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I'm reading a paper right now, and one exctract goes like this:

"Theorem 1: If conditions "...;$|A|<B+1$;..." hold, then "result".

By making additional assumptions, we can weaken the conditions of the theorem and demand "...;|A|<B;..."."

I understand that if we demand less for the same result, then the theorem becomes stronger. But in this case, it appears to me that the conditions have been strengthened and not weakened, since A now lies in a narrower range. Am I confused?

Nick The Dick
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    You are right, if that's what the paper says. You weaken the theorem by strengthening the conditions (if the conclusion is the same). – Robert Israel Nov 29 '21 at 21:06
  • @RobertIsrael i'll try to put it another way, if that can help. The theorem says, that "an economical system can reach an equillibrium (desired state) through a dealing process with the sizes of coalitions restricted by $|A|<|B|+1(0)$". Could this context possibly invert the logic I'm following? It sort of makes sense, because it becomes "easier" when we have to deal with lesser-sized coalitions(by reducing the total number of them to consider), but I'm just not following this logically. I know that it's a specialised context, and I'm sorry if there's not enough info. – Nick The Dick Nov 29 '21 at 21:16
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    From you comment, it sounds analogous to the statement that "any point on the unit disc in $\Bbb{R}^2$ can be joined to the origin by a path of length at most $X$" Here "at most $X$" isn't a condition, i.e., assumption, of the theorem, but part of the conclusion. The statement gets stronger if you make $X$ smaller. If this analogy is accurate, then the terminology being used is very odd as the "condition" is delivered by the theorem not demanded by it. – Rob Arthan Nov 29 '21 at 21:44
  • @RobArthan thanks, it sounds like you're spot on, at least it seems like it to me at the moment. i'll let the author know about that – Nick The Dick Nov 29 '21 at 21:56

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