There are two different Greek words that are used as negatives. Both are translated as "no" or "not". The both negate both single words and sentences. However, these words have very different meanings and translating them simply as "no" or "not", as is almost always the case in the NT, is misleading. To correctly capture the sense of these words, extra words are needed in English to distinguish between them.
The Two Greek Words
The most common Greek negative is ou (οὐ, οὐκ). This is the negative for facts. Most English "no's" fit this definition, but using a simple English "no" to translate this word is not as precise as the Greek.
The other Greek negative is me (μὴ). This is the negative expressing will or thought. It is used in prohibitions and expressions of doubt meaning "not" and "no." This negative expresses an idea that always requires more words in English. Otherwise, the result is misleading.
- οὐ denies, μή rejects;
- οὐ is absolute, μή relative;
- οὐ objective, μή subjective.
These two words are also often used together in the form, ou me, (οὐ μὴ). Unlike the double negative in English, in Greek, the effect is to intensify the meaning. It is sometimes translated as "never" in the NT, but it is also usually translated as a "no" or "not". The sense is "no, you don't even want to think that".
These two words can also appear as part of compound words, for example, the form used in a series to mean "neither...nor" is oude (οὐδὲ) and mede (μηδὲ). These negatives are used with the particle (de) which is a connective. The choice of negative depends if the series is expressing an objective or subjective negative idea.
Translating the Objective Negative
The more common negative, οὐ, is perhaps best translated as "no really" or "in fact, no" rather than simply "no" alone. While "no" usually captures the idea, "no" is a somewhat weaker word in English, since it can also cover opinions and desires. This negative speaks to just facts. This expresses the fact that something cannot happen or cannot exist or didn't happen or didn't exist.
Translating the Subjective Negative
The negative of desire or opinion, me, always needs additional words to clarify its meaning in English. It does not express the idea that something doesn't happen or can't exist. It expresses the idea that someone doesn't want it to happen or think it will happen. To reflect the use of this negative, it should be translated as "you don't want to" or "you don't think".
The Contrasting Negatives
A good verse to see the contrast in meaning between these two negative is Mat 5:17: "Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfill. (KJV)" This verse uses both forms of the negative in a way that is easy to see why the difference is important and why adding extra words add clarity.
When Christ says, "Think not that I am come to destroy the law", he is obviously talking about thinking so the negative of thought is used, me, Μὴ νομίσητε. However, when he says "I am not come", οὐκ ἦλθον, he naturally switches to the objective version, ou, he is stating a fact.
This is not, however, as simple as it might seem. The objective negative can be used to state a fact about thinking. If someone doesn't think something, in Greek, you can say you really didn't think it. For example, if your spouse accuses you of meaning something you didn't mean, you can use the objective negative to say "I really didn't think that". Using the subjective form is much more dangerous since your meaning would be, "I don't think I thought that" or "I didn't want to think that". Ouch! Which one would you use?
So, in translating Mat 5:17, let's clarify those Greek negatives to add the sense that Christ meant and his listeners would have heard. We are also going to fix a few other things, like poorly translated tenses and moods in the process, while keeping the basic translation the same. The added words to complete the negatives are in italics.
"You might not want to think that I've come destroy the laws or the prophets. I really did not come to destroy but to fulfill."
So, even in a verse where the meaning of the negatives is somewhat clear by the words used, it can be made clearer by adding an extra word.
Commands and Verbs of Possibility
One area where Christ violates the "rules" regarding the use of these two negatives is with commands (imperatives) and verbs of possibility (subjunctives). According to the rules, these two verb form should always use the subjective form. Indeed, the subjective form is often called the subjunctive form.
However, Christ completely ignores this rule, using whichever form makes his meaning clear. When the subjective negative is used with the word "to murder" for example, the sense is "do not want to murder" or "do not think to murder". When the objective form is used, the sense is "never murder" or "you cannot murder". This gets a little more complicated because the "command" form is also used with "requests".
This is made worse by Biblical translation, which often translates the form of possibility as the command form. When you read what looks like a command spoken by Christ, it often isn't a command at all. And many of them don't use the subjective form but the objective form. For example, Mat 5:27 ..."You shall not commit adultery", Mat 5:33 ..."You shall not swear falsely", are both verbs of possibility with the objective negative. While Mat 6:8 "Do not be like them:", is a subjective form used with a verb of possibility. In the Lord's Prayer, the line Mat 6:13 "And lead us not into temptation" is a verb of possibility, not command with a subjective negative, even though all of the other requests in the Lord's prayer are in the command form, which is appropriate for requests as well as commands.
Unfortunately, straightening out all the mistranslating of commands or what look like commands in Matthew, would take an almost book-length article so it is just best to check the parsing and explanation of the verse in the articles on this site.