What is the "strong form" of a theorem, or law, or statement?

Answer 1

This varies a lot depending on which theorem, law or statement you are referring to. The 'strong form' would be the version that is either more useful, has fewer hypotheses, or is able to conclude more. For example, the fundamental theorem of algebra says that every non-constant complex polynomial has at least one root. Well and good. But it is true (and not too much harder to prove) that every non-constant complex polynomial has as many roots as the degree of the polynomial if the roots are counted according to multiplicity. This second not only says that roots exist, it actually tells how many there are. This would make it a strong form for the fundamental theorem of algebra and the simple existence version would be the weak form.

Answer 2

I believe that the strong form of a theorem is the if and only if version. For example, consider the following: for every right triangle a^2 +b^2=c^2. This would be a weak statement of the Pythagorean theorem. The strong version would be: For every triangle, a^2+b^2=c^2 if and only if the triangle is right.