"Causing others to stumble" means to cause a fellow Christian to do something he/she considers wrong. It does not mean to do something someone else thinks (or is convinced) is wrong.
What is 'causing others to stumble'?
Paul addresses the issue of causing others to stumble in 1 Corinthians 8. The particular issue then was eating food that had been offered to idols. Some of the Corinthian believers realized that idols were in fact nothing and that food offered to idols was no more contaminated than any other food. Therefore, they had no problem eating food that had been offered to idols. However, other believers who had been saved out of idol worship considered eating food offered to idols as an act of worship of the idols. To them, eating food that had been offered to idols was wrong because it was worship of someone other than Christ. Therefore, if they ate food that had been offered to idols, their consciences condemned them. So Paul commanded those who could eat food offered to idols not to cause those whose consciences did not allow them to eat food offered to idols to eat that food and thus be condemned in their own minds.
In the western world, we generally do not have the problem of food offered to idols. So to use a different example, let's look at drinking alcoholic beverages. There are many Christians who are convinced that it is wrong for them to drink alcoholic beverages. However, Scripture does not speak against drinking alcoholic beverages; it speaks against drunkenness (Ephesians 5:18). So if a Christian looks at the Scripture, realizes that drinking alcoholic beverages is allowed, and therefore enjoys a beer or a glass of wine occasionally, he/she is all right with God to do so. However, suppose that Christian, who is free to drink alcoholic beverages in moderation, invites another Christian who is not free to enjoy alcoholic beverages for dinner. If the first Christian, knowing of his/her invitee's lack of freedom, then has a beer with dinner and gives one to his/her guest, he/she is encouraging the invitee to do something he/she considers to be wrong. If the invitee takes and drinks the beer, he/she has then gone against his/her conscience, and the host has caused the invitee to stumble. A much better approach for the host would be to forego the pleasure of the beer and have no alcoholic drinks available at dinner. For the host to cause the invitee to do what he/she considers sin is for the host to sin.
Jesus commanded us to love each other (John 13:34), and a showing of that love will be a desire not to cause another Christian to do what he/she considers sin, even if we are convinced the particular action is not sin. Love will cause us to want to build each other up, not pull each other down.
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