Different language groups – the rules are usually constant for a whole language group – simply used different conventions how to deal with the negatives.
The Germanic languages are closer to the mathematical logic in the sense that every word like "nicht" flips the meaning of the sentence. It means that one has to count them modulo two.
It doesn't mean that the other language groups are illogical. They simply use different rules for the co-existence of the negation words. As an example of Slavic languages, Czech has very well-defined detailed rules of the negation (and other things). There are three types of negation:
- sentence-wide negation
- negation of a member
- negation of a word
In this list, they are increasingly more "local" negations and each of them has its own rules. The negation of a word is simply a new word with "ne-" ("non-") added: "nechuť" is the "lack of chuť" (appetite), nejeden is "not [just] one" etc.
The negation of a member – "ne" followed by a space – negates the following word(s) as a member of the sentence. "Tell it not to me but to him."
For this question, the most relevant kind of negation is the sentence-wide negation. The rules of Czech, Slavic, and other languages require doing several things at the same moment in order for the negation to be sentence-wide:
- negate the verb, by adding "ne-"
- use negative adverbs if any: "nikdy" ("never") and "nikde" ("nowhere")
- use a negative pronoun if a pronoun is a subject: "nikdo" ("no one")
- add the word "ani" in front of nouns that are subjects that don't exist thanks to the negation: "Ani hlásek se neozval" ("Not even one voice could have been heard.")
It's not only possible but absolutely required that the sentence-wide negation respects all of these pieces. They don't negate each other. Instead, they co-operate to create one sentence-wide negation. Carsten describes Russian that presents some parts of the sentence-wide negation as "intensifiers" but whether one of them is considered a cherry on a pie and less important than others or not, it's the case in most such languages that all of the parts of the negation are needed for a correct sentence-wide negation.
Sentences with four negatives – which are actually just parts of one sentence-wide negative – are common. For example, "Nikdy jsem nikde nikoho nezabil" in Czech means "I have never killed anyone anywhere" but it is literally "Never have I nowhere no one not-killed."