Paul speaks very strongly regarding sexual immorality in the Corinthian Church in today’s Bible reading. He says that a man is committing an act that even unbelievers in Corinth don’t condone: he is having sexual relations with his father’s wife.
Paul recommends that the church deal with this individual in a very strict way: remove him from the congregation (1 Corinthians 5:2, 13) and give him over to Satan (1 Corinthians 5:5). The goal of dealing with this individual is twofold:
- Deal with the individual, aiming to restore him to fellowship with the rest of the church body
- communicate with the church body that sin is serious and should be dealt with seriously. Jesus made a similar statement when he said that it would be better to enter heaven maimed or blind than to go to hell intact. (Matthew 5:29–30)
The terms “sexual immorality” and “sexually immoral” appear twice each in English in this chapter. It refers to the sin being committed and the one who is committing the sin. But both of these terms are based in the same Greek root and we derived our English words “porn” and “fornication” from this Greek root.
Normally when we talk of “porn” we mean pornographic literature and images. But the basis of the word is far broader than those two narrow classifications. Here are definitions from two Greek lexicons:
- to engage in sexual immorality of any kind, often with the implication of prostitution—‘to engage in illicit sex, to commit fornication, sexual immorality, fornication, prostitution.’ 
- fornication, sexual immorality, sexual sin of a general kind, that includes many different behaviors.
In recent years we have seen reports of lawsuits by former church members against their former church for kicking them out of the church. At the core of these lawsuits are libel, slander, and defamation of character. The plaintiff claims that they should be able to live however they want and remain a member of the local church congregation. But based on Paul’s recommendations, the church has a responsibility to deal with sin in order to protect its purity. I believe Paul would say that this can, and must, be done without libel, slander, and defamation of character.
Now, as soon as I typed that next-to-last sentence, I could hear some readers point out (and rightfully so) that the local church, filled with fallen people is far from “pure” and filled with hypocrisy. And I can also hear readers calling out specific (“respectable”) sins that are often tolerated — and even promoted in much of (Western) church culture such as gluttony, lying, and slander to name a few. And I can hear some readers say that “Paul isn’t showing much grace.” Paul already responded to that criticism:
What should we say then? Should we continue in sin so that grace may multiply? Romans 6:1 (CSB)
Should all sin be exposed in the church? Judging from Paul’s example, we would have to say no. Paul didn’t single out
Look, we all sin. I sin. And
What sins do you deal with on a regular basis? Are you grieved by them? Do you feel a need to repent of them in order to walk in deeper intimacy with God? Or do you feel that God isn’t bothered with your sin?
Perhaps you need to follow James’ directive:
Therefore, confess your sins to one another and pray for one another, so that you may be healed. The prayer of a righteous person is very powerful in its effect. James 5:16 (CSB)
Paul is pretty clear in how we should deal with sin: kill it.
Puritan John Owen’s Mortification of Sin in Believers says that we should be killing sin or [sin] will be killing you. John Piper refers to this small book by Owen in two messages, “How to Kill Sin” and “Kill Sin with the Word of God“. I invite you to click those links and listen or read Piper’s messages.
 Louw, Johannes P., and Eugene Albert Nida. Greek-English lexicon of the New Testament: based on semantic domains
 Swanson, James. Dictionary of Biblical Languages with Semantic Domains: Greek (New Testament)