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| width="5%" |  

| width="5%" | <math>\operatorname{En}_\text{sem} :</math>

| width="5%" | <math>\operatorname{En}_\text{sem} :</math>

| width="5%" | [[Image:Rooted Node.jpg|16px]]

| width="5%" align="center" | [[Image:Rooted Node.jpg|16px]]

| width="5%" | <math>\mapsto</math>

| width="5%" | <math>\mapsto</math>

| <math>\operatorname{false},</math>

| <math>\operatorname{false},</math>

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| [[Image:Rooted Edge.jpg|12px]]

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\operatorname{true}.</math>

| <math>\operatorname{true}.</math>

| &nbsp;

| &nbsp;

| <math>\operatorname{Ex}_\text{sem} :</math>

| <math>\operatorname{Ex}_\text{sem} :</math>

| [[Image:Rooted Node.jpg|16px]]

| align="center" | [[Image:Rooted Node.jpg|16px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\operatorname{true},</math>

| <math>\operatorname{true},</math>

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| [[Image:Rooted Edge.jpg|12px]]

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\operatorname{false}.</math>

| <math>\operatorname{false}.</math>

|}

|}

The answer is that the last bastion of 3-adic irreducibility presidios precisely in the duality of the dual interpretations ''En''_sem and ''Ex''_sem. To see this, consider the consequences of there being, contrary to all that we've assumed up to this point, some ultimately compelling reason to assert that the clean slate, the empty medium, the vacuum potential, whatever one wants to call it, is inherently more meaningful of either Falsity or Truth.  This would issue in a conviction forthwith that the 3-adic sign relation involved in this case decomposes as a composition of a couple of functions, that is to say, reduces to a 2-adic relation.

The answer is that the last bastion of 3-adic irreducibility presides precisely in the duality of the dual interpretations <math>\operatorname{En}_\text{sem}</math> and <math>\operatorname{Ex}_\text{sem}.</math>  To see this, consider the consequences of there being, contrary to all that we've assumed up to this point, some ultimately compelling reason to assert that the clean slate, the empty medium, the vacuum potential, whatever one wants to call it, is inherently more meaningful of either Falsity or Truth.  This would issue in a conviction forthwith that the 3-adic sign relation involved in this case decomposes as a composition of a couple of functions, that is to say, reduces to a 2-adic relation.

The duality of interpretation for logical graphs tells us that the empty medium, the tabula rasa, what Peirce called the ''Sheet of Assertion'' (SA) is a genuine symbol, not to be found among the degenerate species of signs that make up icons and indices, nor, as the SA has no parts, can it number icons or indices among its parts.  What goes for the medium must go for all of the signs that it mediates.  Thus we have the kinds of signs that Peirce in one place called "pure symbols", naming a selection of signs for basic logical operators specifically among them.

The duality of interpretation for logical graphs tells us that the empty medium, the tabula rasa, what Peirce called the ''Sheet of Assertion'' (SA) is a genuine symbol, not to be found among the degenerate species of signs that make up icons and indices, nor, as the SA has no parts, can it number icons or indices among its parts.  What goes for the medium must go for all of the signs that it mediates.  Thus we have the kinds of signs that Peirce in one place called "pure symbols", naming a selection of signs for basic logical operators specifically among them.

<blockquote>

Every word is a symbol.  Every sentence is a symbol.  Every book is a symbol.  Every representamen depending upon conventions is a symbol.  Just as a photograph is an index having an icon incorporated into it, that is, excited in the mind by its force, so a symbol may have an icon or an index incorporated into it, that is, the active law that it is may require its interpretation to involve the calling up of an image, or a composite photograph of many images of past experiences, as ordinary common nouns and verbs do;  or it may require its interpretation to refer to the actual surrounding circumstances of the occasion of its embodiment, like such words as 'that', 'this', 'I', 'you', 'which', 'here', 'now', 'yonder', etc.  Or it may be pure symbol, neither 'iconic' nor 'indicative', like the words 'and', 'or', 'of', etc. (C.S. Peirce, ''Collected Papers'', CP 4.447).

</blockquote>

<p>Every word is a symbol.  Every sentence is a symbol.  Every book is a symbol.  Every representamen depending upon conventions is a symbol.  Just as a photograph is an index having an icon incorporated into it, that is, excited in the mind by its force, so a symbol may have an icon or an index incorporated into it, that is, the active law that it is may require its interpretation to involve the calling up of an image, or a composite photograph of many images of past experiences, as ordinary common nouns and verbs do;  or it may require its interpretation to refer to the actual surrounding circumstances of the occasion of its embodiment, like such words as 'that', 'this', 'I', 'you', 'which', 'here', 'now', 'yonder', etc.  Or it may be pure symbol, neither 'iconic' nor 'indicative', like the words 'and', 'or', 'of', etc.</p>

 

<p>(Peirce, ''Collected Papers'', CP 4.447)</p>

Some will recall the many animadversions that we had on this topic, starting here:

Some will recall the many animadversions that we had on this topic, starting here: