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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.

Notes

  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.

Reine Abstraktion
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    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? – Kristian Berry May 25 '23 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. – Kristian Berry May 25 '23 at 13:51
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    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 May 25 '23 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 – Reine Abstraktion May 25 '23 at 17:56
  • You could be, but that wouldn't make it a Gettier case. – Ludwig V May 25 '23 at 17:59
  • You might have a better question if you stated what is a Gettier paradox. – Boba Fit May 25 '23 at 17:59
  • Edited @BobaFit – Reine Abstraktion May 25 '23 at 18:16
  • Being convinced, absolutely or otherwise, is a psychological attitude, it does not amount to justification. You can convert your example into a Gettier case, but that would require inventing a scenario where the false lemma (1+1=3 in your case) would appear justified to an impartial rational agent (you are probably better off with a different false lemma). As such a justification cannot be deductive from true premises, it is impossible to produce a Gettier case with deductive reasoning alone. – Conifold May 25 '23 at 19:20
  • Wouldn't person thinking the dog looking like sheep also have a similar issue @Conifold – Reine Abstraktion May 25 '23 at 19:52
  • I don't understand your analogy at all. However, you are right that Gettier cases get the right answer for the wrong reasons. But that is not enough to call the JTB account of knowledge into question. You need a reason that is good, but not conclusive, such as seeing a sheep-like something in a field. Deductive reasoning is all or nothing, that is, it is conclusive or wrong, so it can't provide what you need. – Ludwig V May 25 '23 at 20:21
  • Alright, so I understood, gettier can't ever occur in deductive reasoning, and when deductive reasoning is misapplied that is not gettier. I guess my main pain point was understanding what constitutes as a good justification. Thanks for your time @LudwigV – Reine Abstraktion May 25 '23 at 21:28
  • You're welcome. I'm glad we could sort it out. You can have deductive reasoning, but if you do, you need false (but reasonable) premisses. If you look up Gettier's original examples, you'll see an example in his second case. I'm sure it'll be in Stanford Encyclopaedia of Philosophy if it isn't in Wikipedia. – Ludwig V May 25 '23 at 21:37
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    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 May 26 '23 at 06:57

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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!).

Kristian Berry
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