Is it a paradox to assert "1+1=2 is true, but I don't know it"? I just thought of this chestnut myself.

  • 4
    It's anomalous but not necessarily contradictory. If we take knowledge defined in the simple "justified true belief" sense, there are numerous ways one could say this truthfully. One might have an unjustified true belief, or might have justification but still not believe, or simply say the sentence frivolously when it turns out the statement is indeed true. For example, "string theory is true, but I don't know it." I have no idea whether string theory is true, but if it is, that sentence is also true.
    – commando
    Commented Jun 16, 2014 at 14:52
  • 1
    I don't think so under any normal definition of paradox.
    – virmaior
    Commented Jun 16, 2014 at 15:22
  • @commando If you posted your comment as an answer, I would accept it.
    – user107952
    Commented Jul 2, 2014 at 2:21

5 Answers 5


Is it a paradox to assert "1+1=2 is true, but I don't know it"?

Assuming that

  • p is "1+1=2 is true",
  • "it" refers to p,
  • your statement is a compound statement A & B,
  • A and B are of the form: propositional attitude + p,

then the crucial question is: What is the propositional attitude wrt p in A?

1) If we assume that it is "I know that p", then your conjunctive statement is stating that

I know that 1+1=2 is true & I don't know that 1+1=2 is true

which is, obviously, a contradiction. (Or, there are two different meaning of "knowing" involved.)

2) If we assume that it is "I assert that p", then your conjunctive statement is stating that

I assert that 1+1=2 is true & I don't know that 1+1=2 is true

by which you say that p in A is true, but that you can't fulfill some further condition required to actually know that p (e.g. p is true but lacks justification). No contradiction here.


According to the following view, such an assertion simply cannot be correct/warranted. The problem is that you cannot know that "1+1=2 is true, but I don't know it".

A third kind of argument comes from considering a knowledge variant of Moore's paradox:

(4) It is raining and I don't know that it is raining.

If asserting (4), the speaker cannot know what she asserts. For if she knows that p&q, she knows that p and she knows that q. And if she knows that q, then q. Applied to (4) this gives the result that the speaker knows that it is raining and also doesn't know that it is raining, i.e. an open contradiction. (This is a special case of the reasoning involved in the so-called knowability paradox, or Fitch's paradox; cf. Sorensen 1988 and Williamson 2000 for overviews).

The idea of the argument is that the strangeness of an assertion of (4) depends on the fact that such an assertion cannot be correct (warranted). That it cannot be correct is explained by the appeal to the idea of the self-representation as knower: I cannot correctly represent myself as knowing what I cannot know. Hence, on this view, I cannot correctly assert (4).

SEP, "Belief and Assertion"

So, this reference shows a contradiction without using Mozibur's KK Axiom, but rather the somewhat minimal requirement that a warranted assertion must (at least) be knowable.

PS: I am ignoring that "1+1=2" has a formal proof and is close to being tautological. I am treating it as a random proposition. Perhaps there is something interesting to be found by treating "1+1=2" (more?) properly.


Is it a paradox to assert "1+1=2 is true, but I don't know it"?

Its not a paradox: Plenty of children when they first learn arithmetic know that 1+1=2 is true, but they don't understand why. They have simply learnt it as as a piece of formal knowledge - parrot fashion (and note that no-one thinks a parrot repeating phrases in the English language understands English); hence one should not call this knowledge (though it is knowledge of a certain kind); when they understand why, it then becomes proper knowledge; one might say that this is a kind of justified true knowledge - that is knowledge with an account as to why its true.

However, its also acceptable to say that 'if I know something, then I know I know this'; and in fact this statement becomes an axiom in the axiomatic formulation of epistemology that is known as epistemic logic - but note that logicians generally acknowledge that it only captures certain aspects of epistemology; one can't really say that epistemology has been reduced to a logic; this is of course typical of all foralisations.

In Suhrawardis philosophy of Illuminationism (Ishraq) true knowledge is that which becomes 'visible' or 'illuminated'; (his theory goes much further than this as it is also a theory of ontology); this form of knowledge which is intuitive, immediate and atemporal is contrasted against that of cognitive reason (aql); in this theory, someone who cognitively understands the formal rules of manipulation of arithmetic does not neccessarily have intuitive knowledge - that is true knowledge.

There are obvious parallels here with Descarte who held that we could be sure of an idea when we had a 'clear and distinct' notion of it.


I'd say no. You can have a correct stance on a position without any understanding on WHY that stance is correct. You may also arrive at a correct answer through incorrect reasoning/faulty data.

Neither of these change the fact that the final assertion is still true.

Example: Rainbows are caused by light refraction through moisture in the air. I don't have direct knowledge of this statement being true (because I have never done the math/experiments to verify), but I take it as a true statement due to the large number of smart people/science books that proclaim it to be so.


I would say that it is not a paradox and actually that the statement is neither true nor false. It seems there is a "free variable" at play, even though it is hidden: 1 + 1 = 2 is not always true. While in our "normal" mathematics, in our "normal" field, it is true, the result is different in Z2, which is the field commonly known as the bit field. In the bit field, there are only two numbers: 0 and 1. And 1 + 1 = 0.

Maybe the question you are asking is something like this: "P ∨ ~P is a tautology, but I do not know it [is a tautology]." Well, that is certainly a plausible statement. It could indeed be true. If a Chinese actor who doesn't happen to speak English reads this from a script, everything he said is true, even though he has no idea what he said. The meaning is literally true to us English speakers, but it is merely noise to everyone else.

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