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Revision as of 16:24, 27 May 2009
, 16:24, 27 May 2009→Note 1: markup
Now there are many ways to dance around this idea, and I feel like I have tried them all, before one gets down to acting on it, and there many issues of interpretation and justification that we will have to clear up after the fact, that is, before we can be sure that it all really makes any sense, but I think this time I'll just jump in, and show you the form in which this idea first came to me.
Now there are many ways to dance around this idea, and I feel like I have tried them all, before one gets down to acting on it, and there many issues of interpretation and justification that we will have to clear up after the fact, that is, before we can be sure that it all really makes any sense, but I think this time I'll just jump in, and show you the form in which this idea first came to me.
Start with a proposition of the form <math>x ~\operatorname{and}~ y,</math> which I graph as two labels attached to a root node, so:
Start with a proposition of the form <math>x ~\operatorname{and}~ y,</math> which is graphed as two labels attached to a root node:
{| align="center" cellpadding="6" width="90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
</pre>
</pre>
|}
Written as a string, this is just the concatenation "<math>x~y</math>".
Written as a string, this is just the concatenation "<math>x~y</math>".
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
{| align="center" cellpadding="6" width="90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
</pre>
</pre>
|}
{| align="center" cellpadding="6" width="90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
</pre>
</pre>
|}
Back to the proposition <math>xy.\!</math> Imagine yourself standing
in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>xy\!</math> is true, as shown here:
{| align="center" cellpadding="6" width="90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| |
| |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
Now ask yourself: What is the value of the
Now ask yourself: What is the value of the proposition <math>xy\!</math> at a distance of <math>dx\!</math> and <math>dy\!</math> from the cell <math>xy\!</math> where you are standing?
proposition xy at a distance of dx and dy
from the cell xy where you are standing?
Don't think about it -- just compute:
Don't think about it — just compute:
{| align="center" cellpadding="6" width="90%"
|
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| (x + dx) and (y + dy) |
| (x + dx) and (y + dy) |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
To make future graphs easier to draw in Ascii land,
To make future graphs easier to draw in ASCII, I will use devices like '''<code>@=@=@</code>''' and '''<code>o=o=o</code>''' to identify several nodes into one, as in this next redrawing:
I will use devices like @=@=@ and o=o=o to identify
several nodes into one, as in this next redrawing:
{| align="center" cellpadding="6" width="90%"
|
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| (x + dx) and (y + dy) |
| (x + dx) and (y + dy) |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
However you draw it, these expressions follow because the
However you draw it, these expressions follow because the expression <math>x + dx,\!</math> where the plus sign indicates addition in <math>\mathbb{B},</math> that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation in logic, parses to a graph of the following form:
expression x + dx, where the plus sign indicates (mod 2)
addition in B, and thus corresponds to an exclusive-or
in logic, parses to a graph of the following form:
{| align="center" cellpadding="6" width="90%"
|
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| x + dx |
| x + dx |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
<pre>
Next question: What is the difference between
Next question: What is the difference between
the value of the proposition xy "over there" and
the value of the proposition xy "over there" and
