It's not only possible, but very common to hold inconsistent opinions and be aware of those. When you are faced with conflicting views, they need not be a question of value of one over another... especially when you are supposed to hold all ethical principles upon which you derive your morals from as equal to one another.
Take for example this excerpt from guidelines on ethics in medicine:
Sure, ultimately, the common person will typically instead ask 'What should I do?', but often times that they arrive at an incorrect conclusion because they have not examined the reasons (principles) they came to that conclusion... and thus in a similar, but different, situation regarding the same set of principles, create a conflict.
That excerpt doesn't illustrate holding inconsistent views. It is possible to reconcile them. And there are differening approaches to striking that balance. This is a type of analysis that courts must engage in all the time when dealing with legal principles.
The idea that formal logic is a useful way of ... well, anything, really, besides dealing with abstract formal logic problems is pretty much the issue that prevents people from addressing this type of issue coherently.
The resolution is developing your mechanism of choice, as the article calls it. What principle takes precedent? Do you balance competing concerns? You can't not reach a decision, and you reach that decision by working through the principles to resolve the inconsistency between them, because each individually mandates a different outcome.
You're conflating the idea of resolving inconsistencies with abandoning the principles, but that's not how it works. If you'd like, I can show you a template analysis for the problem.
But in any event, when you're deciding on a course of action, you are absolutely picking one value over the others.
In this particular case, Scalia exercised his right to read the dissent from the bench. This is uncommon, and only used when a justice feels particularly strongly about the issue.
From the NY Times article
That's an interesting part of US courtroom procedure.