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I have been thinking about the following dialogue for some time. It consists of some arguments I've come up with for both sides, and includes the idea of Russel's teapot.

Suppose X is a proposition which

1) Implies various propositions about the physical universe.

2) There is no empirical justification for X

3) There is no empirical justification that X is false

4) Suppose there are countably many mutually exclusive propositions, one of which is X, and each of which satisfy (1), (2), and (3).

Fred Argument 1:

Fred: It's about time you admitted X is false!

George: I require empirical justification to believe this proposition that X is false. There is none.

Fred: But surely you believe that there is no teapot in orbit of Jupiter? I intuit that this is absurd, yet we agree that there is no empirical justification for or against it, just like proposition X. Do you not then believe that proposition X is false?

George: I believe that there is no teapot in orbit of Jupiter, but not that proposition X is false. I would be justified in saying that such a teapot does not exist because someone would have had to put it there. In this respect, this claim is unlike proposition X.

Fred: How do you know that there is not a non-man-made teapot orbiting Jupiter?

George: I would infer by the laws of thermodynamics that the odds of a teapot assembling by chance alone are quite small.

Fred pauses to think.

Fred Argument 2:

Fred: Ok. I agree that my former argument cannot convince you. But X is a member of a countable set of mutually exclusive propositions, each with no empirical justification for or against. If one assigns a probability to each one, and chooses a threshold to represent "very unlikely", only a finite amount of propositions could not be very unlikely. From this we can infer that proposition X is probably very unlikely.

George: I object that you are not warranted in assigning a probability space to this scenario. Such probabilities would stem from the existence of evidential justification of some sort, which for proposition X there is not. To give you an example of what I mean, consider the specific proposition that there is a fairy with a wing of a certain specific color whose RGB value is (10, 80, 128).

Fred: that would look quite nice.

George: but there are countably many distinct propositions, one for each natural number: there exists a fairy with a wing of certain specific color whose RGB value is (10, 80, 128 2^(-n) ). I reject your inference that one can say anything about how likely each one is, and I do not believe that there are no fairies.

Fred: I see that this will not convince you either then. To me, it is self evident that fairies of any sort do not exist.

The question is, what position is justified on proposition X, if indeed there is enough information to take one? What philosophical views are there that weigh in on this matter? As pointed out by Conifold, An answer to this question necessarily involves giving one's view on epistemic justification. I would like not to suppose that Fred and George share such a view, but for the answer to give possible viewpoints and where they stand in promoting the arguments involved.

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    What was correct or incorrect depends on one's view of epistemic justification. It is not clear what that is here, or if Fred and George share one. They also seem to switch what they are arguing for/against. Then there is confusion between what is "best to believe" and what one is "justified to believe", and between not believing X and believing it to be false. Fred's probabilistic argument is perplexing: it does not seem to use anything specific about teapot orbiting Jupiter, which is an odd way to conclude that that is unlikely. – Conifold Aug 22 '17 at 19:58
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    Ok +1. I have made large edits modifying my question to your concerns. Please tell me if there are still issues with it. – Dean Young Aug 23 '17 at 16:13
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    I edited it to make it more clear. By "a proposition has a mutually exclusive proposition" I mean "there are two propositions, and the relation 'mutually exclusive' applies to them". – Dean Young Aug 23 '17 at 18:16
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    A set of first order formulas A in a language L is 'a mutually exclusive set of formulas' if for every two distinct formulas f, g in A, f implies not g. Make sure to note that this does not mean that for all formulas f, g in A, not f implies g. – Dean Young Aug 23 '17 at 18:49
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    This is still not clear to me. OK so 'The exact center of the universe is closest to x, y' where x and y are integer coordinates on a system centered at the center of the Earth is such a set of propositions. Then X is just one of them -- say the one that says the Earth is the center of the universe. Is that a fair example? – jobermark Aug 23 '17 at 22:06
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From a constructivist notion of logic, like that behind Intuitionistic mathematics, most statements do not need to be true or false. In particular, only those false statements that lead directly into problematic consequences should really even be considered false. The vast majority of statements are better off without a truth value. Even to give them baseless estimates of probability creates the illusion of understanding where it is really not present.

