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If the Principal of a school says that unruly boys will not be allowed to play games and if someone concludes that it is perfectly fine for unruly girls to play games, what is the fallacy called as?

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    Not a fallacy. There is no ambiguity involved. The principal claims that for all x, if x is unruly and x is a boy, x won't be allowed to play games. That's logically independent of (or consistent with) the claim that: some x is such that x is unruly and x is a girl and x is allowed to play games. – Hunan Rostomyan Apr 3 '14 at 10:00
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    "Bad communication." – Andrew Cheong Apr 3 '14 at 11:35
  • @HunanRostomyan, consistency is not enough here. The person hearing this is supposing that the first implies the second when it doesn't. – Addem Apr 3 '14 at 17:04
  • @Addem I read 'it's perfectly fine' as 'it's logically consistent with'. Then, the principal claims [that unruly boys can't play], and the person concludes that [that unruly girls can play] is perfectly fine (i.e. logically consistent) with that. While [that unruly girls can play] does not follow from [that unruly boys can't play], the claim that: { [that unruly girls can play] is consistent with [that unruly boys can't play] } does follow. Had the person concluded [that unruly girls can play], there would be a fallacy. But the claim is only about the two being mutually satisfiable. – Hunan Rostomyan Apr 3 '14 at 18:00
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This would be described by a "correct" usage of the exception that proves the rule. By specifying boys when perhaps you don't need to (you could say "unruly children"), the contrary of your statement applies for those children who are not boys.

Quite often "the exception that proves the rule" is taken to mean "X therefore not X in general", which is logically incorrect, but not what is meant here.

Generally the utterance "boys can't play" does in fact imply "girls can". Because if both could play, or none could play, you either wouldn't say anything or would say "children". There is more information in what you say.

It's less clear in cases where there is an adjective, as it is difficult to determine the assumptions that are implicitly made. If you say "boys can't play" it's clear this is applied with a group of children in mind. But if you say "unruly boys can't play" its unclear whether the group you have in mind: all the children, the unruly ones, the girls, or less likely the not-unruly girls and the unruly boys.

So the "unruly" case is using the exception that proves the rule where perhaps it shouldn't. Is it a fallacy with a name, who knows. The exception-that-proves-the-rule-when-it-doesn't-fallacy?

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    Note: the 'imply' in the third paragraph is meant informally. Since 'implies' already has a formal meaning, we use 'implicates' to mean, roughly, what the informal 'implies' means. We would, for example, say that (i) boys can't play does not imply that girls can, but (ii) boys can't play (perhaps with an added emphasis on 'boys') might implicate that girls can't. – Hunan Rostomyan Apr 3 '14 at 18:52
  • I don't think it's really a proper usage, since there is nothing in the prohibition on unruly boys' playing games which states that they are the only prohibited category. One might infer that, from the standpoint of the person making the statement, it is at least possible that there might exist an unruly girl who is allowed to play, but it would be entirely possible that every single unruly girl had some other disqualifying factor which made her ineligible to play, and that no unruly girls who are not prohibited from playing exist. – supercat May 16 '14 at 6:11
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I'm not sure it has an official name, although it's close to denying the antecedent. The essence of the exchange is that one person says "if (unruly boy) then (no play)" and is taken to imply "if (unruly not boy) then (play)".

Here the denial is not actually of the antecedent. The more exact structure would be something like $(A\land B) \rightarrow C$ therefore $(A\land\negB)\rightarrow \neg C$.

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    denial of the non-antecedent? We're manufacturing fallacies to cover other problems. – virmaior Apr 4 '14 at 2:26

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