Unlike (standard*) English but like the other Romance languages, Portuguese uses what’s called negative concord (concordância negativa in Portuguese). This means that multiple negative elements occurring in the same clause in Portuguese do not cancel one another but rather reinforce each other. This is not at all the same thing as a double negative (dupla negação in Portuguese).
*Some dialects of English do use negative concord, but they are not the standard form anywhere and are generally looked down on, no matter how common they may be.
Because English lacks negative concord, it instead uses negative polarity items such as any, anyone, anything, ever where Portuguese uses actual negative elements. Once you remove the negative from the clause in English, you are left with something that’s ungrammatical if you don’t adjust the negative polarity items as well.
For example, I did not see anything is grammatical but removing the not leaves you with something that is either no longer grammatical in English or which means something else altogether; in any event, it is not the opposite of the original.
Portuguese doesn't work that way. As Carla Fernanda Ferreira Guedes wrote in her masters thesis of 2001 entitled “Sujeitos Negativos e Concordância Negativa em Português numa perspectiva de Sintaxe Comparada”:
3.1. Dupla Negação vs Concordância Negativa – algumas reflexões
Antes de entrarmos propriamente no estudo da Concordância Negativa, justifica-se, uma vez mais, fazer uma aproximação à Dupla Negação.
Como foi já abordado muito brevemente nos capítulos anteriores, existem línguas em que, da co-ocorrência de dois itens negativos no mesmo domínio sintáctico resulta uma interpretação positiva e, por isso, o cancelamento da negação. Este fenómeno está relacionado com o facto de, nestas línguas, a presença de um marcador negativo ou de uma palavra negativa não permitir a co-ocorrência de outros elementos da mesma natureza, dentro do seu domínio sintáctico, requerendo antes a presença de palavras não inerentemente marcadas com o traço [+ neg].
In other words, if we have one negative in English, we cannot use more negatives in that clause for concordance, but must use these other types of words, the negative polarity items. Professor Lawler, our resident emeritus professor of linguistics on English Language & Usage likes to cite Horn’s rule.
Horn’s rule is Simplex Negatio Negat; Duplex Negatio Affirmat; Triplex Negatio Confundit. Single negative negates; double negative affirms; triple negative confuses.
If you don’t immediately think of multiple negatives causing confusion in English, just think of Bilbo’s famous statement that
I don’t know half of you half as well as I should like; and I like less than half of you half as well as you deserve.
That makes Portuguese examples like não vi nada or sem nada mais seem pretty easy, doesn’t it now? :)
Horn and Lawler have both done an impressive lifetime of work on negation; if you’re interested in the subject, I recommend checking out their publications. In fact, Professor Horn is still publishing, and this year (2016) he has published Licensing NPIs: Some Negative (and Positive) Results, which from the small excerpts I’ve read appears to be quite interesting. Here’s an example:
Despite the considerable progress that has been achieved over the last two decades, the bad news is that we know squat about the proper treatment of negation and polarity. But then, by the Law of Excluded Middle, the good news must be that we don’t know squat about the proper treatment of negation and polarity.
If you’re a native speaker of English, you’re sure to recognize that squat is a negative polarity item, so not knowing squat is rather like não sabe [uma] merda in Portuguese but milder because of the minced oath of using squat for the more explicit shit.
As Horn observes, there’s still a lot to learn about these matters.