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In the sentence

there may be no legitimate form of reasoning to the best explanation understood as an alternative to inductive reasoning.

does may be no mean

  • that it is possible that there is no legitimate form of reasoning
  • it is impossible that there is any?
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4 Answers

It means that the scientific community does not accept other explanation than that.

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I read it more in the sense of 'not permissible under normal circumstances', as in 'you may not leave without permission'.

This may surprise some. However, the context is such that you are not supposed to expect/ allow for a possibility, except maybe, in an extraordinary situation.

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'You may not divide by zero'. The next sentence can be: 'The user may, in his ignorance'. (possibility), or, 'So, check denomintor first'. (permissibility). –  Kris Apr 21 '12 at 7:31
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It depends what follows. If the next word is but there will be at least a strong implication that there is no legitimate form of reasoning.

Compare ‘There may be no water in the middle of the desert, but that doesn’t mean that people can’t survive there for short periods.’ That doesn’t really leave open the possibility that there is water in the middle of the desert.

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It means that is it possible. You can use the word "might" here too.

The author uses "may" here because they do not want to make a definitive statement. Wikipedia would call this a "weasel word" because the author is sucking the meaning out of the statement like a weasel sucks an egg. Eventually you have something that looks like a sentence but means nothing.

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Well, yes, the author is hedging his bets when he uses a word like "may", but that hardly means that it renders the sentence meaningless. It is quite reasonable to say "I think that this is probably true, but I cannot say so with 100% certainly, and I want to make that lack of certainty clear." –  Jay Apr 20 '12 at 17:28
    
Very true. By "eventually" I meant "if you use enough weasel words." –  meetar Apr 20 '12 at 18:53
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