These verbs all mean to state positively, as though anticipating or countering argument or skepticism. Whereas assert means to state with some force or conviction, allege means to state without offering proof. It was alleged that he was present at the scene of the crime, but he asserted that he was in Europe at the time. Whatever one asserts he would defend in argument, but whether anyone believes something alleged is beside the point; the aim of alleging is to learn the truth by proving or disproving the claim made.

Affirm means to declare or state positively that something is true; it indicates firm belief or unshakable conviction: to affirm one’s faith in God. Asseverate is nearly synonymous with assert but even more positive; the word is uncommon even in formal writing. Testify means to bear witness, as in a court of law, or declare solemnly to be true: to testify on behalf of the defendant; I can testify to this man’s veracity and good character.

Maintain , as here considered, means to assert something in the face of evidence or arguments to the contrary: in spite of circumstantial evidence pointing to his guilt, the accused maintained that he was innocent. Maintain almost always involves controversy or disagreement; it presupposes a prior statement to which one is adhering, and in this sense is a reaffirming of one’s position.

Aver means to declare confidently as fact. Avouch means to vouch for or affirm positively. Avow means to declare openly: to avow one’s guilt. None of these words is common in contemporary writing; all occur only in formal writing, and avouch seldom even there.