Changes

===Commentary Note 9.4===

===Commentary Note 9.4===

Boole rationalized the properties of what we now dub "boolean multiplication", roughly equivalent to logical conjunction, in terms of the laws that apply to selective operations.  Peirce, in his turn, taking a very significant step of analysis that has seldom been recognized for what it would lead to, much less followed, does not consider this multiplication to be a fundamental operation, but derives it as a by-product of relative multiplication by a comma relative. Thus, Peirce makes logical conjunction a special case of relative composition.

Boole rationalized the properties of what we now dub "boolean multiplication",

roughly equivalent to logical conjunction, in terms of the laws that apply to

selective operations.  Peirce, in his turn, taking a very significant step of

analysis that has seldom been recognized for what it would lead to, much less

followed, does not consider this multiplication to be a fundamental operation,

but derives it as a by-product of relative multiplication by a comma relative.

Thus, Peirce makes logical conjunction a special case of relative composition.

This opens up a very wide field of investigation,

This opens up a very wide field of investigation, "the operational significance of logical terms", one might say, but it will be best to advance bit by bit, and to lean on simple examples.

"the operational significance of logical terms",

one might say, but it will be best to advance

bit by bit, and to lean on simple examples.

Back to Venice, and the close-knit party

Back to Venice, and the close-knit party of absolutes and relatives that we were entertaining when last we were there.

of absolutes and relatives that we were

entertaining when last we were there.

Here is the list of absolute terms that we were considering before,

Here is the list of absolute terms that we were considering before, to which I have thrown in 1, the universe of "anybody or anything", just for good measure:

to which I have thrown in 1, the universe of "anybody or anything",

just for good measure:

1   = "anybody"             = B +, C +, D +, E +, I +, J +, O

 

| 1 || = || "anybody" || = || B +, C +, D +, E +, I +, J +, O

m   = "man"                 = C +, I +, J +, O

 

| m || = || "man"     || = || C +, I +, J +, O

n   = "noble"               = C +, D +, O

 

| n || = || "noble"   || = || C +, D +, O

w   = "woman"               = B +, D +, E

| w || = || "woman"   || = || B +, D +, E

Here is the list of "comma inflexions" or "diagonal extensions" of these terms:

Here is the list of "comma inflexions" or "diagonal extensions" of these terms:

1, = "anybody that is ---" = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

 

| 1,

m, = "man that is ---"     = C:C +, I:I +, J:J +, O:O

| = || "anybody that is ---"

 

| = || B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

n, = "noble that is ---"   = C:C +, D:D +, O:O

 

| m,

w, = "woman that is ---"   = B:B +, D:D +, E:E

| = || "man that is ---"

| = || C:C +, I:I +, J:J +, O:O

| n,

| = || "noble that is ---"

| = || C:C +, D:D +, O:O

| w,

| = || "woman that is ---"

| = || B:B +, D:D +, E:E

One observes that the diagonal extension of 1

One observes that the diagonal extension of 1 is the same thing as the identity relation !1!.

is the same thing as the identity relation !1!.

Inspired by this identification of "1," with "!1!", and because

Inspired by this identification of "1," with "!1!", and because the affixed commas of the diagonal extensions tend to get lost in the ordinary commas of punctuation, I will experiment with using the alternative notations:

the affixed commas of the diagonal extensions tend to get lost

in the ordinary commas of punctuation, I will experiment with

using the alternative notations:

m, = !m!

n, = !n!

| m, || = || !m!

w, = !w!

| n, || = || !n!

| w, || = || !w!

Working within our smaller sample of absolute terms,

Working within our smaller sample of absolute terms, we have already computed the sorts of products that apply the diagonal extension of an absolute term to another absolute term, for instance, these products:

we have already computed the sorts of products that

apply the diagonal extension of an absolute term to

another absolute term, for instance, these products:

m,n = !m!n = "man that is noble"   = C +, O

n,m = !n!m = "noble that is man"   = C +, O

| m,n || = || !m!n || = || "man that is noble"   || = || C +, O

n,w = !n!w = "noble that is woman" = D

w,n = !w!n = "woman that is noble" = D

| n,m || = || !n!m || = || "noble that is man"   || = || C +, O

| n,w || = || !n!w || = || "noble that is woman" || = || D

| w,n || = || !w!n || = || "woman that is noble" || = || D

This exercise gave us a bit of practical insight into

This exercise gave us a bit of practical insight into why the commutative law holds for logical conjunction.

why the commutative law holds for logical conjunction.

Further insight into the laws that govern this realm of logic,

Further insight into the laws that govern this realm of logic, and the underlying reasons why they apply, might be gained by systematically working through the whole variety of different products that are generated by the operational means in sight, namely, the products indicated by {1, ''m'', ''n'', ''w''}‹,›{1, ''m'', ''n'', ''w''}.

and the underlying reasons why they apply, might be gained by

systematically working through the whole variety of different

products that are generated by the operational means in sight,

namely, the products indicated by {1, m, n, w}<,>{1, m, n, w}.

But before we try to explore this territory more systematically,

But before we try to explore this territory more systematically, let us equip ourselves with the sorts of graphical and matrical representations that we discovered to provide us with such able assists to the intuition in so many of our previous adventures.

let us equip ourselves with the sorts of graphical and matrical

representations that we discovered to provide us with such able

assists to the intuition in so many of our previous adventures.

===Commentary Note 9.5===

===Commentary Note 9.5===