by Carl Pierer
The old problem of induction raised the question of how we can justify inferences from singular observations to general statements. In the last century two newer problems were presented. The Ravens Paradox, which I will explain in section II, is due to Carl G. Hempel. The ‘grue problem' was put forth by Nelson Goodman and I shall present it in section III. In section IV I will first compare the two problems and then attempt to show that Hempel's paradox can be solved, whereas Goodman's ‘grue' points to a deeper problem.
Carl Hempel focuses on the question of what counts as evidence for hypotheses. There are two principles of inductive reasoning that we seem to accept.
(i) If all Ps have been observed to be Qs, then this counts as evidence for "All Ps are Qs". Hempel writes: "(…) this hypothesis is confirmed by an object a if a is P and Q; and the hypothesis is disconfirmed by a if a is P, but not Q." (Hempel 1945, p. 18)
(ii) What counts as evidence for one statement, counts as evidence for all logically equivalent statements. So, if all non-B's have been observed to be non-A's, then this counts as evidence for the original statement. It seems plausible to accept the second principle. The statements are equivalent exactly because they are true under the same conditions.
Hempel thinks that this account entails a paradox, namely the Ravens Paradox. Say we start off with the hypothesis: "All ravens are black". We can then construct its logical equivalent: "All non-black things are non-ravens". By (i), everything that we can observe that is not a raven and not black will count as evidence for the second statement. By (ii) we are bound to accept these evidences as evidence for the initial hypothesis. But this seems absurd. We do not believe that my brown jumper does in any way affect the hypothesis that all ravens are black. This fact seems completely irrelevant.
Nelson Goodman formulated a "new riddle of induction". In order to construe it, he needs to come up with a new predicate: the infamous "grue" (Goodman 1954, p 74). Objects are grue if they are observed before time t and are green, and if they are observed after t and they are blue. Now, t is some time in the future, say the 1st November 2020. So, if we have a green emerald before time t, then this certainly supports the hypothesis "All emeralds are green". However, we can also take it as evidence for "All emeralds are grue". The issue, then, is to determine whether we have any reasonable grounds to prefer thinking of emeralds as green over emeralds as grue.
One might object that we have no reason to believe that emeralds will miraculously change colour after the 1st November 2020. But this objection misses the point. Asserting that an object is grue is to say "this object has been observed before t and is green or it has been observed after t and is blue". So if we observe an emerald now and find it to be green, then it will be ‘grue' as well. The hypothesis "All emeralds are grue" is true, for all emeralds so far observed have been found to be green. In order to disprove the theory we would have to wait until after t and then re-examine the emeralds. Yet, this further evidence is not available to us now. Given our present state of evidence, green emeralds support the ‘grue'-hypothesis as well as the green-hypothesis.
Of course, we can point to past observations: all emeralds have been green so far, so we think to have reasons to believe that they will be green in the future as well. However, any evidence that will count in favour of the greenness of emeralds will count in favour of their grueness as well.
These two problems focus on the question of what evidence is and what evidence we consider to support our theories. They avoid the "old question of induction". They do not ask why we should believe in inferences from singular observations to general statements. Rather, they shift attention to the question which singular observations support which general statements. In this section I shall discuss the two problems and what they entail.
Concerning his paradox, Hempel argues that the reasons why we consider this to be a paradox are merely psychological (Hempel 1983, pp. 25). It is not because there is any logical fault with the hypotheses involved but only because we have certain background knowledge that allows us to determine which evidence to count as relevant. He shows that statements such as "All P's are Q's" say something about their logical equivalents. Such statements divide the world into three classes (Hempel 1983, p. 26): (1) P's that are Q's, (2) P's that are non-Q's and (3) non-P's that are non-Q's. We count any member of (1) as a supporting evidence for our hypothesis. If (2) has any member at all, we consider the hypothesis to be refuted. Hempel thinks that any member of (3) should indeed count as evidence and that we lack reason to oppose this.
Since we have some prior knowledge about members of (3), we know at the outset that the hypothesis "All non-black things are non-ravens" is well supported. Re-finding members for (3) does not further support the hypothesis, exactly because we know them beforehand to be members of (3). Therefore, we consider the idea that they should support a statement about (1) paradoxical. Hempel uses an intriguing example to make this point clear: Assume we have the statement "All sodium salts burn yellow". Now we have a block of ice and hold it into a flame. We do not count this as positive evidence for our hypothesis even though the flame does not turn yellow. Hempel thinks this is because "(…) we happen to <<know anyhow>> that ice contains no sodium salt (…)" (Hempel 1945, p. 27). And since we already believe the hypothesis, we do not consider to have found anything new.
However, say we were to test a substance ‘a' of unknown constitution (ibid.). As it happens, the flame does not turn yellow and analysis of ‘a' reveals that it does not contain sodium salts. We have then found a new member for (3). Initially, we did not know to which class ‘a' belonged. It might have turned out to be sodium salt that does not burn yellow, hence a member of (2). So finding ‘a' not to be member of (2) supports the initial hypothesis. We are then compelled to count the fact that ‘a' is not a sodium salt and does not burn yellow as evidence for "all sodium salts burn yellow".
Therefore, Hempel's paradox turns out to be no paradox after all. The problem that persists, however, is that we have to decide which theory is the more interesting and the more informative one. Assuming that we do not yet consider "All non-black things are non-ravens" to be well confirmed, it is difficult to introduce reasons why this statement should be preferred to "All ravens are black".
This is where the ‘grue-problem' ties in. Goodman points out that we lack reasonable grounds for projecting green rather than ‘grue'. The ‘grue-problem', as the Ravens Paradox, demonstrates that background knowledge plays a crucial role in our decision of what we count as evidence. But, while it was possible in Hempel's case to resolve the problem by an appeal to ignore background knowledge, this will not help in the ‘grue' case. On their own, we have found evidence confirming the green-hypothesis to simultaneously confirm the ‘grue'-hypothesis. However, invoking background knowledge to resolve the ‘grue-problem' is not an option either. As John Vickers (Vickers 2013, SEP) points out, it is evident that the ‘grue'-hypothesis is extremely unlikely. But this is the very point of the problem rather than its solution. Goodman shows with this problem that it seems as if any observation can be counted as evidence for any theory whatsoever. (Goodman 1954, p. 75)
In this essay I first presented the two modern problems of induction, the Ravens paradox and the grue-problem. Then I discussed how the two relate to each other. I found that they both raise questions about the status of evidence, but that they approach the concept from different angles. Hempel's paradox pointed to the question "What can count as evidence for a theory?" (Goodman 1954, p. 81) Goodman, in contrast, raises the question: "What hypotheses are confirmed by their evidence?" (ibid.) Eventually, I tried to establish that the two differ crucially, since the ravens paradox is only apparent and can be solved whereas the grue-problem poses a "new riddle of induction" that persists.
Goodman, N. (1954). Fact, Fiction & Forecast. London: University of London – The Athlone Press.
Hempel, C. G. (1983). Studies in the Logic of Confirmation. In P. Achinstein, The Conecpt of Evidence (pp. 10-43). Oxford: Oxford University Press.
Vickers, J. (2013, March 21). The Problem of Induction. Retrieved from Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/archives/spr2013/entries/induction-problem/