If you begin to assign false statements the kind of truth value that allows people to imagine their inverses are true, you are likely to encourage action upon a wider range of bad options. No one can prove that the Earth is the center of the Universe, and acting on that statement is unwise. But if we decide this is false, or even that we can be sure it is highly unlikely, we have emboldened ourselves to act upon the inverse with some measure of certainty. But such action is really not warranted either.

This is why Intuitionism suggests that we discard the Law of the Excluded Middle from mathematics in all cases where the set of options is not finite and identified. It serves little purpose, but to create 'truths' if no value out of whole cloth, and really says nothing productive. In your case, where we know the domain is not finite, we should refrain from deciding anything about the proposition.

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    Wow exactly what I didn't know I was looking for! I just became an intuitionist in mathematics too. +infinity. – Dean Young Aug 23 '17 at 23:27
  • I might be misunderstanding this, but are you saying we should not act, as if there was no teapot? – Chris Wohlert Aug 24 '17 at 10:51
  • If there is a specific action you wish to take that assumes it matters whether or not there is a teapot, you should not take it. Can you actually frame an action where that specifically matters? If so would it matter whether what is there was a teapot or an untraced asteroid? Can we safely assume that there is not an untraced asteroid anywhere in particular that we have not checked? Shouldn't we just plan for a certain quantity of debris, or actually look first? The idea of ruling out stupid questions is not valuable, it just feels good. – jobermark Aug 24 '17 at 14:35
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Your answer includes many common but mistaken ideas about epistemology. Your answer discusses justification: a process that allegedly shows an idea is true or more probable than alternatives or something like that.

In reality, no argument ever justifies any conclusion. If the premise of an argument is true, and the rules of inference are correct, then the conclusion is true. But unless you have some guarantee that the premises and rules of inference are correct, your conclusion is a guess.

At this point, people who believe in justification can retreat into dogmatically asserting that some things are true and that's all there is to it. In that case, this person is intellectually not much different from a creationist who insists that god made the world 6000 years cuz the bible sez so.

Another attempt to retreat from basic logic involves invoking probabilities of theories. This tactic is simply silly. Probabilities are tested using relative frequencies. Since laws of physics, biology etc are universal theories, you would need multiple copies of the whole of physical reality to test probabilities of theories, which makes no sense. All real calculations of probability involve assuming some particular theory that makes probabilistic predictions is true, e.g. - quantum mechanics. You then predict the probability of an event from a theory, not the other way around. The theory is not an event and so doesn't appear anywhere in the event space used to make the prediction and can't be assigned a probability. And any way of assigning probability would just be a guess unless it was justified as certainly true, which is impossible, so at best you would have a guess at the probability.

Karl Popper explained the solution to this problem: all knowledge consists of unjustified guesses controlled by criticism. Yoy notice a problem, guess a solution and criticise the guess. The criticism may include experimental and logical tests, as well as looking at whether the theory is consistent with other knowledge. The criticisms aren't justified either. They, too, are just guesses that can be controlled by further criticisms. See "On the sources of knowledge and of ignorance" in "Conjectures and Refutations" and chapter I of "Realism and the Aim of Science" by Popper. Also worth reading are "The Fabric of Reality" chapters 3 and 7 by David Deutsch and chapters 1-4,10,13,15 and 16 of "The Beginning of Infinity" by Deutsch.

The X in your question includes every idea ever created by anyone. So listing those properties isn't much good as a way of distinguishing between ideas. Is there a teapot in orbit around Jupiter? No. That is inconsistent with epistemology. The knowledge required to make a teapot and put it in orbit around Jupiter requires many rounds of guesswork and criticism. This hasn't happened, so the teapot doesn't exist. Fairies don't exist either. All alleged recordings of fairies have been refuted. And if there is another species capable of creating explanatory knowledge like humans they would be having a noticeable effect on the rest of the world, e.g. - they would be building power stations to help members of their species live in comfort. They would also want to trade with us, so it would make no sense for them to hide.

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