will not; intend not to
Expresses the speaker's firm intention not to do something or a prediction that something will not happen.
This pattern is fairly formal and literary. In everyday conversation, you would usually say ないつもりだ or just use the negative form (e.g., 行かない). But まい appears in written Japanese, speeches, and set phrases. It has two main uses: expressing the speaker's strong intention not to do something, or expressing a negative conjecture about something (probably won't). For example, そんなことは言うまい means 'I will not say such a thing.' When used for conjecture, it often combines with だろう or に違いない: 彼は来るまい (He probably won't come). Note that まい does not change for politeness; it is always plain.