Changes

To consider how a system of logical graphs, taken together as a semiotic domain, might bear an iconic relationship to a system of logical objects that make up our object domain, we will next need to consider what our logical objects are.

To consider how a system of logical graphs, taken together as a semiotic domain, might bear an iconic relationship to a system of logical objects that make up our object domain, we will next need to consider what our logical objects are.

A popular answer, if by popular one means that both Peirce and Frege agreed on it, is to say that our ultimate logical objects are without loss of generality most conveniently referred to as Truth and Falsity.  If nothing else, it serves the end of beginning simply to go along with this thought for a while, and so we can start with an object domain that consists of just two ''objects'' or ''values'', to wit, <math>O = \mathbb{B} = \{ \text{false}, \text{true} \}.</math>

A popular answer, if by popular one means that both Peirce and Frege agreed on it, is to say that our ultimate logical objects are without loss of generality most conveniently referred to as Truth and Falsity.  If nothing else, it serves the end of beginning simply to go along with this thought for a while, and so we can start with an object domain that consists of just two ''objects'' or ''values'', to wit, <math>O = \mathbb{B} = \{ \operatorname{false}, \operatorname{true} \}.</math>

Given those two categories of structured individuals, namely, <math>O = \mathbb{B} = \{ \text{false}, \text{true} \}</math> and <math>S = \{ \text{logical graphs} \},\!</math> the next task is to consider the brands of morphisms from <math>S\!</math> to <math>O\!</math> that we might reasonably have in mind when we speak of the ''arrows of interpretation''.

Given those two categories of structured individuals, namely, <math>O = \mathbb{B} = \{ \operatorname{false}, \operatorname{true} \}</math> and <math>S = \{ \text{logical graphs} \},\!</math> the next task is to consider the brands of morphisms from <math>S\!</math> to <math>O\!</math> that we might reasonably have in mind when we speak of the ''arrows of interpretation''.

With the aim of embedding our consideration of logical graphs, as seems most fitting, within Peirce's theory of triadic sign relations, we have declared the first layers of our object, sign, and interpretant domains.  As we often do in formal studies, we've taken the sign and interpretant domains to be the same set, <math>S = I,\!</math> calling it the ''semiotic domain'', or, as I see that I've done in some other notes, the ''syntactic domain''.

With the aim of embedding our consideration of logical graphs, as seems most fitting, within Peirce's theory of triadic sign relations, we have declared the first layers of our object, sign, and interpretant domains.  As we often do in formal studies, we've taken the sign and interpretant domains to be the same set, <math>S = I,\!</math> calling it the ''semiotic domain'', or, as I see that I've done in some other notes, the ''syntactic domain''.

Returning to <math>\operatorname{En}</math> and <math>\operatorname{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S\!</math> to the object domain <math>O,\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.

Returning to <math>\operatorname{En}</math> and <math>\operatorname{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S\!</math> to the object domain <math>O,\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.

Taking <math>\operatorname{En}</math> and <math>\operatorname{Ex}</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \text{Falsity}, \text{Truth} \},\!</math> let's factor each arrow across the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Cactus Node Big Fat.jpg|16px]],&nbsp;[[Image:Cactus Spike Big Fat.jpg|12px]]<math>\},</math> the domain that consists of a single rooted node plus a single rooted edge.  As a strategic tactic, this allows each arrow to be broken into a purely syntactic part <math>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and its purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : S_0 \to O.</math>

Taking <math>\operatorname{En}</math> and <math>\operatorname{Ex}</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \operatorname{falsity}, \operatorname{truth} \},\!</math> it is possible to factor each arrow across the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Cactus Node Big Fat.jpg|16px]],&nbsp;[[Image:Cactus Spike Big Fat.jpg|12px]]<math>\},</math> the domain that consists of a single rooted node plus a single rooted edge.  As a strategic tactic, this allows each arrow to be broken into a purely syntactic part <math>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and its purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : S_0 \to O.</math>

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:

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On the other side of the ledger, because the syntactic factors, ''En''_syn and ''Ex''_syn, are indiscernible from each other, there is a syntactic contribution to the overall interpretation process that can be most readily investigated on purely formal grounds.  That will be the task to face when next we meet on these lists.

On the other side of the ledger, because the syntactic factors, <math>\operatorname{En}_\text{syn}</math> and <math>\operatorname{Ex}_\text{syn},</math> are indiscernible from each other, there is a syntactic contribution to the overall interpretation process that can be most readily investigated on purely formal grounds.  That will be the task to face when next we meet on these lists.

Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:

Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape: