If the batter makes contact with a batted ball while in the batter's box, it is a "dead" ball and declared foul.
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True enough, sort of. If a batted ball touches the batter while he is...
I don't think so. Rule 6.05 (f) states the batter is out if he attempts to hit a 3rd strike then the ball touches him.
According to MLB Rule 6.08(b), the batter is entitled to first base when:
"He...
If the batter is in fair territory, the batter is out. If the batter is not in fair territory, for example the batter is still in the batter's box, the ball is called foul.