Show that 1+1+1=3

1+1+1 = (1 + (1+1)) = (1+3) =3

The mistake in the inner bracket calculation is that I considered 1+1 to be equal 3, and that of the outer bracket is that is that I considered 1 plus 3 to be equal 3 again.

Would the above constitute as a Gettier paradox, or simply, a misapplication of deductive reasoning?

What is the Gettier Paradox?

I think it's best understood by the following excerpt from wikipedia

In a 1966 scenario known as "The sheep in the field", Roderick Chisholm asks us to imagine that someone, X, is standing outside a field looking at something that looks like a sheep (although in fact, it is a dog disguised as a sheep). X believes there is a sheep in the field, and in fact, X is right because there is a sheep behind the hill in the middle of the field. Hence, X has a justified true belief that there is a sheep in the field.

In analogy to here, the inner most bracket computation and outermost bracket is like thinking they dog in the sheep's clothing is a sheep, and then, the ultimate fact that I get three is like there actually being sheep in the field.


  1. The point is, if I were to use inductive or abductive reasoning, then even application of correct reasoning doesn't neccesiate correct result.

  2. Also, suppose I did a deduction with no mistakes, then I'd imagine that Gettier would never occur. Hence, for title question, only the case of double negative in the deduction need to be considered.

  • 2
    Since a Gettier case occurs when we have a justified true answer to a question, but the truth of the answer is accidental modulo the justification, we'd have to see if mis-adding the terms is yet sufficiently justifying or not. Would we be justified in adding 1 + 1 to get 3 in the first place? E.g. we seem to be trying to use an associative condition to bracket the addends, but if we misapply that condition, is our grasp of the condition itself justified? Commented May 25, 2023 at 13:46
  • I mean, even if we wanted to be overly pedantic about the exact phrase "Gettier case," we could probably go ahead and just define some phrase "quasi-Gettier case" and cover your example by that even if the more "technical" base definition doesn't match. It's not likely at all that the formal concept of a Gettier case must work in only one way. Commented May 25, 2023 at 13:51
  • 1
    I think you have misunderstood what Gettier cases are about. They all involve two contexts. In one of them (the context of the subject's belief/knowledge) a given proposition is false but justified. In the other (the context of objective truth, known only to the reader), it is true and justified. You don't have two contexts at work in your example, so it is not a Gettier case. It is simply a mistaken calculation. You may believe your conclusion, but your belief is false, so there is no question of it being knowledge and no question of it being a Gettier case.
    – Ludwig V
    Commented May 25, 2023 at 17:54
  • Could I not be absolutely convinced that my mistake calculation is actually right? For instance, I have seen many times people get right answers through two answers in Physics, and they think they did it right completely since it is right at the end. @LudwigV Commented May 25, 2023 at 17:56
  • 1
    Actually, understanding what a good, as opposed to conclusive, justification is difficult. Or rather it's difficult to articulate general criteria. Specific examples (like seeing what looks like a sheep) aren't hard to think of.
    – Ludwig V
    Commented May 26, 2023 at 6:57

1 Answer 1


Since questions about proper logical forms/procedures can themselves be open to a startling extent, let us construct an example like so:

Mark is sympathetic to doubts about the universal legitimacy of the excluded-middle law (LEM). One day, Mark is studying an obscure mathematical problem X. He comes up with various (epistemically) possible answers, one of which, y, he derives using double-negation elimination (DNE). He doesn't know that DNE usually requires (or confirms) LEM, so he doesn't know that his argument is unstable on that background level. At any rate, he thinks the derivation of y is stronger than his derivations of other options. Moreover, as it just so happens, y can be derived for significantly different reasons that are much more stable, even with respect to Mark's LEM hesitance. Finally, y happens to be the correct solution to X. Is Mark justified in accepting y, and if so, is his true belief a Gettier case?

For more on the question of mathematical Gettier cases, see the linked-to work by Neil Barton, an established set theorist (apparently this essay was published about a week ago!).

  • True blue philosopher, whoever wrote what was quoted. There's the same error - that Gettier made - in it as well.
    – Hudjefa
    Commented May 26, 2023 at 6:02
  • Would this be an example of synchronicity? :P Commented May 26, 2023 at 9:29

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