How to analyse the fact that , at the age of 6,

(1) I knew that : 1 + 1 =2

but, at the same time,

(2) I did not know that 1² + 1² = square root of 4 ?

Can someone know the first thing without knowing the other? But are there really two different things to know here? Are not these "two" truths the same truth?

  • 4
    These are not "the same thing". One is a consequence of the other (given logic and axioms of arithmetic), and non-ideal reasoners never know all the logical consequences of what they know. The issue is known in epistemic logic as the (lack of) logical omniscience, and we have posts discussing it, see e.g. Is the problem of logical omniscience intractable?
    – Conifold
    Apr 1, 2019 at 3:11

2 Answers 2


The problem is not unique to knowledge. The problem is sometimes called "Frege's Puzzle", and it applies to a variety of mental states. For instance:

Lois believes that Superman is strong. Lois doesn't believe that Clark Kent is strong.

Lois wants to meet Superman. Lois doesn't want to meet Clark Kent.

Lois is curious who Superman is. Lois isn't curious who Clark Kent is.

Lois hopes Clark Kent will succeed as a journalist. Lois doesn't hope Superman will succeed as a journalist.

(where Superman is identical to Clark Kent.)

Frege's idea was that there's a difference between sense and reference. 'Superman' and 'Clark Kent' refer to the same person but they have different senses (or 'modes of presentation'); that is, they refer in different ways. The same goes for '2' and 'the square root of 4': they refer to the same number, but in different ways. Frege thought it is possible to have different beliefs/desires/etc about two concepts with different senses, even when they have the same reference.

There are various attempted solutions to this puzzle. SEP's entry on propositional attitude reports provides a good discussion.

  • 1
    Thanks for the clear analysis of the question and for the conceptualization in terms of sense/reference distinction.
    – user37859
    Apr 1, 2019 at 9:32
  • But Superman isn't identical to Clark Kent in reality (only in fantasy).
    – Bread
    Apr 8, 2019 at 23:55
  • @Bread That doesn't really matter, which you should realize if you understand the answer. Similar cases can be generated with 'Obama' and 'The 44th president of the US', 'Wittgenstein' and 'The author of the Tractatus', etc.
    – Eliran
    Apr 9, 2019 at 1:00

It always comes down to semantics with these problems. We have to be very careful with how we define the words surrounding "knowledge," because its very easy to be inconsistent.

If one accepts the most popular philosophical definition of knowledge, "Justified True Belief," then justification is an important part of the puzzle. If, at the age of 6, you lack the mathematical background to justify saying "1² + 1² = square root of 4," then you would not know it. You may know a semantically identical thing, "1+1=2," but without the justification which shows that they are the same, there's no issue with consistency in the statements.

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