[Inquiry] Peirce's Rules Of Inference

[Jon Awbrey]

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PROI.  Note 2

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Frithjof Dau wrote:
> 
> Hi,
> 
> On Donnerstag 21 Juni 2007 12:46  Jon Awbrey wrote:
> >
> > Responding to messages by John Sowa and Frithjof Dau:
> >
> > One of these issues had to do with a conceptual inefficiency in the
> > representation of logical equality (logical equivalence, iff, <=>).
> >
> > If you look at the topological dual form of Peirce's graph for x <=> y,
> > you get this:
> >
> > y o   o x
> >   |   |
> > x o   o y
> >    \ /
> >     @
> >
> > In other words, (x => y) & (y => x).
> 
> I agree:  The representation of equivalence is IMHO
> a (maybe even _the_ ) major downside of Peirce's notation.

I first ran into this in connection with the ancient (in AI time) dispute
between connectionists and symbolicists over the so-called "XOR problem",
basically turning on the fact that 2 of the 16 boolean functions, namely,
XOR and EQU, are not efficiently implemented by threshold-type neurons.
Here's a somewhat anecdotal discussion of the work that I did on that:

http://forum.wolframscience.com/printthread.php?threadid=630&perpage=11

> > The fact that each variable is mentioned twice turned out
> > to be an efficiency boondoggle under the constraints that
> > I was working, and it remains a conceptual ineffficiency
> > under any conditions.
> >
> > The way that I eventually solved this was by introducing
> > the following cactus form:
> >
> > x   y
> > o---o
> >  \ /
> >   @
> >
> > Under the analogous existential interpretation,
> > this has the meaning x =/= y, that is, x XOR y.
> >
> > From that we get x = y (or x <=> y if you prefer)
> > by negating it, yielding the following cactus form:
> >
> > x   y
> > o---o
> >  \ /
> >   o
> >   |
> >   @
> >
> You might have a look at the works of Pavel Kocura, who works
> on EGs as well (in a quite unsolid way, I have to say ...). Anyhow,
> he uses a new notation for equivalence, which is an oval, separated
> by a vertical line into two halfs.
> 
> Anyhow, you have to alter (extend) the calculus for this new syntax.

Yes, I do know some of the additional axioms that are required,
but I have yet to pin it down to the most elegant or minimal set.

Additional discussion here:

http://www.centiare.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems

The basic construct is generalized from negation
to the so-called "minimal negation operators":

As graphs these are drawn as cactus lobes:

x    x   y    x y z
o    o---o    o-o-o  
|     \ /      \ /
@      @        @       ...

As text expressions by means of parentheses:

(x)  (x, y)  (x, y, z)  ...

The analogue of the existential interpretation assigns the meaning:
"just one false among the corresponding list of arguments", that is:

not x,  x=/=y,  just one false among {x, y, z},  ...

There is some exposition here:

http://www.centiare.com/Minimal_negation_operator

Jon Awbrey

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