Rationalizing with an irrational person is never pleasant. It seems like you can't get your point(s) across. If that individual is a work colleague – boss, employee or customer – how do you ensure a good relationship?
Clearly, finding common ground is critical. Finding common ground requires give and take, and a willingness to hear what the other person is saying, so giving the irrational colleague an ear is a good solution. Take extra time to listen. Rephrase what you hear to ensure you're recognizing what his or her core argument is.
By itself, this will not yield a collaborative outcome, but it's a start. Often, irrational behavior is a desire to be heard, so just giving that person your attention will be extremely helpful, whether it's to address an employee's concern, hear what your boss wants done first today, or solve a customer's installation problem.
Beyond listening, remember that rationality cannot fight irrationality. If someone is acting out, their behavior is probably rooted in an emotional or illogical context. They're not about to listen to reason. In fact, when you approach an irrational situation with logic, you're likely to be shouted down.
So dig into the reasons why – find out why he or she fears having an alarm system installed, or why one of your employees can't get started on Friday mornings and yells at you every time you try to help. Maybe you'll find out Fridays and Saturdays used to be his off days in his previous job. Knowing more information will assist you in developing suggestions to help him get over his Friday phobia.
When you dig deeper, the answers will become more apparent.