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Just to make up a discrete example, let us suppose that the cardinality of this choice is a finite ''n'', and just to make it fully concrete let us say that ''n'' = 5. Figure 1 affords a rough picture of the situation.
Just to make up a discrete example, let us suppose that the cardinality of this choice is a finite ''n'', and just to make it fully concrete let us say that ''n'' = 5. Figure 1 affords a rough picture of the situation.
{| align="center" cellspacing="6" style="text-align:center; width:60%"
|
<pre>
o-------------------------------------------------o
| |
| ? ? ? ? ? |
| o o o o o |
| |
| o o o o o |
| |
| o o o o o |
| |
| o o o o o |
| |
| o o o o o |
| |
| ooooo |
| |
| Ө n = 5 |
| |
o-------------------------------------------------o
Figure 1. Juncture of Degree 5
</pre>
|}
This pictures a juncture, represented by "O", where there are ''n'' options for the outcome of a conduct, and we have no clue as to which it must be. In a sense, the degree of this node, in this case ''n'' = 5, measures the uncertainty that we have at this point.
This pictures a juncture, represented by "Ө", where there are ''n'' options for the outcome of a conduct, and we have no clue as to which it must be. In a sense, the degree of this node, in this case ''n'' = 5, measures the uncertainty that we have at this point.
This is the minimal sort of setting in which a sign can make any sense at all. A sign has significance for an agent, interpreter, or observer because its actualization, its being given or its being present, serves to reduce the uncertainty of a decision that the agent has to make, whether it concerns the actions that the agent ought to take in order to achieve some objective of interest, or whether it concerns the predicates that the agent ought to treat as being true of some object in the world.
This is the minimal sort of setting in which a sign can make any sense at all. A sign has significance for an agent, interpreter, or observer because its actualization, its being given or its being present, serves to reduce the uncertainty of a decision that the agent has to make, whether it concerns the actions that the agent ought to take in order to achieve some objective of interest, or whether it concerns the predicates that the agent ought to treat as being true of some object in the world.
The way that signs enter the scene is shown in Figure 2.
The way that signs enter the scene is shown in Figure 2.
{| align="center" cellspacing="6" style="text-align:center; width:60%"
|
<pre>
o-------------------------------------------------o
| |
| k_1 = 3 k_2 = 2 |
| o-----o-----o o-----o |
| "A" "B" |
| o----o----o o----o |
| |
| o---o---o o---o |
| |
| o--o--o o--o |
| |
| o-o-o o-o |
| |
| ooooo |
| |
| O n = 5 |
| |
o-------------------------------------------------o
Figure 2. Partition of Degrees 3 and 2
</pre>
|}
This illustrates a situation of uncertainty that has been augmented by a classification.
This illustrates a situation of uncertainty that has been augmented by a classification.
As a matter of fact, at least in this discrete type of case, it would be possible to use the degree of the node as a measure of uncertainty, but it would operate as a multiplicative measure rather than the sort of additive measure that we would normally prefer. To illustrate how this would work out, let us consider an easier example, one where the degree of the choice point is 4.
As a matter of fact, at least in this discrete type of case, it would be possible to use the degree of the node as a measure of uncertainty, but it would operate as a multiplicative measure rather than the sort of additive measure that we would normally prefer. To illustrate how this would work out, let us consider an easier example, one where the degree of the choice point is 4.
{| align="center" cellspacing="6" style="text-align:center; width:60%"
|
<pre>
o-------------------------------------------------o
| |
| ? ? ? ? |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| oo oo |
| |
| O n = 4 |
| |
o-------------------------------------------------o
Figure 3. Juncture of Degree 4
</pre>
|}
Suppose that we contemplate making another decision after the present issue has been decided, one that has a degree of 2 in every case. The compound situation is depicted in Figure 4.
Suppose that we contemplate making another decision after the present issue has been decided, one that has a degree of 2 in every case. The compound situation is depicted in Figure 4.
{| align="center" cellspacing="6" style="text-align:center; width:60%"
|
<pre>
o-------------------------------------------------o
| |
| o o o o o o o o |
| \ / \ / \ / \ / |
| o o o o n_2 = 2 |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| oo oo |
| |
| O n_1 = 4 |
| |
o-------------------------------------------------o
Figure 4. Compound Junctures of Degrees 4 and 2
</pre>
|}
This illustrates the fact that the compound uncertainty, 8, is the product of the two component uncertainties, 4 times 2. To convert this to an additive measure, one simply takes the logarithms to a convenient base, say 2, and thus arrives at the not too astounding fact that the uncertainty of the first choice is 2 bits, the uncertainty of the next choice is 1 bit, and the compound uncertainty is 2 + 1 = 3 bits.
This illustrates the fact that the compound uncertainty, 8, is the product of the two component uncertainties, 4 times 2. To convert this to an additive measure, one simply takes the logarithms to a convenient base, say 2, and thus arrives at the not too astounding fact that the uncertainty of the first choice is 2 bits, the uncertainty of the next choice is 1 bit, and the compound uncertainty is 2 + 1 = 3 bits.
A set of signs enters on a setup like this as a system of ''middle terms'', a collection of signs that one may regard, aptly enough, as constellating a ''medium''.
A set of signs enters on a setup like this as a system of ''middle terms'', a collection of signs that one may regard, aptly enough, as constellating a ''medium''.
{| align="center" cellspacing="6" style="text-align:center; width:60%"
|
<pre>
o-------------------------------------------------o
| |
| k_1 = 3 k_2 = 2 |
| o-----o-----o o-----o |
| "A" "B" |
| o----o----o o----o |
| |
| o---o---o o---o |
| |
| o--o--o o--o |
| |
| o-o-o o-o |
| |
| ooooo |
| |
| O n = 5 |
| |
o-------------------------------------------------o
Figure 5. Partition of Degrees 3 and 2
</pre>
|}
The ''language'' or ''medium'' here is the set of signs {"A", "B"}. On the assumption that the initial 5 outcomes are equally likely, one may associate a [[frequency distribution]] (''k''<sub>1</sub>, ''k''<sub>2</sub>) = (3, 2) and thus a [[probability distribution]] (''p''<sub>1</sub>, ''p''<sub>2</sub>) = (3/5, 2/5) = (0.6, 0.4) with this language, and thus define a communication ''[[channel (communications)|channel]]''.
The ''language'' or ''medium'' here is the set of signs {"A", "B"}. On the assumption that the initial 5 outcomes are equally likely, one may associate a [[frequency distribution]] (''k''<sub>1</sub>, ''k''<sub>2</sub>) = (3, 2) and thus a [[probability distribution]] (''p''<sub>1</sub>, ''p''<sub>2</sub>) = (3/5, 2/5) = (0.6, 0.4) with this language, and thus define a communication ''[[channel (communications)|channel]]''.
