In 1 Corinthians 14:22 Paul says 'tongues are a sign for unbelievers, but prophesying is a sign for believers'. Yet he goes on, in verses 23, 24 and 25 and makes an argument which seems to me to suggest the opposite. That tongues are for the believers, and prophesying for the unbelievers. My question is... what is Paul saying?
Wherefore tongues are for a sign, not to them that believe, but to them that believe not: but prophesying serveth not for them that believe not, but for them which believe. If therefore the whole church be come together into one place, and all speak with tongues, and there come in those that are unlearned, or unbelievers, will they not say that ye are mad? But if all prophesy, and there come in one that believeth not, or one unlearned, he is convinced of [convicted by] all, he is judged of [by] all: And thus are the secrets of his heart made manifest; and so falling down on his face he will worship God, and report that God is in you of a truth. (1 Corinthians 14:22-25)
You have read well. Yours is a very good question. On the face of it, in English, you are right. It appears that Paul contradicts himself. Actually, though, he does not.
The first verse you mention points out the difference in the functions of languages and prophecy. Those who had the gift of speaking in human languages they had not learned in the normal way were able to teach the gospel to those who might not otherwise have heard it because of language difficulties. (Speaking in tongues in the Bible is always speaking in human languages that a hearer who spoke that language could understand.) Prophecy, on the other hand, is for those who are already believers. It makes no sense to an unbeliever, because he doesn’t believe there is a God who gives his message to man. The point of prophecy (giving a message directly from God, usually not having to do with foretelling the future) is to give clearer explanation of doctrine to those who are already believers.
Now for the part that seems contradictory. If a person who is an unbeliever comes into an assembly and everyone was speaking a variety of foreign languages, his reaction is not to hear his language but to say that everyone must be crazy. If someone who had the gift of his language were to take him aside and explain what was happening that would be a different matter from what Paul is describing. What Paul is really saying is that the place for speaking in tongues was in teaching unbelievers, not in the assembly of the believers.
If, however, an unbeliever comes into an assembly and hears one gifted with prophecy talking about sin, and that is one of his sins, then he is convicted of his sin by all of those who are prophesying. When his sin is known, he is judged to be a sinner by all who know it. Therefore he falls on his face (translated worship) and then goes out and tells others that this group must be from God because they knew his sin without him telling it. A similar passage comes from the story of Jesus talking to a woman at a well in Samaria. After he had talked about her life, she said, “I see you are a prophet.” (John 14:9) Then, “The woman then left her waterpot, and went her way into the city, and saith to the men, Come, see a man, which told me all things that ever I did: is not this the Christ?” (John 4:28-29)
Where some translations say, “he is convinced of all, he is judged of all,” do not make the mistake that it is saying that he is convinced that all they say is true. The “of all” means by all the people, and has nothing to do with his belief or unbelief in what they are saying. He could still be an unbeliever, but walk away acknowledging that these people speak for their God.
The prophecy is primarily for believers, because they will receive and understand the message. Nevertheless, it will convict unbelievers. They will not go away thinking these Christians are all crazy, but knowing that there is something different about these people, in a positive sense.
So Paul is not contradicting himself. He is saying that tongues and prophecy had specific audiences. When they are used in a way not consistent with those uses tongues failed to be effective, but prophecy still had some effect. If anything, his argument is the same that he earlier makes, that tongues were the least valuable of the miraculous gifts when those gifts were available to men (in the first, and possibly the early second, century).