Here’s something with which I’ve noticed intro to philosophy students tend to struggle: the difference between it being possible that a theory is true, and the theory’s being plausible. Example:
In the course I’m TAing this semester, one of the topics we discussed is the nature of value. In virtue of what does anything have value, or what is the source or cause of things being valuable? Most of the students have never taken philosophy before, and it seems that many are pre-theoretical subjectivists about value. Very roughly, subjectivism is the view that things get their value from people valuing them. The main alternative to subjectivism is objectivism about value. Objectivism, again roughly, is the view that all or some things just are valuable, independent of anyone’s beliefs, attitudes, choices, etc.
So, in the class, we do this thought experiment where the professor taps students’ intuitions by asking them about the status of a beautiful planet containing a well-functioning ecosystem. By stipulation, no one knows that this planet exists. Is it better that the planet exists than if it didn’t exist? If you had the power and choice to bring such a planet into existence at no cost to yourself and then would forget about it forever, do you have any reason to choose to create the planet?
You’re supposed to get the intuition that you should create the planet. The bigger conclusion to draw is that it’s at least possible that the planet, and maybe other things, could just be valuable, period, regardless of whether anyone cares or even knows about them. Whether subjectivism or objectivism about value is the better theory, all things considered, is a separate question.
But, when asked: “Is it possible that something could be valuable even if no one values it?,” quite a few students, presupposing subjectivism, say no – it is only possible for things to get value from people valuing them. This is false. What they mean is that it is only plausible to think that things get value from people valuing them.
To correct this, you have to teach the student what “possible” means, philosophically speaking. And before you know it, you’re talking about alternate possible worlds, and the laws of physics, and mathematical truths, and it all goes downhill from there. While I always do my best to help students in this sort of case, I can’t help but conclude that the way critical thinking is handled by conventional educational institutions is just far too little, far too late.