Is it possible in any logical system to know what one doesn't know literally. I don't mean the daily usage of the phrase like "Sam doesn't know physics", where you are just ignorant about some topic or you know a little about it, etc. What I mean is, suppose I know a collection of propositions, let's call the collection C, then can I know something which is not in C (The answer, should I say 'intuitively' sounds no). Any articles, proofs, etc will be welcome.
Basically, the 'unknown unknowns'!
Let's model this conundrum with sets.
Let's assume to start with that you are aware of the contents of one set, call it set C. But your universe also contains sets A, B, D, E, F, ...
If you do not know that those extra sets exist, then you don't know what your areas of ignorance are.
Now if you do know that those other sets exist but you do not know what their contents are then it could be asserted that you do know what you don't know.
The true statements in a logical system are fixed as soon as you say what the rules are and the starting assumptions are.
Related post on Math Stackexchange : Is Mathematics one big tautology?
However, still, it is not trivial to search for the meaningful true statements in a system and that's why we have so much Mathematics. So, while the truths are fixed , we have to learn and find them. But, I suppose this is not what you meant.
Now, one could also go on the meta level of asking what would be the truths of a different starting assumptions and rules too.
Analysis is also related to an important school of thought known as post modernism/post structuralism. Here is a nice discussion of it on youtube.
Is it possible to know what you don't know?
Yes. If you were to come in to the room with a bag, and tell me you have something in the bag, I would know two things:
I can justify both conclusions easily by citing the fact that people lie, and that when they the truth, my knowledge about the state of affairs in the world (like the content of your bag), are limited. So, knowing what we don't know is easy because we can justify our belief that we don't know things.
But, you have also invoked the question of unknown unknowns, and here's where things get more interesting. How do come to a state of knowledge about what we don't know. In the example of the bag, your having a bag and asking about the contents of the bag defines the scope of belief and knowledge. Specifically, you ask the question 'What is in the bag?', and that begins our journey of reason. But what about in the general situation?
First, we have to start with a recognition that reason is defeasible (SEP). From the article:
In philosophy of logic, defeasible reasoning is a kind of provisional reasoning that is rationally compelling, though not deductively valid.1 It usually occurs when a rule is given, but there may be specific exceptions to the rule, or subclasses that are subject to a different rule. Defeasibility is found in literatures that are concerned with argument and the process of argument, or heuristic reasoning.
So, our first strategy in determining unknown unknowns is to write out what we believe or know. Then we can use our intuition to explore the boundary of the known and unknown. For instance, do I know anything about the bag? Do I know anything about you who carries the bag? Do I know about what you have access to put in the bag? Do I know about what is possible regarding the bag holding particular contents? This can help separate unknowns that are and can be known from those that cannot. This might be considered an exploration of facts and defeaters.
But the more important question is, are there limits to this process? And the answer is yes. There are limits to knowledge, and therefore we are always confronted with the possibility of unknown unknowns. This rests on the basis that one cannot know everything. That there is no omniscience for human beings. What justification?
First, intuitively, through our own experiences, we are constantly discovering that our model of the world is inaccurate. We witness such an imperfection in others. We can examine people's mistakes in reasoning, and see that time and time again, people come to the wrong conclusion because they are confronted with unknown unknowns. We can examine the tentative nature of inductive and abductive logic. Is there an explanation? Perhaps. Perhaps we human beings are systems of physical computation (SEP) who have material limits to our ability to know. For instance, we can only know when we are alive, and we live a finite period. Whatever the justification, the conclusion helps to support the idea that knowledge is fallible (IEP). From the article:
Fallibilism is the epistemological thesis that no belief (theory, view, thesis, and so on) can ever be rationally supported or justified in a conclusive way. Always, there remains a possible doubt as to the truth of the belief.
If human reason is defeasible (it is), and there are inescapably unknown unknowns (there are), then reason itself is fallible (it is).
Goedel's incompleteness theorem can be thought of as knowledge of what one doesn't know, namely that it (that which one doesn't know) exists. Namely, in a logical system L incorporating the Peano Arithmetic, there will necessarily be an assertion that can be neither proved nor disproved by L.
