For beloved see comments on Jude JUD.1.3.

Now is emphatic; its function is to emphasize that it is not one letter but two that have been written. This may stress Peter’s deep concern for his readers, or the importance of the subject matters he is dealing with in the letters.

Referring to this letter as the second letter almost certainly implies that the earlier one is The First Letter from Peter. Both letters serve as “reminders.” This reminds us of 1.13, where the same Greek word is used, and where it is established that the present letter is a testament. The testament form of writing, however, has been abandoned for the whole of chapter 2, so that here, at the beginning of chapter 3, it is Peter’s intention to go back to it. If the first of the two letters referred to is indeed 1 Peter, this suggests that both letters are intended to be testaments. The use of the word reminder also suggests that the readers already know what they are being reminded of. Them clearly refers to the two letters, hence Good News Translation “both letters.”

Aroused can also be “stimulated,” “awakened,” or “stirred up.” Sincere translates an adjective found only here and in Phil 1.10, and which can be translated literally “examined by the sun’s light and found to be genuine,” and hence “pure,” “without blemish,” “free from falsehoods.” In the present context it points to moral and ethical purity, uncontaminated by sensual passion and the other vices of the false teachers. Sincere mind can therefore be translated as “pure thoughts” (Good News Translation), “wholesome thinking” (New International Version), “true understanding” (Jerusalem Bible), “minds uncontaminated by error” (Phillips). It should be noted that Revised Standard Version can be interpreted to mean that the minds are already sincere, and all that is needed is for these sincere minds to be aroused. The meaning of the Greek text is reflected more accurately in Good News Translation, where it is clear that the purpose of the two letters is to stir up pure thoughts in the readers’ minds. The clause I have aroused your sincere mind by way of reminder may therefore be rendered as “I have reminded you of these things in order to stimulate your minds to think pure (or, clean) thoughts” or “I have … in order to stir up your minds to think pure….”

Quoted with permission from Arichea, Daniel C. and Nida, Eugene A. A Handbook on The Second Letter from Peter. (UBS Handbook Series). New York: UBS, 1980. For this and other handbooks for translators see here .