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In the sense of why the Barber in the Barber Paradox doesn't go mad or enter an infinite decision loop. What makes our minds paradox proof?

Can an artificial intelligence be made paradox proof?

I know that the exact functioning of mind is not known yet, but I'd appreciate any insight scratching the surface.

  • Im not sure our minds are paradox-proof. We can handle the barber paradox because it is contrived and something o a trick with words, but I doubt we could handle a true paradox. This would require the existence of true contradiction, something that has never been found. Indeed, one of the argument for this being a mind-created world is that it is free of contradictions. But your question is still interesting since a machine may not be able to see through paradoxes like the barber whether it really is one or not. . – PeterJ Apr 15 '18 at 12:25
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I think this is a good question, also it may be a bit too broad.

One philosophical movement that concerned itself with such questions was existentialism. However, existentialism doesn't assume that the human mind is paradox immune, and instead accepts madness as one possible outcome. But of course, the paradox has to be relevant to the individual in a way "concerning its existence". In a way, the human mind is mostly immune to irrelevant paradoxes. (There exists psychological disorders which impact this ability of the human mind, and it makes life much more difficult for the affected persons.) It is certainly a good idea to program artificial intelligence in a way that it doesn't hang on the first irrelevant inconsistency. This may not be trivial, but it certainly isn't impossible.

More recently, Roger Penrose tried to argue that computers can't reproduce the abilities of the human mind, because they can't overcome paradoxes by reflection like a human mind. In this context, my guess is that human mind can overcome paradoxes, because it has an infinite history and past experience to draw from, which make it different from an idealized computer (where the infinite past experience was intentionally omitted from the idealized model).

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    You don't make clear what the existentialism has got to do with paradox. Maybe you can also add some sources on Penrose? – iphigenie Sep 24 '12 at 11:14
  • Thanks, Just wondering about deterministic, no runtime-exception design of human mind. At least we can "wonder" about a paradox. – user2411 Sep 24 '12 at 12:11
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    @iphigenie My view on existentialism is based on reading some works of Camus, Kierkegaard and Dostoyevsky. They are not concerned with logical paradoxes, but with what an individual perceives as paradox (or absurd). With respect to Penrose, I'm thinking about his book "The Emperor's New Mind" (and the follow up discussions including his books "Shadows of the Mind" and "The Large, the Small, and the Human Mind"). – Thomas Klimpel Sep 24 '12 at 13:02
  • Great answer, but I don't think you need to bring up Penrose. As Hilary Putnam says in his review of "Shadows of the Mind" Penrose's books are based on "an argument that all experts in mathematical logic have long rejected as fallacious." – Wandering Logic May 1 '14 at 23:20
  • @WanderingLogic The fact that Penrose's arguments why computers in principle can't reproduce the abilities of the human mind are not convincing doesn't change the observation that todays computers are in fact unable to reproduce the abilities of the human mind. My answer already indicates that my guess is that this is not a question of computational power (as Penrose suggests), but of "past experiences". An old slow computer with the software you need for your job can be more useful than a new fast computer missing the required software. – Thomas Klimpel May 2 '14 at 13:32
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The key to our being immune to paradoxes is that we recognize and abandon them. Therefore, so long as we can program artificial intelligence to recognize when something is a paradox, the AI should not have difficulty with paradoxes.

The first time someone considers "this statement is false", they'll eventually come to the conclusion that it's a paradox; then they will stop trying "true then false then true then false..." So, as long as a computer can do the same, it really won't have issue with paradoxes.

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Well, you can always limit the number of iterations or timeout to recover from "infinite loops" in decision procedures. You may not get the correct answer in finite time if there exists one (Goedels incompleteness), but you can always recover from paradoxes by not answering them :)

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There is no such thing as a paradox in nature. Paradoxes in logic often arise from removing the temporal element from reality, attempting to flatten the universe into a time-invariant model of truth. In so doing, time-varying phenomena which are possible in nature become impossible, i.e. paradoxical in our model. The limitations are in the model of using a flat logic to model higher-dimensional realities. Contradictions of this sort can be visualized in similar fashion as the impossibility of non-overlapping arcs in a K-connected planar or N-dimensional graph for certain infinite set combinations of K and N.

