Paul ends this fist section of First Timothy with this:

This command I entrust to you, Timothy, my son, in accordance with the prophecies previously made concerning you, that by them you fight the good fight, keeping faith and a good conscience, which some have rejected and suffered shipwreck in regard to their faith. Among these are Hymenaeus and Alexander, whom I have handed over to Satan, so that they will be taught not to blaspheme. (1 Timothy 1:18-20)

Note, again, how Paul's greeting meshes with his message. In the greeting he writes, "Paul, an Apostle ... according to the commandment of God." He now writes to Timothy, "This command I entrust to you..." Leadership is not autonomous. Leadership is in the service of those that are led and works to accomplish the goals of the Head, which is Jesus Christ.

Paul told Timothy that the goal of instruction is "love from a pure heart and a good conscience and a sincere faith" and here Paul urges him to keep the faith and a good conscience. He now warns that not keeping a good conscience can lead to the shipwreck of faith. We need to teach the truth and be honest with our failings to follow. That is integrity and maintains a good conscience. To teach the truth and live the lie in secret is very dangerous indeed. 

Whereas Paul was anonymous regarding those who taught strange doctrines and dallied in speculative non-essentials, he names names when it comes to bad doctrine. Hymenaeus and Alexander have blasphemed. Paul wants Timothy and others to know their names so that they can avoid their teaching. In Second Timothy, we get a clue regarding what Hymenaeus taught:

and their talk will spread like gangrene. Among them are Hymenaeus and Philetus, men who have gone astray from the truth saying that the resurrection has already taken place, and they upset the faith of some. (2 Timothy 2:17-18)

So leaders: Be wary of speculating about things on which the Scriptures say little. Promote those things about which the Scriptures have much to say. Command, teach, and live in a way that promotes love, faith, and a good conscience.

<>< Test everything. Cling to what is good. ><>