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 call ''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, 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, "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.

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.

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

Back to Venice, and the close-knit party 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, to which I have thrown in 1, the universe of "anybody or anything", just for good measure:

Here is the list of absolute terms that we were considering before, to which I have thrown in <math>\mathbf{1},</math> the universe of ''anything'', just for good measure:

:{| cellpadding="4"

{| align="center" cellspacing="6" width="90%"

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

|

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

\mathbf{1}

& =     & \text{anything}

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

& =     & \mathrm{B}

& +\!\!, & \mathrm{C}

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

& +\!\!, & \mathrm{D}

& +\!\!, & \mathrm{E}

& +\!\!, & \mathrm{I}

& +\!\!, & \mathrm{J}

& +\!\!, & \mathrm{O}

\mathrm{m}

& =     & \text{man}

& =     & \mathrm{C}

& +\!\!, & \mathrm{I}

& +\!\!, & \mathrm{J}

& +\!\!, & \mathrm{O}

\mathrm{n}

& =     & \text{noble}

& =     & \mathrm{C}

& +\!\!, & \mathrm{D}

& +\!\!, & \mathrm{O}

\mathrm{w}

& =     & \text{woman}

& =     & \mathrm{B}

& +\!\!, & \mathrm{D}

& +\!\!, & \mathrm{E}

|}

|}