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{{DISPLAYTITLE:Semiotic Information}}
{{DISPLAYTITLE:Semiotic Information}}
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
'''Semiotic information''' is the information content of signs as conceived within the [[semeiotic]] or [[sign-relational]] framework developed by [[Charles Sanders Peirce]].
'''Semiotic information''' is the information content of signs as conceived within the [[semeiotic]] or [[sign-relational]] framework developed by [[Charles Sanders Peirce]].
The average uncertainty reduction per sign of the language is computed by taking a ''weighted average'' of the reductions that occur in the channel, where the weight of each reduction is the number of options or outcomes that fall under the associated sign.
The average uncertainty reduction per sign of the language is computed by taking a ''weighted average'' of the reductions that occur in the channel, where the weight of each reduction is the number of options or outcomes that fall under the associated sign.
:* The uncertainty reduction of <math>(\log 5 - \log 3)</math> gets a weight of 3.
:* The uncertainty reduction of <math>(\log 5 - \log 3)\!</math> gets a weight of 3.
:* The uncertainty reduction of <math>(\log 5 - \log 2)</math> gets a weight of 2.
:* The uncertainty reduction of <math>(\log 5 - \log 2)\!</math> gets a weight of 2.
Finally, the weighted average of these two reductions is:
Finally, the weighted average of these two reductions is:
: <math>{1 \over {2 + 3}}(3(\log 5 - \log 3) + 2(\log 5 - \log 2))</math>
: <math>{1 \over {2 + 3}}(3(\log 5 - \log 3) + 2(\log 5 - \log 2))\!</math>
Extracting the general pattern of this calculation yields the following worksheet for computing the capacity of a 2-symbol channel with frequencies that partition as ''n'' = ''k''<sub>1</sub> + ''k''<sub>2</sub>.
Extracting the general pattern of this calculation yields the following worksheet for computing the capacity of a 2-symbol channel with frequencies that partition as <math>n = k_1 + k_2.\!</math>
{| cellspacing="6"
{| cellspacing="6"
| Capacity
| Capacity
| of a channel {"A", "B"} that bears the odds of 60 "A" to 40 "B"
| of a channel {“A”, “B”} that bears the odds of 60 “A” to 40 “B”
|-
|-
|
|
| <math>=\quad {1 \over n}(k_1(\log n - \log k_1) + k_2(\log n - \log k_2))</math>
| <math>=\quad {1 \over n}(k_1(\log n - \log k_1) + k_2(\log n - \log k_2))\!</math>
|-
|-
|
|
| <math>=\quad {k_1 \over n}(\log n - \log k_1) + {k_2 \over n}(\log n - \log k_2)</math>
| <math>=\quad {k_1 \over n}(\log n - \log k_1) + {k_2 \over n}(\log n - \log k_2)\!</math>
|-
|-
|
|
| <math>=\quad -{k_1 \over n}(\log k_1 - \log n) -{k_2 \over n}(\log k_2 - \log n)</math>
| <math>=\quad -{k_1 \over n}(\log k_1 - \log n) -{k_2 \over n}(\log k_2 - \log n)\!</math>
|-
|-
|
|
| <math>=\quad -{k_1 \over n}(\log {k_1 \over n}) - {k_2 \over n}(\log {k_2 \over n})</math>
| <math>=\quad -{k_1 \over n}(\log {k_1 \over n}) - {k_2 \over n}(\log {k_2 \over n})\!</math>
|-
|-
|
|
| <math>=\quad -(p_1 \log p_1 + p_2 \log p_2)</math>
| <math>=\quad -(p_1 \log p_1 + p_2 \log p_2)\!</math>
|-
|-
|
|
| <math>=\quad - (0.6 \log 0.6 + 0.4 \log 0.4)</math>
| <math>=\quad - (0.6 \log 0.6 + 0.4 \log 0.4)\!</math>
|-
|-
|
|
| <math>=\quad 0.971</math>
| <math>=\quad 0.971\!</math>
|}
|}
* [[Charles Sanders Peirce (Bibliography)]]
* [[Charles Sanders Peirce (Bibliography)]]
* Peirce, C.S. (1867), "Upon Logical Comprehension and Extension", [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].
* Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].
==See also==
==See also==
* [[Boolean function]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Boolean-valued function]]
{{col-break}}
* [[Logical graph]]
* [[Logical graph]]
* [[Logical matrix]]
* [[Logical matrix]]
* [[Propositional calculus]]
* [[Propositional calculus]]
* [[Semeiotic]]
* [[Semeiotic]]
{{col-break}}
* [[Sign relation]]
* [[Sign relation]]
* [[Triadic relation]]
* [[Triadic relation]]
{{col-end}}
{{col-end}}
===Related articles and projects===
===Related articles===
{{col-begin}}
* [[Semiotic Theory Of Information]]
{{col-break}}
* [[Directory:Jon_Awbrey/Papers/Information_%3D_Comprehension_%C3%83%E2%80%94_Extension|Information = Comprehension × Extension]]
* [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_Logic_Of_Information Peirce's Logic Of Information]
* [[Inquiry Driven Systems]]
* [http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension × Extension]
* [[Introduction to Inquiry Driven Systems]]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [[Peirce’s Logic Of Information]]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [[Pragmatic Theory Of Information]]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
* [[Propositional Equation Reasoning Systems]]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}
[[Category:Charles Sanders Peirce]]
[[Category:Charles Sanders Peirce]]
