Changes

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:

{| align="center" cellpadding="10" width="90%"

{| cellpadding="6"

| <math>\operatorname{En}_\text{sem} :</math> &nbsp; [[Image:Rooted Node.jpg|16px]] &nbsp; <math>\mapsto \operatorname{false},</math> &nbsp; [[Image:Rooted Edge.jpg|12px]] &nbsp; <math>\mapsto \operatorname{true}.</math>

| width="5%" | &nbsp;

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

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

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

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

| &nbsp;

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

| &nbsp;

| <math>\mapsto</math>

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

|-

|-

| <math>\operatorname{Ex}_\text{sem} :</math> &nbsp; [[Image:Rooted Node.jpg|16px]] &nbsp; <math>\mapsto \operatorname{true},</math> &nbsp; [[Image:Rooted Edge.jpg|12px]] &nbsp; <math>\mapsto \operatorname{false}.</math>

| &nbsp;

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

| <math>\mapsto</math>

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

|}

|}

|}

|}

The interpretation maps ''En'', ''Ex'' : ''Y'' &rarr; ''X'' are factored into a shared syntactic part:

The interpretation maps <math>\operatorname{En}, \operatorname{Ex} : Y \to X</math> are factored into (1) a common syntactic part and (2) a couple of distinct semantic parts:

: ''En''_syn = ''Ex''_syn = ''E''_syn : ''Y'' &rarr; ''Y''₀

 

<math>\begin{array}{ll}

 

: ''En''_sem, ''Ex''_sem : ''Y''₀ &rarr; ''X''

\operatorname{En}_\text{syn} = \operatorname{Ex}_\text{syn} = \operatorname{E}_\text{syn} : Y \to Y_0

\end{array}</math>

\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : Y_0 \to X

\end{array}</math>

The functional images of the syntactic reduction map ''E''_syn : ''Y'' &rarr; ''Y''₀ are the two simplest signs or the most reduced pair of expressions, regarded as rooted trees taking the shapes @ and |, and these may be treated as the canonical representatives of their respective equivalence classes.

The functional images of the syntactic reduction map <math>\operatorname{E}_\text{syn} : Y \to Y_0</math> are the two simplest signs or the most reduced pair of expressions, regarded as the rooted trees [[Image:Rooted Node.jpg|16px]] and [[Image:Rooted Edge.jpg|12px]], and these may be treated as the canonical representatives of their respective equivalence classes.

The more Peirce-systent among you, on contemplating that last picture, will 1st or 2nd or 3rd-naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"

The more Peirce-sistent among you, on contemplating that last picture, will naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"

{| align="center" style="text-align:center; width:90%"

{| align="center" style="text-align:center; width:90%"