Changes

==Selection 5==

==Selection 5==

<pre>

<blockquote>

| The Signs for Multiplication

<p>'''The Signs for Multiplication'''</p>

 

| I shall adopt for the conception of multiplication

<p>I shall adopt for the conception of multiplication 'the application of a relation', in such a way that, for example, 'l'w shall denote whatever is lover of a woman.  This notation is the same as that used by Mr. De Morgan, although he apears not to have had multiplication in his mind.</p>

| 'the application of a relation', in such a way that,

 

| for example, 'l'w shall denote whatever is lover of

<p>'s'(m +, w) will, then, denote whatever is servant of anything of the class composed of men and women taken together.  So that:</p>

| a woman.  This notation is the same as that used by

 

| Mr. De Morgan, although he apears not to have had

<p>'s'(m +, w)  =  's'm +, 's'w.</p>

| multiplication in his mind.

 

<p>('l' +, 's')w will denote whatever is lover or servant to a woman, and:</p>

| 's'(m +, w) will, then, denote whatever is

 

| servant of anything of the class composed

<p>('l' +, 's')w  =  'l'w +, 's'w.</p>

| of men and women taken together.  So that:

 

<p>('sl')w will denote whatever stands to a woman in the relation of servant of a lover, and:</p>

| 's'(m +, w)  =  's'm +, 's'w.

 

<p>('sl')w  =  's'('l'w).</p>

| ('l' +, 's')w will denote whatever is

 

| lover or servant to a woman, and:

<p>Thus all the absolute conditions of multiplication are satisfied.</p>

 

| ('l' +, 's')w  =  'l'w +, 's'w.

<p>The term "identical with ---" is a unity for this multiplication.  That is to say, if we denote "identical with ---" by !1! we have:</p>

 

| ('sl')w will denote whatever stands to

<p>'x'!1!  =  'x',</p>

| a woman in the relation of servant of

| a lover, and:

| ('sl')w  =  's'('l'w).

| Thus all the absolute conditions

| of multiplication are satisfied.

| The term "identical with ---" is a unity

| for this multiplication.  That is to say,

| if we denote "identical with ---" by !1!

| we have:

| 'x'!1!  =  'x',

</pre>

Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of "information" to explain what it is that signs convey in the process.  By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn. So let's take a moment to draw out a few of these characters.

Peirce in 1870 is five years down the road from the Peirce of 1865-1866

who lectured extensively on the role of sign relations in the logic of

scientific inquiry, articulating their involvement in the three types

of inference, and inventing the concept of "information" to explain

what it is that signs convey in the process.  By this time, then,

the semiotic or sign relational approach to logic is so implicit

in his way of working that he does not always take the trouble

to point out its distinctive features at each and every turn.

So let's take a moment to draw out a few of these characters.

Sign relations, like any non-trivial brand of 3-adic relations,

[[Sign relations]], like any non-trivial brand of [[3-adic relations]], can become overwhelming to think about once the cardinality of the object, sign, and interpretant domains or the complexity of the relation itself ascends beyond the simplest examples. Furthermore, most of the strategies that we would normally use to control the complexity, like neglecting one of the domains, in effect, projecting the 3-adic sign relation onto one of its 2-adic faces, or focusing on a single ordered triple of the form ‹&nbsp;''o'',&nbsp;''s'',&nbsp;''i''&nbsp;› at a time, can result in our receiving a distorted impression of the sign relation's true nature and structure.

can become overwhelming to think about once the cardinality of

the object, sign, and interpretant domains or the complexity

of the relation itself ascends beyond the simplest examples.

Furthermore, most of the strategies that we would normally

use to control the complexity, like neglecting one of the

domains, in effect, projecting the 3-adic sign relation

onto one of its 2-adic faces, or focusing on a single

ordered triple of the form <o, s, i> at a time, can

result in our receiving a distorted impression of

the sign relation's true nature and structure.

I find that it helps me to draw, or at least to imagine drawing,

I find that it helps me to draw, or at least to imagine drawing, diagrams of the following form, where I can keep tabs on what's an object, what's a sign, and what's an interpretant sign, for a selected set of sign-relational triples.

diagrams of the following form, where I can keep tabs on what's

an object, what's a sign, and what's an interpretant sign, for

a selected set of sign-relational triples.

Here is how I would picture Peirce's example of equivalent terms:

Here is how I would picture Peirce's example of equivalent terms: ''v'' = ''p'', where "''v''" denotes the Vice-President of the United States, and "''p''" denotes the President of the Senate of the United States.

v = p, where "v" denotes the Vice-President of the United States,

and "p" denotes the President of the Senate of the United States.

o-----------------------------o-----------------------------o

o-----------------------------o-----------------------------o

|  Objective Framework (OF)  | Interpretive Framework (IF) |

|  Objective Framework (OF)  | Interpretive Framework (IF) |

|                                                          |

|                                                          |

o-----------------------------o-----------------------------o

o-----------------------------o-----------------------------o

Depending on whether we interpret the terms "v" and "p" as applying to

Depending on whether we interpret the terms "''v''" and "''p''" as applying to persons who hold these offices at one particular time or as applying to all those persons who have held these offices over an extended period of history, their denotations may be either singular of plural, respectively.

persons who hold these offices at one particular time or as applying to

all those persons who have held these offices over an extended period of

history, their denotations may be either singular of plural, respectively.

As a shortcut technique for indicating general denotations or plural referents,

As a shortcut technique for indicating general denotations or plural referents, I will use the "elliptic convention" that represents these by means of figures like "o&nbsp;o&nbsp;o" or "o&nbsp;&hellip;&nbsp;o", placed at the object ends of sign relational triads.

I will use the "elliptic convention" that represents these by means of figures

like "o o o" or "o ... o", placed at the object ends of sign relational triads.

For a more complex example, here is how I would picture Peirce's example

For a more complex example, here is how I would picture Peirce's example of an equivalence between terms that comes about by applying one of the distributive laws, for relative multiplication over absolute summation.

of an equivalence between terms that comes about by applying one of the

distributive laws, for relative multiplication over absolute summation.

o-----------------------------o-----------------------------o

o-----------------------------o-----------------------------o

|  Objective Framework (OF)  | Interpretive Framework (IF) |

|  Objective Framework (OF)  | Interpretive Framework (IF) |