquiry.
>
> John_ _ _ _ _ _ _ _ _ _
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