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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 02:56, 6 May 2008
, 02:56, 6 May 2008→Note 12: markup
===Note 12===
===Note 12===
'''Notes on Cactus Language'''
I happened on the graphical syntax for propositional calculus that
I now call the "cactus language" while exploring the confluence of
I happened on the graphical syntax for propositional calculus that I now call the ''cactus language'' while exploring the confluence of three streams of thought. There was C.S. Peirce's use of operator variables in logical forms and the operational representations of logical concepts, there was George Spencer Brown's explanation of a variable as the contemplated presence or absence of a constant, and then there was the graph theory and group theory that I had been picking up, bit by bit, since I first encountered them in tandem in Frank Harary's foundations of math course, ''c.'' 1970.
three streams of thought. There was C.S. Peirce's use of operator
variables in logical forms and the operational representations of
logical concepts, there was George Spencer Brown's explanation of
a variable as the contemplated presence or absence of a constant,
and then there was the graph theory and group theory that I had
been picking up, bit by bit, since I first encountered them in
tandem in Frank Harary's foundations of math course, c. 1970.
More on that later, as the memories unthaw, but for the moment
More on that later, as the memories unthaw, but for the moment I want very much to take care of some long-unfinished business, and give a more detailed explanation of how I used this syntax to represent a popular exercise from the PDP literature of the late 1980's, McClelland's and Rumelhart's "Jets and Sharks".
I want very much to take care of some long-unfinished business,
and give a more detailed explanation of how I used this syntax
to represent a popular exercise from the PDP literature of the
late 1980's, McClelland's and Rumelhart's "Jets and Sharks".
The knowledge base of the case can be expressed as a single proposition.
The knowledge base of the case can be expressed as a single proposition. The following display presents it in the corresponding text file format.
The following display presents it in the corresponding text file format.
<pre>
File "jas.log". Jets and Sharks Example
File "jas.log". Jets and Sharks Example
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
Let's start with the simplest clause of the conjoint proposition:
Let's start with the simplest clause of the conjoint proposition:
<blockquote><code>
( jets , sharks )
</code></blockquote>
Drawn as the corresponding cactus graph, we have:
Drawn as the corresponding cactus graph, we have:
<pre>
jets sharks
jets sharks
o-----o
o-----o
\ /
\ /
@
@
</pre>
According to my earlier, if somewhat sketchy interpretive suggestions,
According to my earlier, if somewhat sketchy interpretive suggestions, we are supposed to picture a quasi-neural pool that contains a couple of quasi-neural agents or ''units'', that between the two of them stand for the logical variables ''jets'' and ''sharks'', respectively. Further, we imagine these agents to be mutually inhibitory, so that settlement of the dynamic between them achieves equilibrium when just one of the two is ''active'' or ''changing'' and the other is ''stable''or ''enduring''.
we are supposed to picture a quasi-neural pool that contains a couple
of quasi-neural agents or "units", that between the two of them stand
for the logical variables "jets" and "sharks", respectively. Further,
we imagine these agents to be mutually inhibitory, so that settlement
of the dynamic between them achieves equilibrium when just one of the
two is "active" or "changing" and the other is "stable" or "enduring".
===Note 13===
===Note 13===