Changes

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Evidently what Peirce means by the associative principle, as it applies to this type of product, is that a product of elementary relatives having the form (R:S:T)(S:T)(T) is equal to R but that no other form of product yields a non-null result.  Scanning the implied terms of the triple product tells us that only the following case is non-null:  J = (J:J:D)(J:D)(D).  It follows that:

Evidently what Peirce means by the associative principle, as it applies to this type of product, is that a product of elementary relatives having the form <math>(\mathrm{R}:\mathrm{S}:\mathrm{T})(\mathrm{S}:\mathrm{T})(\mathrm{T})\!</math> is equal to <math>\mathrm{R}\!</math> but that no other form of product yields a non-null result.  Scanning the implied terms of the triple product tells us that only the case <math>(\mathrm{J}:\mathrm{J}:\mathrm{D})(\mathrm{J}:\mathrm{D})(\mathrm{D}) = \mathrm{J}\!</math> is non-null.  It follows that:

:{| cellpadding="4"

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

| 'l','s'w

|

| =

| "lover and servant of a woman"

\mathit{l},\!\mathit{s}\mathrm{w}

& = &

| &nbsp;

\text{lover and servant of a woman}

| =

| "lover that is a servant of a woman"

& = &

\text{lover that is a servant of a woman}

| &nbsp;

| =

& = &

| "lover of a woman that is a servant of that woman"

\text{lover of a woman that is a servant of that woman}

| &nbsp;

& = &

| =

\mathrm{J}

| J

|}

|}