American law specifying that a man can beat his wife with a stick no thicker than his thumb?

answer: 

Overall, there doesn't seem to be one concrete answer on this that everyone agrees on, but the following references might shed some light on it for you...

Freyd, J. & Johnson J.Q. (1998). Commentary: Domestic Violence, Folk Etymologies, & "Rule of Thumb". Retrieved May 8, 2008, from http://dynamic.uoregon.edu/~jjf/essays/ruleofthumb.html

  • This essay discusses the differences in opinion on the legal history of the term "rule of thumb".

According to:
Service, N. N. (1994, July 26). Word experts overrule; feminists on 'rule of thumb'. Plain Dealer (Cleveland, Ohio), pp. 9E.

  • "But the phrase 'rule of thumb,' as it turns out, has more to do with carpentry and beer-making than domestic violence.
    Those who attributed the phrase to an old rule of law that supposedly condoned wife-beating are having second thoughts. The latest capitulation came a few days ago, when Rep. Charles Schumer, D-N.Y., who often invoked the phrase when talking about domestic violence, started to use it again during a news conference outside the Capitol, then stopped midsentence. He explained that Sen. Barbara Boxer, D-Calif., had told him the background of the phrase was 'apocryphal.' Which is a politician's way of saying something is not true."

From:
Feminists' facts challenge questionable, amount to a backlash.(1994, June 20). The Oregonian (Portland, Oregon), pp. B06.

  • "The 1874 case State vs. Oliver states: ``We assume that the old doctrine that a husband had the right to whip his wife provided that he used a switch no larger than his thumb is not the law in North Carolina.'' This common-law rule was derived from the 177th New Constitution of Roman Emperor Justinian I. See Chapter 14. Thus The Oregonian, Seebach and Sommers are off by 1,447 years."

Rhodes, E. (1991, May 5). Brutality at home has long history. The Seattle Times, pp. A11.

  • This source describes the history of wife-beating a bit with a very brief mention of what you're looking for.
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