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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'!

J D
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Siddharth Chakravarty
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  • Well, the true statements are fixed as soon as you say what the rules are and the starting assumptions are – Reine Abstraktion Jul 09 '23 at 17:40
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    Your question needs to be clarified. In some ways, philosophy is all about knowing what you don't know. The identification and elimination of wrong or incomplete claims to knowledge begins with Socrates and extends to the Hegelian dialectic, at least. Similarly, the hard sciences can be seen not as certain knowledge, but as the elimination false claims. There are also, as Rumsfeld rightly quipped, the "known unknowns," such as the variable in an algebraic equation or the precise blank shape on a map designating unexplored territory. – Nelson Alexander Jul 09 '23 at 17:48
  • Do you mean something like, "I know that I don't know the answer to question *X*?" for some question *X*? C.f. the question of [epistemic closure](https://plato.stanford.edu/entries/closure-epistemic/). – Kristian Berry Jul 09 '23 at 18:20
  • Of course it is. Some research might require some distinct facts to be assembled, and as you go through the list you know what you know, and you know what you don't know (need to know). You might like to listen to [Donald Rumsfeld](https://www.youtube.com/watch?v=REWeBzGuzCc) exploring the theme. – Weather Vane Jul 09 '23 at 18:45
  • Actual conversation from a school classroom. Boy (knowing of the teacher's passion) "Please sir, what do you know about model railways?" Teacher (after a pause): "Well, after 26 years involvement I can say that I know what I know." We all laughed and thought what a fool he was, but only later did I realise what he meant. – Weather Vane Jul 09 '23 at 18:57
  • We build on existing knowledge: sentences, theories, methods, to acquire new knowledge. This is the way school and training works. – Mauro ALLEGRANZA Jul 10 '23 at 06:05
  • @KristianBerry I modified the question to make it more clear. – Siddharth Chakravarty Jul 10 '23 at 17:01
  • @SiddharthChakravarty do you mean that we might have knowledge of a specific kind of one set of propositions, and then knowledge of another specific kind for a different set of propositions? Perhaps one might think of having multiple flavors of *k* operators on propositions (it seems often, maybe even always, possible to multiply fundamental operators). – Kristian Berry Jul 10 '23 at 17:06
  • @KristianBerry No, what I mean is can I know a proposition which is not in my set of propositions. (using the same **k** operator) – Siddharth Chakravarty Jul 10 '23 at 17:16
  • @SiddharthChakravarty what is separating the one set of propositions from the singleton of the other proposition? Offhand it seems as if you could use union or conjunction to merge the sets. But if the sets are held apart, perhaps something about what holds them apart would prevent you from applying *k* to the outlier. – Kristian Berry Jul 10 '23 at 17:35
  • @KristianBerry the problem is not doing a conjunction, had I done conjunction it just means I have learned or have now known the thing, I am saying is it possible to learn without learn, like do both at the same time, know and not know. – Siddharth Chakravarty Jul 10 '23 at 17:47
  • @SiddharthChakravarty some take the knower paradox to involve an example of a sentence that is both known and unknown, but that means dialethism/paraconsistency, so whether such a thing is possible depends on how logical possibility is calibrated modulo paraconsistent logic. One can have, "If dialethism were true, contradictions would be possibly true," which might itself be a counterpossible conditional, though (these issues are rather thorny). – Kristian Berry Jul 10 '23 at 17:53
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    See: 'How do I know what I don't know?' https://philosophy.stackexchange.com/questions/99682/how-do-i-know-what-i-dont-know/99689#99689 – CriglCragl Jul 10 '23 at 20:58
  • What do we know? Nothing. What are we very sure of? A lot of things. – Scott Rowe Jul 10 '23 at 22:38
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    This question's importance grows with every passing second. I guess the shape of water varies ... for better/worse *hypothesis non fingo*. – Agent Smith Jul 28 '23 at 10:08
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    @AgentSmith I feign no apotheosis. – Scott Rowe Jul 29 '23 at 13:04

4 Answers4

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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.

niels nielsen
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Depends on what you mean by 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.

Reine Abstraktion
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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:

  1. I don't know if there is actually something in the bag.
  2. If there is something in the bag, I don't know what it is.

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).

J D
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  • "That there is no omniscience for human beings. ", and why won't it hold for any entity, would it render omniscience as self-refuting or, there could be some resolution for it. – Siddharth Chakravarty Jul 28 '23 at 18:00
  • @SiddharthChakravarty That's an entirely separate question contingent upon the belief of entities that are sufficiently different from humans in physical or intellectual capacity to whom the question applies predicated upon the belief of their existence. As I do see reason to believe such entities exist at this time, I haven't given it much thought. IOW, I can't answer questions like is 'God omniscient' because the question 'Who or what is God?' seems to me too metaphysicalu speculative and the source of too much consensus to be of value to discussion .That being said... you're in the place! – J D Jul 28 '23 at 19:54
  • https://philosophy.stackexchange.com/q/3252/40730 comes from the knowledge base and there are others – J D Jul 28 '23 at 19:54
  • This is what makes airport security so challenging. People carrying bags, etc. – Scott Rowe Jul 29 '23 at 13:11
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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.

Mikhail Katz
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    Short n sweet. Except that I'd make a trinity: Gödel's incompleteness + Turing noncomputability + [Tarski's Undefinability](https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem) – Rushi Jul 28 '23 at 09:37