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According to Bayesian inference/confirmation theory, your confidence in a hypothesis increases as you observe more and more evidence predicted by that hypothesis (according to bayes theorem and the laws of the probability calculus). These updates happen in a very specific way. Although there are some other problems associated with this version of confirmation theory, this is at least in principle quite intuitive.

However, in the face of a grue hypothesis, this idea falls apart. I could suggest some hypothesis "All A's are B's". Every time there is a new instance of an A being a B, my confidence in that hypothesis improves. However, we can form a grue-like hypothesis: "All A's are B's until time t". This hypothesis predicts the same thing as the former but there is a divergence at some arbitrary time as to what really happens.

Does our confidence in these two hypotheses really increase in the same way?

Is there any way to solve this problem which doesn't amount to arbitrary selection of prior probabilities to favour the former type of hypothesis?

  • The more or less standard solution is to restrict the class of predicates allowed to be featured in a hypothesis. Swinburne restricts it to qualitative predicates, like green, that can be tested for without knowing spatiotemporal location of the sample. This already rules out grue. Quine restricts it further to "natural kinds", which he then attempts to define in terms of "similarity", see New Riddle of Induction. – Conifold May 18 '18 at 20:43
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    There has been a lot of wok on this question. E.g., Sober, "No Model No Inference: A Bayesian Primer on the Grue Problem". fitelson.org/confirmation/sober_grue.pdf – Mark Andrews May 19 '18 at 2:32
  • @Conifold I've just read Swinburne's paper 'Grue' where he outlines the difference between 'green' and 'grue' in terms of being a qualitative predicate. I understand and agree with the distinction. Though, what about this distinction means that we should prefer hypotheses that project green over hypotheses that project grue? I'm new to reading up on these problems by the way, so bare with me if the answer to that question is somehow obvious. – Joe Lee-Doktor May 19 '18 at 11:29
  • According to Swinburne, only qualitative predicates are projectible for the purposes of law-like generalizations that are subject to confirmation. So we should apply Bayesian approach to hypotheses that use them only. In a sense, this is Swinburne's formalization of what Hume and Mill called "uniformity of nature", and what, according to Goodman, we evolved to do instinctively by association. – Conifold May 22 '18 at 5:18
  • @Conifold Yeah, okay. I thought that was what Swinburne was arguing but I can't remember if what he wrote in "Grue" made it very explicit that he thinks some principle of uniformity should just be applied. Many thanks. – Joe Lee-Doktor May 22 '18 at 7:17
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I think you have at least 3 hypotheses.

h1. All A are B. 
h2. h1 will be correct until at least time t. 
h3. At time t some A will stop being B.

Your confidence in h1 will increase as you discover A that are B (and fail to discover any A that are not B)

Your confidence in h2 will increase as you approach time t and fail to observe any A that are not B

You may never observe any evidence of A becoming not B before t, so, in the absence of evidence, your confidence in h3 remains the same.

  • I have considered this 'solution' before. Couldn't you equally say that you have these hypotheses: h1. A's are B's until time t h2. A's are B's after time t h3. A's are something else after time t. Or am I misunderstood such that your answer is more 'proper' than what I've constructed? – Joe Lee-Doktor May 18 '18 at 20:21
  • You will never get evidence of your H2 and H3 until after t, so your confidence in them will remain the same under the inference/confirmation theory. There's no conflict, it's just that 0 evidence = 0 increase in confidence. – JeffUK May 18 '18 at 20:59
  • So only our confidence in the hypothesis "A's are B's until time t" can increase. Since 't' can be any arbitrary time, doesn't that mean that we can never gain confidence in a hypothesis that makes a claim about the future? – Joe Lee-Doktor May 19 '18 at 8:36
  • This is too simplistic. I'm hungry; I'm about to add some noodles to a pot of boiling water on the stove. As it happens, these noodles when cooked sink. "h1. All noodles are uncooked." That's true; I just put them in. "h2. h1 will be correct until at least time t." t varies here, but roughly 8 to 9 minutes in the noodles start to sink every time I've done it; however, these noodles aren't done yet. "h3. At time t some A will stop being B." Well, 12 minutes in, all of my noodles are going to be cooked. I don't have to wait 8 minutes to know this though. – H Walters May 19 '18 at 15:08
  • ...and sure, this is a restricted universe (because I'm only counting my noodles here), but we could consider other forms of evidence about cosmological facts that have yet to occur (heat death, proton decay, etc). IOW, one doesn't have to wait until time t to get evidence of any generic thing. – H Walters May 19 '18 at 15:12

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