TimG said:
> Not so clear in many four-card major systems.
I think take-out doubles apply to any system, except perhaps goulash systems. But maybe that's just me.
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> But, support doubles are not.
That is indeed correct. Support doubles are not vital, although they help in some circumstances, like 5-card majors strong NT systems (sayc, 2/1, french std).
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> Or, how about this classic responsive double auction:
> (1♥)-DBL-(2♥)-DBL? Do you really need this takeout double?
> (1♥)-DBL-(2♥)-DBL? Do you really need this takeout double?
YES you need it a lot. It is for the minors. One could NOT use 2NT as a replacement because 2NT is needed for a bunch of things, like for instance to distinguish between a competitive 3m bid from a stronger one (e.g. Lebenshol). Even if you skip Lebenshol, you might want to use 2NT as natural instead of for the minors.
There is of course some arbitrarity in dbl and 2NT (e.g. 2NT could be for the minors and double Lebenshol) but you cannot do away with double without lessening the effectiveness of your competitive bidding.

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