Natural deduction is used to try to prove that some reasoning is correct (``to check the validity of a sequent'', says theory). Example:
I tell you: ``In summer it's warm, and now we're in summer, so now it's warm''. You start doing calculations, and finally reply: ``OK, I can prove that the reasoning you just made is correct''. That is the use of natural deduction.
But it's not always so easy: ``if you fail a subject, you must repeat it. And if you don't study it, you fail it. Now suppose that you aren't repeating it. Then, or you study it, or you are failing it, or both of them''. This reasoning is valid and can be proven with natural deduction.
Remark that you don't have to believe nor understand what you are told. For example, I say that: ``Thyristors are tiny and funny; a pea is not tiny, so it isn't a thyristor''. Even if you don't know what am I talking about, or think that it is stupid (which it really is), you must be completely sure that the reasoning was correct.
So, given a supposition ``if all this happens, then all that also happens'', natural deduction allows us to say ``yes, that's right''. In logical language: if you are given a sequent , you can conclude at the end that it is (valid). Then we write ( has as consequence ).
Daniel Clemente Laboreo 2005-05-17