Hi Ben, Yes, and the key issue is how to get from 3 to 4, it seems to me. By this question I mean to be staying within formal logic, not broaching Howard's issue about levels of abstraction from natural language.
Cheers, Cathy On Wed, Feb 25, 2015 at 2:49 AM, Benjamin Udell <[email protected]> wrote: > Argh, error, I said "How to get from 3. to 4.?" I meant "How to get from > 2. to 3.?" Corrected below. - Best, Ben > > > On 2/24/2015 7:45 PM, Benjamin Udell wrote: > > 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 2. to 3.?* 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 > > > > > > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L > but to [email protected] with the line "UNSubscribe PEIRCE-L" in the > BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm > . > > > > > >
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