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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
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|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
</pre>
==Commentary Note 11.1==
<pre>
We have reached in our reading of Peirce's text a suitable place to pause --
We have reached in our reading of Peirce's text a suitable place to pause --
actually, it is more like to run as fast as we can along a parallel track --
actually, it is more like to run as fast as we can along a parallel track --
Therefore, let us inaugurate a systematic study of relational composition,
Therefore, let us inaugurate a systematic study of relational composition,
general enough to explicate the "generative potency" of Peirce's 1870 LOR.
general enough to explicate the "generative potency" of Peirce's 1870 LOR.
</pre>
==Commentary Note 11.2==
<pre>
Let's bring together the various things that Peirce has said
Let's bring together the various things that Peirce has said
about the "number of function" up to this point in the paper.
about the "number of function" up to this point in the paper.
Peirce's presentation that I am presently trying to avoid as much as possible.
Peirce's presentation that I am presently trying to avoid as much as possible.
Thus, I will need to experiment with various sorts of compromise formations.
Thus, I will need to experiment with various sorts of compromise formations.
</pre>
==Commentary Note 11.3==
<pre>
Having spent a fair amount of time in earnest reflection on the issue,
Having spent a fair amount of time in earnest reflection on the issue,
I cannot see a way to continue my interpretation of Peirce's 1870 LOR,
I cannot see a way to continue my interpretation of Peirce's 1870 LOR,
as spied from a combinatorial point of view, that I hope will
as spied from a combinatorial point of view, that I hope will
serve in reeding Peirce's text, if we draw on it judiciously.
serve in reeding Peirce's text, if we draw on it judiciously.
</pre>
==Commentary Note 11.4==
<pre>
The task before us now is to get very clear about the relationships
The task before us now is to get very clear about the relationships
among relative terms, relations, and the special cases of relations
among relative terms, relations, and the special cases of relations
in the set R of real numbers and whose correlate lies in the set S
in the set R of real numbers and whose correlate lies in the set S
of syntactic terms.
of syntactic terms.
</pre>
==Commentary Note 11.5==
<pre>
It always helps me to draw lots of pictures of stuff,
It always helps me to draw lots of pictures of stuff,
so let's extract the somewhat overly compressed bits
so let's extract the somewhat overly compressed bits
Clearly, then, E cannot qualify as a pre-function
Clearly, then, E cannot qualify as a pre-function
or a function on either of its relational domains.
or a function on either of its relational domains.
</pre>
==Commentary Note 11.6==
<pre>
Let's continue to work our way through the rest of the first
Let's continue to work our way through the rest of the first
set of definitions, making up appropriate examples as we go.
set of definitions, making up appropriate examples as we go.
So, E_3 is a pre-function e_3 : X ~> Y,
So, E_3 is a pre-function e_3 : X ~> Y,
and E_4 is a pre-function e_4 : X <~ Y.
and E_4 is a pre-function e_4 : X <~ Y.
</pre>
==Commentary Note 11.7==
<pre>
We come now to the very special cases of 2-adic relations that are
We come now to the very special cases of 2-adic relations that are
known as functions. It will serve a dual purpose on behalf of the
known as functions. It will serve a dual purpose on behalf of the
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
==Commentary Note 11.8==
<pre>
Now let's re-examine the "numerical incidence properties" of relations,
Now let's re-examine the "numerical incidence properties" of relations,
concentrating on the definitions of the assorted regularity conditions.
concentrating on the definitions of the assorted regularity conditions.
We observe that F is 3-regular at G and 1-regular at H.
We observe that F is 3-regular at G and 1-regular at H.
</pre>
==Commentary Note 11.9==
<pre>
Among the vast variety of conceivable regularities affecting 2-adic relations,
Among the vast variety of conceivable regularities affecting 2-adic relations,
we pay special attention to the c-regularity conditions where c is equal to 1.
we pay special attention to the c-regularity conditions where c is equal to 1.
and we record this fact in either of
and we record this fact in either of
the manners F : X <- Y or F : Y -> X.
the manners F : X <- Y or F : Y -> X.
</pre>
==Commentary Note 11.10==
<pre>
In the case of a 2-adic relation F c X x Y that has
In the case of a 2-adic relation F c X x Y that has
the qualifications of a function f : X -> Y, there
the qualifications of a function f : X -> Y, there
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
==Commentary Note 11.11==
<pre>
The preceding exercises were intended to beef-up our
The preceding exercises were intended to beef-up our
functional literacy skills to the point where we can
functional literacy skills to the point where we can
Let's see how much easy mileage we can get in our exploration
Let's see how much easy mileage we can get in our exploration
of functions by adopting the above templates as a paradigm.
of functions by adopting the above templates as a paradigm.
</pre>
==Commentary Note 11.12==
<pre>
Since functions are special cases of 2-adic relations, and since the space
Since functions are special cases of 2-adic relations, and since the space
of 2-adic relations is closed under relational composition, in other words,
of 2-adic relations is closed under relational composition, in other words,
Bur proofs after midnight must be checked the next day.
Bur proofs after midnight must be checked the next day.