Regarding whether a purported AI could achieve this same capability, it would have to rely on a model fully adequate to overcoming the shortsightedness of painting itself rationally into a corner using an insufficient model of reality.

However, as we try to get a machine to approximate limitless human creativity, I believe it would be impossible to impute all of the wisdom and introspection regarding such blind spots without constant human intervention, model correction, and introduction of algorithms for handling such inconsistencies as they arise and become humanly detectable.

  • +1 Good point about flattening the universe into a time-invariant model of truth. – Frank Hubeny Apr 14 '18 at 17:16
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We're paradox free because of the differing registers of both rational & irrational thinking we're capable of.

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If you hear a nice paradox like "this sentence is wrong" for the first time, you think "well, if this sentence is correct, then it is wrong because that's what it says". Then you think "But if the sentence is wrong then it is correct". (You can argue this two different ways: If the sentence is wrong, then its statement "this sentence is wrong" agrees with reality, so the sentence is correct. The other way: If the sentence is wrong that means it's statement "this sentence is wrong" is wrong. Since "this sentence is wrong" is wrong, the opposite "this sentence is correct" is correct).

Now you can start with the same argument again. But you are not going to do this infinitely, because one of two things happen: Either you figure out that after two steps of the argument you come back to the original, so you end up saying: "'This sentence is wrong' is a very strange sentence, because whether I assume it's right or wrong, I always conclude the opposite". Or you don't figure it out, but you get bored with the whole situation, and you say "This is too difficult for me, I've had enough of it, I'll think about something else".

Anyone trying to create a high-powered artificial intelligence would be aware of this and make sure that in difficult cases the AI doesn't get stuck in an infinite loop but either detects that there is a loop, or just gives up. Same as a human would do. So asking such an AI "is the sentence 'this sentence is wrong' right or wrong" wouldn't result in a smoking pile of metal containing a former AI, it would result in a quick answer "This is a paradox, because whatever I assume leads to the opposite" or a slower answer "I thought about it very hard for five seconds, and I can't figure it out".

(What would be hard is to decide when the AI should give up. This sentence looks easy enough, so you could think that if the AI can't figure it out in five seconds, it never will. But if you ask an AI to figure out a way to solve the problem of food supplies in third world countries, that's a hard problem, where finding a solution is expected to take a long time, so the AI should be allowed or allow itself to think about it for a very long time).

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The barber paradox is easily resolved: Assuming the initial claim (the statement about the barber) to be true leads to a contradiction, therefore the statement is not true (that is, it is not true that the barber shaves all those and only those who are not shaving themselves). Of course that's not the only way to resolve the paradox; another resolution would be to declare that there's no barber (that's essentially how ZFC resolves Russell's paradox), therefore any statement about the barber is vacuous. Indeed, quite a few paradoxes can be resolved by simply assuming some information given when telling the paradox is false.

There are a few paradoxes which cannot be resolved that way (like the "distilled" version of the liar paradox, "this sentence is wrong"). Such paradoxes are "resolved" by understanding that there are no resolutions, and then simply ignoring them, as they are not relevant for our decisions anyway (remember, our rational thinking ability did not evolve to think about logic puzzles, but to make decisions in the real world which help us to survive).

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The human brain does not go through every logical option when making a decision. Doing so takes too long and results in sub-optimal decisions. Neuroeconomics suggests that (1) our decision making process follows a diffusion approach, where data accumulates until one option passes a threshold, and that option is taken. It also suggests that we rely on emotions as a heuristic. Because if that, we might make what appear to be paradoxical decisions, but because we don't parse through every option until we find the right one, our brain doesn't fall into the infinite loop issue.

That being said, it is possible that the universe itself is not logically consistent. While inconsistent mathematics may seem like a purely academic line of thinking, it is possible that the universe itself has actual paradoxical conditions. Take relativity and quantum mechanics. They are incompatible, and we are trying to find a grand unification theory. But maybe there isn't one. Maybe the universe obeys both, and we just live in a logically inconsistent universe where both of these theories, or something very close to them, are indeed true.

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