I have “heard” that there are other letters Paul wrote that are not in the Bible. For example, there was a Third letter to the Church of Corinth that was not included in the Bible — Why not?
It is hard to respond to the general question about “other letters.” There are many “other letters” you might be referring to, so if you ask about specifics, I will happily respond. Feel free to do some of your own research and if there is a specific “other” book you want me to comment on, I will be happy to. Otherwise, my suggestion is that you read my book “Reasons for Belief” which has an entire chapter devoted to questions about the creation of the Bible (www.ipibooks.com).
Now, about 3rd Corinthians. There is no manuscript evidence for a third letter by Paul. In fact, there is literally zero manuscript evidence for any other books by Paul. From what we know of him, we can assume that Paul wrote MANY letters. However, it is apparent that certain letters were written as general letters, to be read in all the churches. These are the ones which ended up in the New Testament. Of course, this is reasonable speculation and is not proved by direct evidence. Specifically, many scholars believe that Paul wrote a third letter to the Corinthians, because he talks about another letter he wrote to them in 2 Corinthians (2 Corinthians 7:8; “I see that I hurt you with my letter”). Some, but definitely not all scholars believe that the letter he is referring to is not the one we have–ie. 1 Corinthians. When I read 2 Corinthians 7:8, it certainly seems to me to be a reference to 1 Corinthians, but scholars speculate differently. Bottom line, this is scholarly speculation. We can assume that Paul wrote other letters, but the fact is that we have literally zero physical or even ancient written evidence of other letters.
Still, we can ask, if Paul did write other letters, why are they not in the Bible and the ones we have are included? Such questions require reasoned speculation. We simply do not know. I assume that it was the consensus view of the very early church that certain letters were not only inspired, but intended by Paul for general circulation. Possibly it was Paul himself who helped to decide which letters he wrote were to be circulated in the churches. Were these speculated other letters also inspired? Perhaps yes and perhaps no. We simply do not know. It is possible that Paul wrote other inspired letters which are not in the Bible? If so, why not? Again, this requires speculation. Perhaps, even though Paul wrote other authoritative letters, it was generally recongnized that the purpose of the letters was too specific to a particular place and situation, and, even though from the Holy Spirit, still did not need to be circulated widely. In the final analysis I, as a Christian, have a basic faith that trusts that if God wanted a particular letter in the New Testament, he would have seen to its being included and if he wanted a certain letter to not be in the New Testament he, by his divine providence, would have seen to it not being included. Of course, I am speaking by faith and not by evidence when I say this.
About 3rd Corinthians, there is an actual document which is known as 3rd Corinthians. Scholars are unanimous that this letter is pseudepigraphical. This is a very fancy word which means that the letter was written as if it was from the apostle Paul, but the claim of Pauline authorship is fake. To us in the Western world, it is quite shocking to think of writing a letter under a false name–a name of a person with much greater reputation than the actual writer. To us this would be blatant lying and a total scandal. In the ancient world, the attitude toward pseudepigraphal letters was somewhat different. There was 1 Enoch, which was clearly not written by Enoch. Many other books were written in the tradition of a biblical character were penned under that person’s name. The Jews were not offended by this, as they assumed that everyone knew it was not by the actual person Enoch, for example, but it was written under his name to explain the nature of the letter and to draw attention to it.
Let us return to the question. There is a letter known as “3rd Corinthians.” What do we know about this book? The evidence points toward its being written in the middle of the second century. An official Roman Catholic source gives the date to be AD 160-170. The contents reveal that it was written to oppose Gnostic thought. As far as I know, no serious student of the topic believes that this letter was written by Paul. The purpose of the letter was to “correct” some of the false Gnostic interpretations of 1 Corinthians, especially 1 Corinthians 15, which is the section on the resurrection. The text of 3rd Corinthians was included in a larger document known as the Acts of Paul. Many believe that this letter was written in response to a text known as the Letter of the Corinthians to Paul. This was a Gnostic document. We have many gnostic writings available to us, including the Gospel of Thomas and the Gosple of Judas. Specifically, 3rd Corinthians was written to oppose the false interpretation in the Letter of the Corinthians to Paul concerning the phrase “flesh and blood cannot inherit the kingdom of God.” (1 Cor 15:50). The Gnostics taught that this meant that the resurrection was not physical. The text known as 3rd Corinthians was actually accepted by the Syraian and the Armenian Church for a while as canonical, but there is no evidence it was ever accepted by the Latin West or the Greek Eastern church. We can safely conclude that it was not written by Paul and that it was not an inspired book.
I hope this helps.