12,089
edits
Changes
MyWikiBiz, Author Your Legacy — Tuesday September 09, 2025
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Cactus Language (view source)
Revision as of 14:44, 27 May 2009
, 14:44, 27 May 2009→Differential Logic: segment
===Differential Logic===
===Differential Logic===
====Note 1====
One of the first things that you can do, once you have a really decent calculus for boolean functions or propositional logic, whatever you want to call it, is to compute the differentials of these functions or propositions.
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 x & y, which
I graph as two labels attached to a root node, so:
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| x and y |
| x and y |
o---------------------------------------o
o---------------------------------------o
</pre>
Written as a string, this is just the concatenation "x y".
Written as a string, this is just the concatenation "<math>x~y</math>".
The proposition xy may be taken as a boolean function f(x, y)
The proposition <math>xy\!</math> may be taken as a boolean function <math>f(x, y)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that <math>0\!</math> means <math>\operatorname{false}</math> and <math>1\!</math> means <math>\operatorname{true}.</math>
having the abstract type f : B x B -> B, where B = {0, 1} is
read in such a way that 0 means "false" and 1 means "true".
In this style of graphical representation,
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.
the value "true" looks like a blank label
and the value "false" looks like an edge.
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| true |
| true |
o---------------------------------------o
o---------------------------------------o
</pre>
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| false |
| false |
o---------------------------------------o
o---------------------------------------o
</pre>
<pre>
Back to the proposition xy. Imagine yourself standing
Back to the proposition xy. Imagine yourself standing
in a fixed cell of the corresponding venn diagram, say,
in a fixed cell of the corresponding venn diagram, say,
Have to break here -- will explain later.
Have to break here -- will explain later.
</pre>
====Note 2====
Note 2
<pre>
We have just met with the fact that
We have just met with the fact that
the differential of the "and" is
the differential of the "and" is
Center Cell,
Center Cell,
Chateau Dif.
Chateau Dif.
</pre>
====Note 3====
Note 3
<pre>
Last time we computed what will variously be called
Last time we computed what will variously be called
the "difference map", the "difference proposition",
the "difference map", the "difference proposition",
as telling you what you have to do from each point
as telling you what you have to do from each point
of U in order to change the value borne by f(x, y).
of U in order to change the value borne by f(x, y).
</pre>
====Note 4====
Note 4
<pre>
We have been studying the action of the difference operator D,
We have been studying the action of the difference operator D,
also known as the "localization operator", on the proposition
also known as the "localization operator", on the proposition
of its meaning. We will encounter more and
of its meaning. We will encounter more and
more of these alternative readings as we go.
more of these alternative readings as we go.
</pre>
====Note 5====
Note 5
<pre>
The enlargement operator E, also known as the "shift operator",
The enlargement operator E, also known as the "shift operator",
has many interesting and very useful properties in its own right,
has many interesting and very useful properties in its own right,
as telling us all the different ways to reach
as telling us all the different ways to reach
a model of f from any point of the universe U.
a model of f from any point of the universe U.
</pre>
====Note 6====
Note 6
<pre>
To broaden our experience with simple examples, let us now contemplate the
To broaden our experience with simple examples, let us now contemplate the
sixteen functions of concrete type X x Y -> B and abstract type B x B -> B.
sixteen functions of concrete type X x Y -> B and abstract type B x B -> B.
By way of initial orientation, Table 0 lists equivalent expressions for the
By way of initial orientation, Table 0 lists equivalent expressions for the
sixteen functions in a number of different languages for zeroth order logic.
sixteen functions in a number of different languages for zeroth order logic.
Table 0. Propositional Forms On Two Variables
Table 0. Propositional Forms On Two Variables
| | | | | | |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
o---------o---------o---------o----------o------------------o----------o
The next four Tables expand the expressions of Ef and Df
The next four Tables expand the expressions of Ef and Df
Notice that the functions are given in a different order,
Notice that the functions are given in a different order,
here being collected into a set of seven natural classes.
here being collected into a set of seven natural classes.
