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Revision as of 19:22, 7 January 2008
, 19:22, 7 January 2008→Commentary Note 11.22: markup
===Commentary Note 11.22===
===Commentary Note 11.22===
Let's look at that last example from a different angle.
Let's look at that last example from a different angle.
<blockquote>
<p>So if men are just as apt to be black as things in general:</p>
: <p>[''m'',][''b''] = [''m'',''b'']<p>
<p>where the difference between [''m''] and [''m'',] must not be overlooked.<p>
<p>(Peirce, CP 3.76).
</blockquote>
In different lights the formula [''m'',''b''] = [''m'',][''b''] presents itself as an "aimed arrow", "fair sample", or "independence" condition.
"men are just as apt to be black as things in general".
The example apparently assumes a universe of "things in general", encompassing among other things the denotations of the absolute terms ''m'' = "man" and ''b'' = "black". That suggests to me that we might well illustrate this case in relief, by returning to our earlier staging of 'Othello' and seeing how well that universe of dramatic discourse observes the premiss that "men are just as apt to be black as things in general".
Here are the relevant data:
The "fair sampling" or "episkeptral arrow" condition is tantamount to this:
:{| cellpadding="4"
"Men are just as apt to be black as things in general are apt to be black".
| 1
In other words, men are a fair sample of things in general with respect to
| =
the factor of being black.
| B +, C +, D +, E +, I +, J +, O
|-
| ''b''
| =
| O
|-
| ''m''
| =
| C +, I +, J +, O
|-
| 1,
| =
| B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
| ''b'',
| =
| O:O
|-
| ''m'',
| =
| C:C +, I:I +, J:J +, O:O
|}
The "fair sampling" or "episkeptral arrow" condition is tantamount to this: "Men are just as apt to be black as things in general are apt to be black". In other words, men are a fair sample of things in general with respect to the factor of being black.
Should this hold, the consequence would be:
Should this hold, the consequence would be:
[m,b] = [m,][b].
: [''m'',''b''] = [''m'',][''b''].
When [b] is not zero, we obtain the result:
When [''b''] is not zero, we obtain the result:
[m,] = [m,b]/[b].
: [''m'',] = [''m'',''b'']/[''b''].
Once again, the absolute term b = "black" is most felicitously depicted
Once again, the absolute term ''b'' = "black" is most felicitously depicted by way of its idempotent representation ‘''b''’ = ''b'', = "black that is ---", and thus it can be taken as a selective from the universe of discourse.
by way of its idempotent representation 'b' = b, = "black that is ---",
and thus it can be taken as a selective from the universe of discourse.
Here is the bigraph for the composition:
Here is the bigraph for the composition:
m,b = "man that is black",
: ''m'',''b'' = "man that is black",
here represented in the equivalent form:
here represented in the equivalent form:
m,b, = "man that is black that is ---".
: ''m'',''b'', = "man that is black that is ---".
<pre>
B C D E I J O
B C D E I J O
o o o o o o o 1
o o o o o o o 1
o o o o o o o 1
o o o o o o o 1
B C D E I J O
B C D E I J O
</pre>
Thus we observe one of the more factitious facts
Thus we observe one of the more factitious facts that hold in this universe of discourse, namely:
that hold in this universe of discourse, namely:
m,b = b.
: ''m'',''b'' = ''b''.
Another way of saying that is:
Another way of saying that is:
b -< m.
: ''b'' –< ''m''.
That in itself is enough to puncture any notion
That in itself is enough to puncture any notion that ''b'' and ''m'' are statistically independent, but let us continue to develop the plot a bit more.
that b and m are statistically independent, but
let us continue to develop the plot a bit more.
Putting all of the general formulas and particular facts together,
Putting all of the general formulas and particular facts together, we arrive at following summation of situation in the Othello case:
we arrive at following summation of situation in the Othello case:
If the fair sampling condition holds:
If the fair sampling condition holds:
[m,] = [m,b]/[b] = [b]/[b] = `1`,
: [''m'',] = [''m'',''b'']/[''b''] = [''b'']/[''b''] = `1`,
In fact, however, it is the case that:
In fact, however, it is the case that:
[m,] = [m,1]/[1] = [m]/[1] = 4/7.
: [''m'',] = [''m'',1]/[1] = [''m'']/[1] = 4/7.
In sum, it is not the case in the Othello example that
In sum, it is not the case in the Othello example that "men are just as apt to be black as things in general".
"men are just as apt to be black as things in general".
Expressed in terms of probabilities: P(m) = 4/7 and P(b) = 1/7.
Expressed in terms of probabilities: P(''m'') = 4/7 and P(''b'') = 1/7.
If these were independent we'd have: P(mb) = 4/49.
If these were independent we'd have: P(''mb'') = 4/49.
On the contrary, P(mb) = P(b) = 1/7.
On the contrary, P(''mb'') = P(''b'') = 1/7.
Another way to see it is as follows: P(b|m) = 1/4 while P(b) = 1/7.
Another way to see it is as follows: P(''b''|''m'') = 1/4 while P(''b'') = 1/7.
===Commentary Note 11.23===
===Commentary Note 11.23===
