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II Peter 1:19-21 "Moreover, we possess the prophetic message that is altogether reliable. You will do well to be attentive to it, as to a lamp shining in a dark place, until day dawns and the morning star rises in your hearts. Know this first of all, that there is no prophecy of scripture that is a matter of personal interpretation, for no prophecy ever came through human will; but rather human beings moved by the holy Spirit spoke under the influence of God."
II Peter 3:15-16 "And consider the patience of our Lord as salvation, as our beloved brother Paul, according to the wisdom given to him, also wrote to you, speaking of these things as he does in all his letters. In them there are some things hard to understand that the ignorant and unstable distort to their own destruction, just as they do the other scriptures."
Peter says that "we" possess the prophetic message which is reliable. Who is the "we" he is talking about? That is the question which must be answered to properly understand this passage about who has this authority of interpretation. Back up to the beginning of the letter and read II Peter 1:1-2 "Symeon Peter, a slave and apostle of Jesus Christ, to those who have received a faith of equal value to ours through the righteousness of our God and savior Jesus Christ: may grace and peace be yours in abundance through knowledge* of God and of Jesus our Lord." Who is this letter addressed to? It is addressed to those who "have received a faith of equal value to ours..." Peter is addressing his letter to people who are already Christians. So that rules out unbelievers from the possible candidates. So who is the "we" that Peter is referring to? Peter is an Apostle. Peter is referring to himself and the other Apostles here as the "we" as differentiated from all the other Christians who are not Apostles. This helps us to understand the passage. He says that "you" - again talking to Christians - "will do well to be attentive to it...that there is no prophecy of scripture that is a matter of personal interpretation". This statement is addressed to CHRISTIANS - not unbelievers. So it is to Christians that he goes on to say "for no prophecy ever came through human will; but rather human beings moved by the holy Spirit spoke under the influence of God." What is to be inferred? That because the scriptures originated from God through men influenced by the Holy Spirit to do so, only those who are influenced by the Holy Spirit to do so may interpret scripture. In other words, Peter is reminding his Christian readers that "we" - the Apostles - retain the sole authority to properly interpret scripture and not all Christians.
Now some may object that "Indeed only people moved by the Holy Spirit may interpret the scriptures, but all Christians have the Holy Spirit. Therefore all Christians may interpret the scripture." To that objection I submit I Corinthians 12:27-31 which says "Now you are Christ's body, and individually parts of it. Some people God has designated in the Church to be, first apostles; second, prophets; third, teachers; then, mighty deeds; then, gifts of healing, assistance, administration, and varieties of tongues. Are all apostles? Are all prophets? Are all teachers? Do all work mighty deeds? Do all have gifts of healing? Do all speak in tongues? Do all interpret? Strive eagerly for the greatest spiritual gifts." It is true that all Christians have the Holy Spirit bestowed to them in baptism, but not all Christians have the same gifts. Only those to whom God has specifically given this gift have the authority to authentically interpret the scriptures.
Later in chapter 3 Peter reiterates this by saying that "the ignorant and unstable distort [them] to their own destruction, just as they do the other scriptures." Christians were the recipients of Paul's letters to whom Peter referred, but they were still being misinterpreted by the "ignorant and unstable."
So we can see from scripture that simply being a Christian is not enough, by itself, to authentically interpret scripture. One specifically has to have the authority specifically given by God - as Peter and the other Apostles were - to accurately interpret scripture. I John 4:6 reiterates this as well by saying "We are from God; he who knows God listens to us; he who is not from God does not listen to us. By this we know the spirit of truth and the spirit of error." We know truth by listening to "us" - the Apostles - and not by our own interpretation.
Scripture citations were taken from
http://www.usccb.org/bible/1corinthians/12
http://www.usccb.org/bible/2peter/1
http://www.usccb.org/bible/2peter/3
http://www.usccb.org/bible/1john/4