Most entries, like 8x8, in the multiplication table I know off the top of my head. But some may require a quick calculation. If you ask me what 7x8 is, I may do 49+7=56, and if you ask me what 6x9 is, I might do 70-6=54.
But is there really an important distiction here? You ask me what 8x4 is. Just about right away, 32 comes to mind. But what if, in fact, my mind (or brain?) unconsciously calculated it: 64/2=32? After all, I am more confident of my knowledge of 8x8 than of 8x4, and I think it comes to mind faster and more naturally. And even in my 7x8 calculation, there were unconscious elements. I don't have to consciously think: 7x8 = 7x(7+1) = 7x7 + 7x1 = 49+7 = 56. I just consciously think: 7x7=49, 49+7=56. So along with the conscious processing, there is unconscious application of the distributive law. And hence it is quite a reasonable hypothesis that when I am asked about 8x4, I might indeed be making a quick unconscious calculation. And that would help explain why I can call to mind 8x8 faster than 8x4. Furthermore, if the brain is at all like a computer and if the brain is where memories are housed, information is stored in some encoded and maybe even compressed form. There will thus always be a computational process of some sort when making stored data usable.
Actually, in the above I wasn't completely correct. I think I actually do have 7x8 and 6x9 memorized. But normally (though not now, since the examples are fresh in mind) recalling them from memory takes more time and effort, and I feel it is less reliable than doing the quick calculations. However, one could easily imagine that I don't have them memorized at all, and in the following I will counterfactually assume that.
Now, it is tempting to say that I don't believe that 7x8=56 if I have to actually compute it. But if computation is involved in almost all processes of recall, then it seems we believe very little at all, except the things we're occurrently thinking. And that's absurd. For one, it seems plausible that beliefs are needed for justification, and so if we have so few beliefs, and yet many of our beliefs require many other beliefs for their justification, then fewer of our beliefs end up justified than is right.
Perhaps, then, we should say that there is a difference between calling and recalling to mind. But that distinction is going to be hard to draw.
So maybe what we should say is this. When calling something to mind would involve an unconscious process, then we have a case of belief, but when the process would be conscious, there is no belief until the proposition is called to mind. Now the idiot savant who can do very big arithmetical calculations unconsciously counts as believing all the answers ahead of calculation. That doesn't seem intuitively right. However, whether we call it a belief or not, I do think we should not in any important way distinguish the case of unconscious arithmetical computation from more ordinary cases of recall. And once we realize that unconscious computation can be very complex, we really shouldn't distinguish conscious from unconscious computability in any normatively important way.
Here are some potential consequences. First, we might be pushed to some sort of reliabilism, perhaps of a proper function sort. For there ought to be a distinction between justified and unjustified belief, and if we do not distinguish belief from what one has a skill to compute, then we need a similar account of the justification of the outputs of that skill. But that account, very likely, will involve the reliability of the skill. Second, if we want to maintain some distinction between non-occurrent belief and skill at generating occurrent belief, this distinction is likely to be a vague one, involving the amount of computation. In particular, I suspect that the distinction may not match up with what we want to say about knowledge. So it may be that knowledge doesn't entail belief—maybe knowledge merely entails the possession of a skill of calling to mind.
I am not particularly attached to the conclusions. I just want to provoke some discussion about the phenomena.