</pre>
==Commentary Note 11.13==
<pre>
As we make our way toward the foothills of Peirce's 1870 LOR, there
As we make our way toward the foothills of Peirce's 1870 LOR, there
is one piece of equipment that we dare not leave the plains without --
is one piece of equipment that we dare not leave the plains without --
in the J columns, then the constraint in the K row is automatically true.
in the J columns, then the constraint in the K row is automatically true.
That is one way of understanding the equation: J(L(u, v)) = K(Ju, Jv).
That is one way of understanding the equation: J(L(u, v)) = K(Ju, Jv).
</pre>
==Commentary Note 11.14==
<pre>
First, a correction. Ignore for now the
First, a correction. Ignore for now the
gloss that I gave in regard to Figure 19:
gloss that I gave in regard to Figure 19:
|
|
| J(L(u, v)) = K(Ju, Jv)
| J(L(u, v)) = K(Ju, Jv)
</pre>
==Commentary Note 11.15==
<pre>
I'm going to elaborate a little further on the subject
I'm going to elaborate a little further on the subject
of arrows, morphisms, or structure-preserving maps, as
of arrows, morphisms, or structure-preserving maps, as
distributive multiplication of a sum (u + v) by a factor J.
distributive multiplication of a sum (u + v) by a factor J.
Hence another popular name for a morphism: a "linear" map.
Hence another popular name for a morphism: a "linear" map.
</pre>
==Commentary Note 11.16==
<pre>
I think that we have enough material on morphisms now
I think that we have enough material on morphisms now
to go back and cast a more studied eye on what Peirce
to go back and cast a more studied eye on what Peirce
in the direction T : U -> V, since we are counting only those
in the direction T : U -> V, since we are counting only those
teeth that ideally occupy one and only one mouth of a creature.
teeth that ideally occupy one and only one mouth of a creature.
</pre>
==Commentary Note 11.17==
<pre>
I think that the reader is beginning to get an inkling of the crucial importance of
I think that the reader is beginning to get an inkling of the crucial importance of
the "number of" map in Peirce's way of looking at logic, for it's one of the plancks
the "number of" map in Peirce's way of looking at logic, for it's one of the plancks
as "proper subsumption", that is, excluding the equality case,
as "proper subsumption", that is, excluding the equality case,
while we read the "<" on the right as the usual "less than".
while we read the "<" on the right as the usual "less than".
</pre>
==Commentary Note 11.18==
<pre>
There is a comment that I ought to make on the concept of
There is a comment that I ought to make on the concept of
a "structure preserving map", including as a special case
a "structure preserving map", including as a special case
that is not required in order to call the map 'v' "order-preserving",
that is not required in order to call the map 'v' "order-preserving",
or what is also known as an "order morphism".
or what is also known as an "order morphism".
</pre>
==Commentary Note 11.19==
<pre>
Up to this point in the LOR of 1870, Peirce has introduced the
Up to this point in the LOR of 1870, Peirce has introduced the
"number of" measure on logical terms and discussed the extent
"number of" measure on logical terms and discussed the extent
In sum, therefor, it also serves that only preserves
In sum, therefor, it also serves that only preserves
a due respect for the function of a vacuum in nature.
a due respect for the function of a vacuum in nature.
</pre>
==Commentary Note 11.20==
<pre>
We arrive at the last, for the time being, of
We arrive at the last, for the time being, of
Peirce's statements about the "number of" map.
Peirce's statements about the "number of" map.
['t'] = ['t'f]/[f], is every bite as true as the defining
['t'] = ['t'f]/[f], is every bite as true as the defining
equation in this circumstance, namely, ['t'] = ['t'm]/[m].
equation in this circumstance, namely, ['t'] = ['t'm]/[m].
</pre>
==Commentary Note 11.21==
<pre>
One more example and one more general observation, and then we will
One more example and one more general observation, and then we will
be all caught up with our homework on Peirce's "number of" function.
be all caught up with our homework on Peirce's "number of" function.
interpret [m,] = "the average number of men per things in general" as
interpret [m,] = "the average number of men per things in general" as
P(m) = the probability of a thing in general being a man. Seems okay.
P(m) = the probability of a thing in general being a man. Seems okay.
</pre>
==Commentary Note 11.22==
<pre>
Let's look at that last example from a different angle.
Let's look at that last example from a different angle.
Another way to see it is as follows: P(b|m) = 1/4 while P(b) = 1/7.
Another way to see it is as follows: P(b|m) = 1/4 while P(b) = 1/7.
</pre>
==Commentary Note 11.23==
<pre>
Let me try to sum up as succinctly as possible the lesson
Let me try to sum up as succinctly as possible the lesson
that we ought to take away from Peirce's last "number of"
that we ought to take away from Peirce's last "number of"
As Hamlet discovered, there's a lot to be learned from turning a crank.
As Hamlet discovered, there's a lot to be learned from turning a crank.
</pre>
==Commentary Note 11.24==
<pre>
And so we come to the end of the "number of" examples
And so we come to the end of the "number of" examples
that we found on our agenda at this point in the text:
that we found on our agenda at this point in the text:
the main themes that I aim to elaborate throughout the rest of
the main themes that I aim to elaborate throughout the rest of
this inquiry.
this inquiry.
</pre>
==Selection 12==
<pre>
| The Sign of Involution
| The Sign of Involution
|
|