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Revision as of 03:37, 31 July 2009
, 03:37, 31 July 2009→Categories of structured individuals: TeX markup
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: ''En''<sub>syn</sub> = ''Ex''<sub>syn</sub> = ''E''<sub>syn</sub> : ''Y'' → ''Y''<sub>0</sub>+ +and a couple of differential semantic parts:+ +: ''En''<sub>sem</sub>, ''Ex''<sub>sem</sub> : ''Y''<sub>0</sub> → ''X''+
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> [[Image:Rooted Node.jpg|16px]] <math>\mapsto \operatorname{false},</math> [[Image:Rooted Edge.jpg|12px]] <math>\mapsto \operatorname{true}.</math>
+| width="5%" |
+| width="5%" | <math>\operatorname{En}_\text{sem} :</math>
+| width="5%" | [[Image:Rooted Node.jpg|16px]]
+| width="5%" | <math>\mapsto</math>
+| <math>\operatorname{false},</math>
+|-
+|
+|
+| [[Image:Rooted Edge.jpg|12px]]
+| <math>\mapsto</math>
+| <math>\operatorname{true}.</math>
+|-
+|
+| <math>\operatorname{Ex}_\text{sem} :</math>
+| [[Image:Rooted Node.jpg|16px]]
+| <math>\mapsto</math>
+| <math>\operatorname{true},</math>
|-
|-
−| <math>\operatorname{Ex}_\text{sem} :</math> [[Image:Rooted Node.jpg|16px]] <math>\mapsto \operatorname{true},</math> [[Image:Rooted Edge.jpg|12px]] <math>\mapsto \operatorname{false}.</math>
+|
+|
+| [[Image:Rooted Edge.jpg|12px]]
+| <math>\mapsto</math>
+| <math>\operatorname{false}.</math>
|}
|}
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|}
|}
−The interpretation maps ''En'', ''Ex'' : ''Y'' → ''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:
−{| align="center" cellpadding="10" width="90%"
−|
−<math>\begin{array}{ll}
−1. &
−\operatorname{En}_\text{syn} = \operatorname{Ex}_\text{syn} = \operatorname{E}_\text{syn} : Y \to Y_0
+\end{array}</math>
+|-
+|
+<math>\begin{array}{ll}
+2. &
+\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : Y_0 \to X
+\end{array}</math>
+|}
−The functional images of the syntactic reduction map ''E''<sub>syn</sub> : ''Y'' → ''Y''<sub>0</sub> 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%"
