I should probably know this, but you're saying the exception:
EXCEPTION: If the incorrect batter is called out as a result of their time at bat and is scheduled to be the proper batter, skip that player and the next person in the line-up will be the batter.

only applies with in the inning. Don't have my rule book here, can somebody post that part of the rule for me? I would definitely have kicked this by bring B4 up to start the next inning.
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California Dispensary

B3 does not bat next in ASA because B3 made an out which stood in the previous inning. Since that out (back end of the double play) stands, B3 loses that turn. You cannot have a scoresheet that both marks an out to that batting spot and then allows it to bat again.

B3 does bat in NFHS and NCAA, because that out is negated in both rule sets.

Here's a cut-and-paste from the ASA clarifications:

Play 2: With one out, B7 is scheduled to bat, however B8 bats. B8 hits a fly ball that is caught for the second out of the inning. The defense appeals that B8 batted out of order and the umpire calls B7 out for the third out of the inning. Who is the leadoff batter in the next inning?

Ruling: B8. In this case, since the second out of the inning was made by B8 and the third out was made by B7 for failure to bat in the proper order, B8 is now the leadoff batter in the next inning. When the batter declared out is the third out of the inning the correct batter to leadoff the next inning is the player who would have come to bat had the player been put out by ordinary play. (Rule 7, Section 2 D 2 d)

Apparently the scorekeeper has to use an asterisk or draw an arrow or something.

There are no rules I'm aware of that depend on how something looks in the scorebook... and in MANY BOO situations, you have exactly that - a scoresheet that both marks an out to a batting spot and then allows it to bat again.

AUS (another one I'm loath to disagree with!), what RULE tells you that B3 doesn't bat next in ASA in the inning AFTER BOO happened? And what does the clarification mean if that's true?

I admit I had forgotten about that "clarification"; it makes no sense to me. Frankly, I think it is wrong, but it comes from the official source.

Exact same play as the "clarification" with no outs, and B9 bats with two outs. So, how can it be correct, be logical, or even make sense that B8 leads off the next inning in the same play with one out?

Oh well, I must be wrong.

It apparently means that another clarification is required! ;)

it makes no sense to me

me, either

Well, just add it to the pile of rules that make no sense, like the one that allows the offense to benefit by committing deliberate interference.

'bout time you got back in the game.:cool:

This is one of those things that is not supposed to happen in a game. It used to be so simple and then they decided to keep the outs. That was a good thing and I had the rule applied in my first game of that year. Two batters, three outs. Gotta love it.

But then things like this came about. Personally, not having the batter which already made an out bat again makes complete sense. The player had the chance and failed to reach base safely, same as the batter who had their opportunity to bat and did reach base safely.

So I'm sort of curious as to why anyone would think a player should get a second opportunity to achieve success, or failure just because someone on their team (maybe even themself) screwed up.

Honestly, none of this was about what SHOULD be.. just what is.

Personally, I think that if someone bats out of order, their at bat should go away... completely... including their out or peripheral outs - we should add an out to the offense for the screw up, and the MISSING batter --- the original PROPER batter, should be up to bat. Especially for youth ball. MOST of the time the screwup is not the kid's fault, yet that's the one kid that misses out on batting.

Once you get to HS age, I think Fed has it right, as players that age should be able to follow a lineup regardless of coaching screw ups, and players that age can live with missing an at bat.

What is the excuse when adults BOO? Or NCAA? Or high-level JO ball?
Who cares why it is BOO? Here's a line-up. Find your name and bat after the player whose name is above yours. Failure to read at a 3rd grade level is not an acceptable excuse for screwing up the batting order. Sorry, no sympathy from me, It is their team and their game and their problem.

And what about the other team? People tend to forget there are two teams out there. Why should the defense be denied of the outs which THEY earned by executing a play properly? It isn't their fault the wrong batter hit into an out.