dumbrope wrote:$$W reasonable attack and defense
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . a 2 . . .
$$ | . . . c b . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
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- Click Here To Show Diagram Code
[go]$$W reasonable attack and defense
$$ ------------------
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$$ | . . X . a 2 . . .
$$ | . . . c b . . . .
$$ | . . 1 . . . . . .
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$$ | . . . . . . . . .[/go]
Why not :w2: at 'a'
(You mean "Black 2" rather than White 2".)
Positional judgement according to my method relies on principles for the nature of reduction sequences. The purpose of setting principles is to ensure the same kind of moves for evaluation of all regions. I think that such consistency for all regions is good, because regions are treated likewise regardless of their shapes.
The principle used for the decision between Black 2 or Black A is [4]: "Subject to the other conditions, the defender's territory is minimised by the attacker and maximised by the defender." Here, the defender moves, so he maximises his territory, while abiding by the other principles' conditions. You might ask why "maximised". It is derived from ordinary strategy due to the game aim: a player wants to maximise his own territory, while the opponent wants to minimise it. Black 2 maximises, while Black A would not maximise.
If Black 2 is one space farther to the right, then Black does not maximise his territory, but offers White a chance to invade at A.
A few of the other conditions I have already explained in earlier messages. Direction change to maintain life plays a role here.
or 'b', or even 'c'
Black B (or C) does not maximise Black's territory, because then White can better reduce as follows:
$$W Black wants too much, Black 10 at x
$$ ------------------
$$ | . . . 8 7 9 . . .
$$ | . . . . 6 5 . . .
$$ | . . X x 4 3 . . .
$$ | . . . . X . . . .
$$ | . . O . . . . . .
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- Click Here To Show Diagram Code
[go]$$W Black wants too much, Black 10 at x
$$ ------------------
$$ | . . . 8 7 9 . . .
$$ | . . . . 6 5 . . .
$$ | . . X x 4 3 . . .
$$ | . . . . X . . . .
$$ | . . O . . . . . .
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$$ | . . . . . . . . .[/go]
Black maximises better if instead he plays at 2.
if one is not concerned about strengthening white for this purpose?
Such considerations are considered immaterial in territorial positional judgement. Imagine supporting white stones according to these my principles [4]:
"The life of string-connected or directly connected reduction plays does not need further justification.
If the attacker has only remote support by friendly stones, his early reduction plays are accordingly reasonable.
If the attacker has only remote support by friendly stones, his later reduction plays rely on imagining supporting stones that can reasonably be assumed to have been played earlier. Later, more such supporting stones can be imagined."
:w2: isn't even the most common pro joseki response in this situation (that would 'a')
(You mean "Black 2" rather than White 2".)
Territorial positional judgement does not care for empirical frequency. Quite like life and death status analysis does not care for whether plays elsewhere on the board might be more urgent than locally defending the life of a small group. You also would not question life and death status analysis just because it does not answer the question for globally perfect play, wouldn't you? Analysis sequences serve the purpose of enabling answering analysis questions. They do not (necessarily) suggest perfect play on the whole board. This difference in objectives does not make analysis invalid. We use analyses all the time, because we believe that answering analysis questions is helpful for better developing strategy. Answering analysis questions for territorial positional judgement is a special form of helpful analysis.
To me, it seems as if the choice of :w2: is every bit as arbitrary as Lee Changho's lines to the side.
Lee, in his book [1], does not offer any justification for his line drawing. I, for my method [4], use principles, such as those mentioned, and explain why those principles make sense for the purpose of making territorial PJ. Principles and explanation are much less arbitrary than no principles and no explanation.
permitted to dance off in the other direction to get more eyespace? If that is allowed, why is it allowed only when the eyespace is too small and not when it is larger?
The purpose of territorial PJ is assessment of existing territories. Always allowing direction changes would... (I have already explained in earlier messages.)
In my previous post I already showed that passive answers to attacks to the two-space extension result in a dead shape for white.
This is indeed a question one can come up with. The two-space extension or similar shapes require an exception to one's thinking. The usual go theory thinking about the 2-space extension applies also in PJ: it is considered alive because of having 1 eye and several running directions. The imagined reduction plays do not change the life potential: running directions are replaced by break-through directions.
By the argument you give, white will now dance off in a different direction,
No, because White need not, because the white group is still alive. Dancing off is an exception only for the last moment of defending threatened life.
at 'a' or 'b', or even 'c' if one is not concerned about strengthening white for this purpose? 
unreasonable in the 1st diagram but 
is reasonable in the 2nd one? They are the same armpit hit shape.
because we don't know who has sente. After this, the peaceful responses to attacks result in 10 points. Just average with 4 point result if white gets to reduce first. That's 7 points. (Plus lower order terms as maybe after 
)