List,
Does anybody know an example which justifies intuitionistic logic, so in which classical logic fails? I think Jon, A.S., you once gave me the following example:
"Every unicorn is pink" is false, but "There is no unicorn that is not pink" is true.
"Every unicorn is pink" is false, because it means "If it is a unicorn, then it is pink", and "If it is a unicorn" implies, that unicorns exist. So it is equal with "Unicorns exist, and if it is a unicorn, it is pink". Because unicorns donot exist, the proposition is false.
"There is no unicorn that is not pink" sounds true, because there are no unicorns at all, so there are no non-pink unicorns too. But if it would be so, that this form of proposition too implied the existence-claim, it would be false as well. Is that so? Is in classical logic "There is no unicorn that is not pink" equal with "Unicorns exist, and there is no unicorn that is not pink"?
This might be so e.g. due to the fact alone, that the term "Unicorn" has been mentioned. For EGs, it would mean, that every term written in any place is a possible too in the blank sheet. Meaning, that it generally exists. Otherwise it would not signify anything, it would e.g. be like "NOT &/(", senseless. But this would mean, that the term "existential" in "Existential Graphs" means, that only existing things are allowed in them.
Another way to classically synchronize the two propositions might be to say, that if a term signifies a nonexistent thing, it automatically signifies its phantasy-concept instead. Then "Every unicorn is pink" is false, because in some animated movie by Disney occurs a white unicorn. "There is no unicorn that is not pink" then is false for the same reason. This explanation is somewhat smoother than the first, but requires this said automatism: If A does not physically exist, then A is the existing concept of A.
Best
Helmut
19. Mai 2021 um 06:58 Uhr
"John F. Sowa" <s...@bestweb.net>
wrote:
"John F. Sowa" <s...@bestweb.net>
wrote:
Gary R,
I'm glad you asked.
GR> 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.Intuitionistic logic is a restriction on the permissible rules of inference. That makes it impossible to use many widely accepted theories of mathematics -- among them, the theory that there are hierarchies of infinities.
Peirce was one of the mathematicians who discovered a proof of that point independently of Georg Cantor. And it's the foundation for his theory of continuity -- which Abraham Robinson proved was consistent in 1960.
In applications to science and engineering, especially computer science, nobody uses intuitionistic logic. The reason why is that it "blocks the way" of using the most convenient, efficient, and flexible methods of reasoning.
The mainstream mathematicians don't stop intuitionists from developing their own pet theories. They just ignore them.
John
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