A person can be commanded to do something (“Eat matzah on Pesach!”) or not to do something (“Don’t eat shellfish!”). A mitzvah is either positive or negative, but a single statement can’t be both. A negation, however, is not a separate mitzvah. (We should mention that Rambam uses the Hebrew word sh’lilah.)
What’s a negation? It’s simply the reversal of what a statement would otherwise say. “I did not sleep last night” is the negation of “I slept last night.” It’s not a command at all. In Hebrew, a negation would commonly (but not exclusively) occur when the word “lo” (“no” or “not”) is added. “Lo achalti” (“I did not eat) is the negation of “achalti” (“I ate.”)
Compilers of the mitzvos before the Rambam occasionally erred by including certain negations as negative mitzvos. For example, Exodus 21:7 says that a female slave is not released the same way that a male slave is released. This is not a prohibition, it’s simply a statement of “You know that mitzvah we just said? It doesn’t apply in the following circumstances.”
The Rambam gives many such examples. Another is that a married woman who is raped is not executed for adultery. That’s not a negative mitzvah, it’s just the Torah telling us that the case of coercion does not fall under the definition of adultery.
The way to tell the difference between a negation and an actual prohibitive command, the Rambam tells us, is by the context of the statement in question. (The phraseology tends to be identical, so that’s not much help.) He gives an example from Talmud Zevachim (65b-66a). The Torah says that a kohein slits the throat of sacrificial birds but he does not sever the head altogether. The question is whether the words “lo yavdil” (Leviticus 5:8) mean “he need not sever” or “he may not sever.” They’re very different! In one case a bird with a severed head is acceptable as a sacrifice and in the other, it isn’t! The topic is discussed and the conclusion is reached based upon the context of this statement vis-a-vis the bigger picture of the sacrifices. (The details are far beyond the scope of this summary.)
Summing up, the Rambam provides four words used to identify prohibitions: “hishamer” (guard), “pen” (lest), “al” and “lo” (both meaning “don’t”). The Talmud in Zevachim 106a says that anything that says hishamer, pen or al is a prohibition. (“Lo,” as we’ve said, is a broader stroke.) But even seeming negations may actually be prohibitions! For example, from the fact that a person says, “I did not eat it (referring to maser sheini, the second tithe) in a state of mourning or ritual impurity” (Deut. 26:14), we can infer that it is prohibited to do so.