We were discussing different proposals for what made mental states, properties, qualities, and so on distinctive: features that they all had, and all non-mental states lacked.
One approach to this project is to focus on different ways that our mental lives are thought to be uniquely private, that is, you have a kind of special or priveleged access to your own mental life.
This will take some explaining.
Many of our mental states — especially the “conscious” ones, whatever that amounts to — many of these states are ones we can know about in an especially direct way that isn’t based on evidence, observation, or inference. You can just tell whether and when you’re thinking about elephants.

Other people have to infer what you’re thinking, from your behavior and what you say. So you have some kind of special access to your own mind that other people lack. And you don’t seem to have this access to their mind, either. Nor do you seem to have it to many facts about bodies or brains or your physical environment. You can’t tell how much your brain weighs, or whether your body has paint on it, without looking at or touching or measuring them (or Googling it). You can’t tell what your shoe size or height are in this special direct way — at least not for the first time. Also, other people could in principle be in better positions to know these things than you are.
That last point touches on a different idea, that our access to our own mental states is always better than anyone else’s, or our access to other facts. A label that’s sometimes used here is incorrigibility, meaning no one can be in a better position to know, and so tell you that you’re making a mistake.
This idea that you can’t make mistakes about your own mind is sometimes unpacked using the notion of infallibility. Being infallible about a subject matter means that you can’t have false beliefs. If you believe something, it’s true. Some philosophers think we’re infallible about (at least some of) our own mental states.
A different kind of property is that a mental state “can’t be hidden.” Some philosophers think your own mental states can’t be hidden from you in the sense that if the state is there, then you’ll know it’s there. This property goes by a variety of names, including luminosity, self-evidence, self-intimatingness, and transparency. In some discussions, the last label is used to describe other phenomenon instead.
When philosophers talk about our minds, or parts of our mental life, being private, they usually have one or more of the above ideas in minds.
It’s controversial whether our access to our own minds is incorrigible, infallible, or luminous. Arguably, there are at least some facts about our own minds that we can make mistakes about, and that other people may be in a better position to see. For example, other people may be in a better position to know whether you’re jealous of your friend’s success. (You haven’t admitted it to yourself yet.) If Freudians are right, some of our deepest beliefs and desires are hidden. Cognitive science also seems to posit mental states that people aren’t ordinarily aware of having.
One strategy is to say, ok, maybe we can be wrong about some of our own mental states. Alex thought she wanted to study law, but really she didn’t. Bill thought he was angry, but really he was just hungry. Clara thought she trusted her sister, but really she had doubts she wasn’t admitting to herself.
Still, you might think, there’s some core parts of your mind you can’t be mistaken about. You might conceal those parts, or mislead other people about them, but you’d always be able to know for yourself whether you had those mental properties or not. At least, you’d be able to know this if you’re honest with yourself and reflect carefully enough.
It might be hard to say exactly which mental states belong to the “core parts of your mind,” in this sense, but still, this line of thinking is a natural one. Notice though that there’s some tension between the italicized sentence at the end and the properties discussed in (a) above. Can you tell whether you’re being honest and careful enough, without appealing to any evidence, observation, or inference?
The details are controversial, but many philosophers would agree that our access to (at least some parts of) our own minds is “special” in at least some of these ways — the least controversial are (a) and (e). We don’t seem to have the same special access to anyone else’s mind, nor to facts about our brains or bodies or physical environments.
Some philosophers have proposed that this special access can give us a distinctive mark of the mental — that is, that something counts as a mental state if and only if the person who’s in that state has this special kind of access to it.
But as we’ve seen, this is not something that many philosophers would be prepared to accept. It’s not clear whether we really have special access to all our own mental states. (Some philosophers argue that we have it to very few of them.) And some of these kinds of specialness have been proposed about our access to other facts, too, beyond our own minds.
All that’s really uncontroversial is that for some of our mental states, we’re usually in a better position to know about them than other people are, and in a different kind of position, with the result that other people are liable to make some kinds of mistakes that we’re not. These are interesting and important facts. But exactly what the betterness and differentness of our position amounts to isn’t yet settled; and it’s doubtful that these properties are had by all our own mental states and nothing else.