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I have often seen expressions like "this statement is stronger (or weaker) than that". I know what it means. If $A\Rightarrow B$ but $B\nRightarrow A$, then $A$ is stronger than $B$.

But from time to time I read sentences like, say, "Picard's theorem is stronger than Liouville's theorem". But both theorems are true, so they are logically equivalent. Yeah, Picard's theorem is much harder to prove and it implies Liouville's easily. But I find one just as strong as the other.

My question: is there a serious definition of the terms 'strong' and 'weak' in maths, or are they just a way to speak?

ajotatxe
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2 Answers2

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Picard's theorem and Liouville's theorem are equivalent in the sense that they have the same truth value, which is all that we require for two statements to be equivalent.

There is a subtle difference in what we mean when we say "the theorem A is stronger than the theorem B," and "the statement A is stronger than B."

You might object that these are the same, but this is a matter of terminology, and the word "strong" is used differently depending on the context.

"the theorem A is stronger than the theorem B" means "A has a better/more powerful conclusion and/or less stringent hypotheses."

"the statement A is stronger than B" means what you've said in your post: A implies B, but B does not imply A.

Noah Caplinger
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In this context, it means that the second statement follows easily from the first one, but the first one is not a consequence of the second one.

José Carlos Santos
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  • But it is a consequence. Not easy, not obvious. But if $A$ and $B$ are true, then $A\to B$ and $B\to A$. – ajotatxe Nov 07 '19 at 15:30
  • You are thinking in terms of Logic. That is not appropriate in this context. I was explaining what “stronger than” means in the context that you mentioned it. It has not the same meaning as in Logic. – José Carlos Santos Nov 07 '19 at 15:33
  • I'm thinking a way to put this concept in logic. For two any true statements $p,q$ let $\lambda(p,q)$ the minimal number of steps to derive $q$ from $p$. Then we can say that $p$ is stronger than $q$ if $\lambda(p,q)<\lambda(q,p)$. – ajotatxe Nov 07 '19 at 15:36