"John F. Sowa" <s...@bestweb.net>
wrote:
Jon A, Helmut R, Terry R, Jon AS, List,
JA> I can't imagine why anyone would bother with Peirce's logic if it's just Frege and Russell in another syntax, which has been the opinion I usually get from FOL fans.
That is true. But the EG structure and rules of inference are elegant, and the
algebraic structure is klutzy. For a mathematician, that is a huge difference..
What makes EGs elegant is the simplicity of the structure, minimum of primitives, and symmetry of the rules.
As a result of that structure, note how eg1911 generalizes and relates Gentzen's two systems of natural deduction and sequent calculus. As a result, an unsolved research problem from 1988 is almost trivial in terms of the EG rules. See http://jfsowa.com/talks/ppe.pdf .
JA> Peirce's 1870 Logic of Relatives is already far in advance of anything we'd see again for a century, in principle in most places, in practice in many others, chock full of revolutionary ideas...
I agree. But those ideas are part of the ontology rather than the logic.
HR> I think that "implication, imagination, or belief" mostly do not sit in the symbols of notation such as cuts, but in the variables
I agree that variables are problematical. Three-dimensional graphs show direct connections. But 2-D graphs are forced to use klutzy features like selectives or bridges. The word 'cut' by itself is not bad. But it is a reminder of the recto/verso terminology, which Peirce said was "as bad as it could be".
In eg191, Peirce talks about 'shading'. Although that word takes six letters, the people who the read and write EGs should forget the words and think directly in terms of the diagrams. When doing subtraction, for example, nobody thinks of the words 'minuend' and 'subtrahend'. The words are useful for talking about math, but they should never intrude on the structure of the math.
TR> FOL doesn’t accommodate possible-world semantics, which is necessary (and sufficient) to resolve the paradoxes of material conditionality that persist in FOL. Moreover, possible-world semantics for modalities (necessity, possibility) and intensional (vs. extensional) conditionality are prerequisites for expressing causal laws.
That's true. For the semantics of modal logic, an ontology about possible worlds or something like Peirce's three universes (possibilities, actualities, and the necessitated) must be added. Work on modal semantics during the century after Peirce shows that FOL can be used to define such theories. See http://jfsowa.com/pubs/5qelogic.pdf and http://jfsowa.com/pubs/worlds.pdf .
Peirce was not happy with the earlier versions of his modal EGs. What he intended for Delta graphs is unknown, but any version of FOL (including eg1911) could be used to state a theory of possible worlds that is sufficent to specify a semantics for Delta graphs.whatever that might be.
JAS> As Peirce explains in R 490... "if A then B" is not logically equivalent to "not (A and not-B)".
No. Don Roberts (1973:154) defined a scroll as "Two cuts, one within the other". That makes it exactly equivalent to "not (A and not-B)". That is the way Jay Zeman, Ahti, and many others have defined it, and every EG proof that Peirce wrote is based on that definition. Any ambiguous comments about scrolls are irrelevant.
It's true that in some MSS, Peirce used a horribly contorted definition of negation in terms of a scroll. But in June 1911 (R670), he remembered that his permissions (rules of inference) depend only on whether an area is shaded or unshaded. Since a scroll is limited to two levels, it's just a special case. In R670, he wrote that Figure 10 with scrolls is identical to Figure 11 with nested areas. In L231 and later, he never mentioned the word 'scroll'. The word 'scroll' is just a redundant term for a nest of two ovals, and the way of drawing it cannot be generalized to 3-D.
JFS> Unless any MSS later than December 1911 are found which say anything to the contrary, the version in L231 must be considered definitive.
JAS> No one has the unilateral authority to declare that anything Peirce wrote "must be considered definitive,"
The only authority is Peirce's available MSS. The semantics of first-order EGs in June 1911 is consistent with earlier versions, and it's simpler, more precise, and more complete (full classical FOL with a structure that could be extended to metalanguage, second-order logic, and modal logic by borrowing fatures from earlier versions). Peirce continued to use that version until December 1911, and no later version has bee found.
JAS> Peirce begins his December 1911 letter to Risteen (RL 376) by stating, "I mentioned to you, while you were [here] last year, that I have a diagrammatic syntax which analyzes the syllogism into no less than six inferential steps. I now describe its latest state of development for the first time."
Exactly! Note that L231 shows the six steps of the syllogism from the starting Figure 11 to the concluding Figure 17. If his latest version includes that example, it would be a version or extension of L231. It might even include his Delta graphs.
Unless and until that version or any later is found, L231 must remain his most definitive known version.
John
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