Changes

Peirce is here observing what we might dub a "contingent morphism" or a "skeptraphotic arrow", if you will.  Provided that a certain condition, to be named and, what is more hopeful, to be clarified in short order, happens to be satisfied, we would find it holding that the "number of" map ''v'' : ''S'' → '''R''' such that ''vs'' = [''s''] serves to preserve the multiplication of relative terms, that is as much to say, the composition of relations, in the form:  [''xy''] = [''x''][''y''].

Peirce is here observing what we might dub a "contingent morphism" or a "skeptraphotic arrow", if you will.  Provided that a certain condition, to be named and, what is more hopeful, to be clarified in short order, happens to be satisfied, we would find it holding that the "number of" map ''v'' : ''S'' → '''R''' such that ''vs'' = [''s''] serves to preserve the multiplication of relative terms, that is as much to say, the composition of relations, in the form:  [''xy''] = [''x''][''y''].

So let us try to uncross Peirce's manifestly chiasmatic encryption of the condition that is called on in support of this preservation.

So let us try to uncross Peirce's manifestly chiasmatic encryption

 

of the condition that is called on in support of this preservation.

 

| there are just as many x's per y

| as there are 'per' things<,>

<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>

| things of the universe ...

: <p>[''t''][''f''] =  [''tf'']</p>

So let us resort to the example:

<p>holds arithmetically.  (CP 3.76).</p>

| holds arithmetically.  (CP 3.76).

Now that is something that we can sink our teeth into,

Now that is something that we can sink our teeth into, and trace the bigraph representation of the situation. In order to do this, it will help to recall our first examination of the "tooth of" relation, and to adjust the picture that we sketched of it on that occasion.

and trace the bigraph representation of the situation.

In order to do this, it will help to recall our first

examination of the "tooth of" relation, and to adjust

the picture that we sketched of it on that occasion.

Transcribing Peirce's example, we may let m = "man" and 't' = "tooth of ---".

Transcribing Peirce's example, we may let m = "man" and ''t'' = "tooth of ---". Then ''v''(''t'') = [''t''] = [''tm'']/[''m''], that is to say, in a universe of perfect human dentition, the number of the relative term "tooth of ---" is equal to the number of teeth of humans divided by the number of humans, that is, 32.

Then 'v'('t') = ['t'] = ['t'm]/[m], that is to say, in a universe of perfect

human dentition, the number of the relative term "tooth of ---" is equal to

the number of teeth of humans divided by the number of humans, that is, 32.

The 2-adic relative term 't' determines a 2-adic relation T c U x V,

The 2-adic relative term ''t'' determines a 2-adic relation ''T''&nbsp;&sube;&nbsp;''U''&nbsp;&times;&nbsp;''V'', where U and V are two universes of discourse, possibly the same one, that hold among other things all of the teeth and all of the people that happen to be under discussion, respectively.  To make the case as simple as we can and still cover the point, let's say that there are just four people in our initial universe of discourse, and that just two of them are French.  The bigraphic composition below shows all of the pertinent facts of the case.

where U and V are two universes of discourse, possibly the same one,

that hold among other things all of the teeth and all of the people

that happen to be under discussion, respectively.  To make the case

as simple as we can and still cover the point, let's say that there

are just four people in our initial universe of discourse, and that

just two of them are French.  The bigraphic composition below shows

all of the pertinent facts of the case.

T_1    T_32  T_33    T_64  T_65    T_96  T_97    T_128

T_1    T_32  T_33    T_64  T_65    T_96  T_97    T_128

  o  ...  o    o  ...  o    o  ...  o    o  ...  o      U

  o  ...  o    o  ...  o    o  ...  o    o  ...  o      U

     o            o            o            o          V = m = 1

     o            o            o            o          V = m = 1

     J            K            L            M

     J            K            L            M

Here, the order of relational composition flows up the page.

Here, the order of relational composition flows up the page. For convenience, the absolute term ''f'' = "frenchman" has been converted by using the comma functor to give the idempotent representation ‘''f''’ = ''f'', = "frenchman that is ---", and thus it can be taken as a selective from the universe of mankind.

For convenience, the absolute term f = "frenchman" has been

converted by using the comma functor to give the idempotent

representation 'f' = f, = "frenchman that is ---", and thus

it can be taken as a selective from the universe of mankind.

By way of a legend for the figure, we have the following data:

By way of a legend for the figure, we have the following data: