Cathy, list,
Sometimes Peirce speaks of lines of identity as crossing a cut.
Elsewhere he insists that a line of identity can only abut a cut,
meeting up with another line of identity from the cut's other side. In
those cases the line of identity is a graph, and the ligature formed of
the abutting lines of identity is called a ligature and is not a graph.
I'm not sure what he came to mean about graphs crossing cuts.
Barcan rules, - I think they're better known as 'rules of passage'.
Regarding equivalence of
"There is some married woman who will commit suicide in case her husband
fails in business."
and
"if every married man fails in business some married woman will commit
suicide"
in "Prolegomena"
You once proved the similar example in "On an Improvement on the Gamma
Graphs" and your proof was much quicker than mine here. I was so slow to
get it, I remember!
Anyway:
"/Wxy/" ≡ "/x//y/ are wife and husband together" (two people uniquely
paired in ordered relation)
"/By/" ≡ "/y/ is bankrupt"
"/Sx/" ≡ "/x/ commits suicide"
1. "There is some married woman who will commit suicide in case her
husband fails in business."
1. ∃/x/∃/y/[/Wxy/ & (/By/ → /Sx/)]
2. ∃/x/∃/y/[/Wxy/ & (~/By/ *∨* /Sx/)]
3. [∃/x/∃/y/(/Wxy/ & ~/By/)] *∨* [∃/x/∃/y/(/Wxy/ & /Sx/)]
4. [∀/x/∀/y/(/Wxy/ → /By/)] → [∃/x/∃/y/(/Wxy/ & /Sx/)]
4. "If every married man fails in business some married woman will
commit suicide"
How to get from 3. to 4.? I'm too rusty to remember.
Take the scenic route, expand the variables. This is worth doing at
least once anyway, to see how the alternative among the values of the
variables /x, y/ mixes with the alternative between the predicates ~/B/
and /S/.
Four-person universe:
/e/ = Edith /g/ = Gerald
/f/ = Fanny /h/ = Harold
∃/x/∃/y/[/Wxy/ & (~/By/ ∨ /Sx/)]
Hence,
[/Weg/ & (~/Bg/ *∨* /Se/)]
*∨*
[/Weh/ & (~/Bh/ *∨* /Se/)]
*∨*
[/Wfg/ & (~/Bg/ *∨* /Sf/)]
*∨*
[/Wfh/ & (~/Bh/ *∨* /Sf/)]*.*
Hence,
[/Weg/ & ~/Bg/] *∨* [/Weg/ & /Se/] *∨*
[/Weh/ & ~/Bh/] *∨* [/Weh/ & /Se/] *∨*
[/Wfg/ & ~/Bg/] *∨* [/Wfg/ & /Sf/] *∨*
[/Wfh/ & ~/Bh/] *∨* [/Wfh/ & /Sf/]*.*
Hence,
[∃/x/∃/y/(/Wxy/ & ~/By/)] ∨ [∃/x/∃/y/(/Wxy/ & /Sx/)]
You can see that it will work in any larger universe.
Best, Ben
On 2/24/2015 4:29 PM, Catherine Legg wrote:
Hello all - sorry to come late onto this thread.
I'm very interested to hear about Peirce's late shift in view as to
the meaning of his cut - from simple univocal falsity, to varieties of
possibility (which may divide into further kinds). I am reminded to
Wittgenstein's Tractatus where logical space must 'contain' false
states of affairs as well as true ones, otherwise we could not
understand the meaning of false statements.
I'm confused though about Peirce's big announcement about now being
able to give a meaning to graphs which cross a cut. This crossing
cannot just mean lines of identity, right? As those have always
crossed cuts. He must mean something else, but what? Can someone show
me an example?
I once tried to prove Peirce's famous two statements about the
suiciding wife and the man who fails in business equivalent in regular
FOL, but couldn't do it. Are people sure they're equivalent in FOL, as
in the beta graphs?
The case of ∃(A or B) ≡ ∃A or ∃B is one of the Barcan rules if I'm
not mistaken. These are controversial in modal logic, and metaphysics.
Cheers, Cathy
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