Table 1. Ef Expanded Over Ordinary Features {x, y}
Table 1. Ef Expanded Over Ordinary Features {x, y}
| | | | | | |
| | | | | | |
o------o------------o------------o------------o------------o------------o
o------o------------o------------o------------o------------o------------o
Table 2. Df Expanded Over Ordinary Features {x, y}
Table 2. Df Expanded Over Ordinary Features {x, y}
| | | | | | |
| | | | | | |
o------o------------o------------o------------o------------o------------o
o------o------------o------------o------------o------------o------------o
Table 3. Ef Expanded Over Differential Features {dx, dy}
Table 3. Ef Expanded Over Differential Features {dx, dy}
| | | | | |
| | | | | |
o-------------------o------------o------------o------------o------------o
o-------------------o------------o------------o------------o------------o
Table 4. Df Expanded Over Differential Features {dx, dy}
Table 4. Df Expanded Over Differential Features {dx, dy}
| | | | | | |
| | | | | | |
o------o------------o------------o------------o------------o------------o
o------o------------o------------o------------o------------o------------o
If the medium truly is the message,
If the medium truly is the message,
the blank slate is the innate idea.
the blank slate is the innate idea.
</pre>
====Note 7====
<pre>
If you think that I linger in the realm of logical difference calculus
If you think that I linger in the realm of logical difference calculus
out of sheer vacillation about getting down to the differential proper,
out of sheer vacillation about getting down to the differential proper,
it is probably out of a prior expectation that you derive from the art
it is probably out of a prior expectation that you derive from the art
three group elements fix 4 propositions each, and so we get:
three group elements fix 4 propositions each, and so we get:
Number of orbits = (4 + 4 + 4 + 16) / 4 = 7. Amazing!
Number of orbits = (4 + 4 + 4 + 16) / 4 = 7. Amazing!
</pre>
====Note 8====
Note 8
<pre>
We have been contemplating functions of the type f : U -> B
We have been contemplating functions of the type f : U -> B
studying the action of the operators E and D on this family.
studying the action of the operators E and D on this family.
for j = 1 to k. The proposition (u_1, ..., u_k) indicates the disjunctive
for j = 1 to k. The proposition (u_1, ..., u_k) indicates the disjunctive
region consisting of the cells that are just next door to u_1 · ... · u_k.
region consisting of the cells that are just next door to u_1 · ... · u_k.
</pre>
====Note 9====
Note 9
<pre>
| Consider what effects that might conceivably have
| Consider what effects that might conceivably have
| practical bearings you conceive the objects of your
| practical bearings you conceive the objects of your
More on the pragmatic maxim as a representation principle later.
More on the pragmatic maxim as a representation principle later.
</pre>
====Note 10====
Note 10
<pre>
| Consider what effects that might conceivably have
| Consider what effects that might conceivably have
| practical bearings you conceive the objects of your
| practical bearings you conceive the objects of your
behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus,
behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus,
and I reckon you know where that leads.
and I reckon you know where that leads.
</pre>
====Note 11====
Note 11
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
Your paraphrastic interpretation of what this all
Your paraphrastic interpretation of what this all
means would come out precisely the same as before.
means would come out precisely the same as before.
</pre>
====Note 12====
Note 12
<pre>
Erratum
Erratum
2. The inessential difference between two conventions of presenting matrices.
2. The inessential difference between two conventions of presenting matrices.
I will sort this out and correct it later, as need be.
I will sort this out and correct it later, as need be.
</pre>
====Note 13====
Note 13
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
I think that will serve to fix notation
I think that will serve to fix notation
and set up the remainder of the account.
and set up the remainder of the account.
</pre>
====Note 14====
Note 14
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
So I still have a few wrinkles to iron out before
So I still have a few wrinkles to iron out before
I can give this story a smooth enough consistency.
I can give this story a smooth enough consistency.
</pre>
====Note 15====
Note 15
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
This is consistent with the convention that Peirce uses in
This is consistent with the convention that Peirce uses in
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
</pre>
====Note 16====
Note 16
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
I will have to leave that question as it is for now,
I will have to leave that question as it is for now,
in hopes that a solution will evolve itself in time.
in hopes that a solution will evolve itself in time.
</pre>
====Note 17====
Note 17
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
And now I need to go out of doors and weed my garden for a time ...
And now I need to go out of doors and weed my garden for a time ...
</pre>
====Note 18====
Note 18
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
and so the two representations have the very
and so the two representations have the very
same effects on each point of their bearing.
same effects on each point of their bearing.
</pre>
====Note 19====
Note 19
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
(CP 4.227-323), with particular reference to the section
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).
that treats of "Trichotomic Mathematics" (CP 4.307-323).
</pre>
===Work Area===
====Note 20====
Note 20
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
| | | | | | |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o---------o
</pre>
====Note 21====
Note 21
<pre>
| Consider what effects that might 'conceivably'
| Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| have practical bearings you 'conceive' the
