"Jon Alan Schmidt" <jonalanschm...@gmail.com>
wrote:
JFS: For an explanation and demonstration of that point, see the attached file NatDeduction.pdf--it's just one page from an article I'm writing.
CSP: I soon discovered, upon a critical analysis, that it was absolutely necessary to insist upon and bring to the front, the truth that a mere possibility may be quite real. That admitted, it can no longer be granted that every conditional proposition whose antecedent does not happen to be realized is true ...[T]he verso of the sheet of Existential Graphs represents a universe of possibilities. This, taken in connection with other premisses, led me back to the same conclusion to which my studies of Pragmatism had already brought me, the reality of some possibilities. This is a striking proof of the superiority of the System of Existential Graphs to either of my algebras of logic. For in both of them the incongruity of this strange rule is completely hidden behind the superfluous machinery which is introduced in order to give an appearance of symmetry to logical law, and in order to facilitate the working of these algebras considered as reasoning machines. I cannot let this remark pass without protesting, however, that in the construction of no algebra was the idea of making a calculus which would turn out conclusions by a regular routine other than a very secondary purpose. (CP 4.580-581, 1906)
John, Jon,Thank you! My first idea is, that to negate each step and reverse the order only is valid with the excluded middle, so exclusion of the middle should be necessary for both ways of proof too. But I will read your links, thank you for your patience! I think, the concept of truth is somewhat holy for me, and I saw it being blasphemically blurred by intuitionism, or something like that. But it is all about formalism, OK!Happy Christmas, HelmutHelmut,
I basically agree with the short summary by JAS in his last note, but there are some related issues that I'd like to add.
The first point is that Brouwer, the founder of intuitionistic logic, was a mathematician, and he did not generalize his arguments beyond formal mathematical issues. For all the gory details and citations, see the Stanford article: https://plato.stanford.edu/entries/intuitionism/
Second, intuitionism is a special case of constructivism: the preference for a constructive proof that begins with a hypothesis and some appropriate axioms and constructs a proof.
The opposite of a constructive proof is a proof by contradiction: start with the proposed theorem, negate it, and derive a contradiction. Most mathematicians will accept a proof by contradiction, but they prefer a constructive proof.
One of the very nice properties of Peirce's rules of inference, as he stated them in 1911 is that every proof by contradiction can be converted to a constructive proof by a very simple method: negate each step of the proof and reverse the order.
For an explanation and demonstration of that point, see the attached file NatDeduction.pdf -- it's just one page from an article I'm writing.
That page is from Section 6 of an article that says a lot more about Peirce's EGs and rules of inference. Anyone who would like a review of those issues, see the tutorial http://jfsowa.com/talks/egintro.pdf .
That file has 53 slides, but the first 10 slides are sufficient for an overview of the notation. If you're already familiar with the notation, skip to slide 31 through 35. That is sufficient background to understand the one-page discussion in NatDeduction.pdf.
And by the way, this example is just one of many reasons for preferring Peirce's 1911 version of EGs. It has just 3 pairs of rules of inference, which are very easily reversible. In 1906, he stated 11 rules, for which reversibility is possible, but only with a great deal of complexity.
John
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