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What is this thing...? Lecture 8. Puzzles about evidence
How the problems about evidence look initially: we have all sorts of questions, and theories that might answer them, and we look for data that confirms some theories and disconfirms others.
Why did dinosaurs like Tyrannosaurus rex go extinct? Why do some inherited traits 'skip a generation'? Why substances containing iron often red? What is the charge on a single electron? How many teenagers smoke cigarettes? etc.
If you are a logical empiricist, all this sort of thing reduces to a search for generalizations (or 'laws'). The generalizations are (or might be - there was some uncertainty...) about patterns in our experience. And the problem is fundamentally one about using particular cases to confirm or disconfirm
the generalizations. You have seen a lot of black ravens, and none of any other colors. Does that confirm the hypothesis that all ravens are black? If so, how does this work?
* I think just about everything in the paragraph is a mistake. But it is interesting to think about the relations between generalizations and cases that fit (and don't fit) them. This is one part of science.
Today: Ravens problem and Goodman's problem. Two problems that can be (and were) presented with very simple toy cases, but have deep implications.
What might an "inductive logic" look like? Perhaps: every observed case of an F that is also G provides some support for the generalization that all Fs are G. That looks like a start.
Ravens Problem (Hempel) Three assumptions that each look fine, considered individually:
(1) All observations of particular black ravens confirm the generalization that all ravens are black.
(2) Any evidence that confirms a hypothesis also confirms any hypothesis that is logically equivalent to it.
(3) "All ravens are black" is logically equivalent to "All non-black things are not ravens."
* Logical equivalence: If two sentences are logically equivalent then it's not possible for one to be true and the other to be false. ("Possible" is meant here in the broadest sense). "Sam is here and Jo is away" is logically equivalent to "Jo is away and Sam is here." They say exactly the same thing, using different forms of words.
Put the three assumptions together. A white shoe is a non-black thing, and a non-raven. So it is an instance of this generalization: "All non-black things are not ravens." But that hypothesis is logically equivalent to the hypothesis that all ravens are black. So a white shoe confirms the hypothesis: All ravens are black.
You can either accept the conclusion or discard at least one of (1)-(3) above.
* A bit more terminology. An 'instance' of a generalization is a case that fits it.
I.J. Good: reject (1). Whether an instance of a generalization confirms it depends on background knowledge.
Various people, including me: reject (1). It depends on the way the observations were made. Was there a genuine test? See T&R.
You can also reject (2) or (3).
Or you can accept the conclusion that observations of white shoes confirm that all ravens are black.
Goodman's problem
* Introduce it in reverse order from Chapter 3 of T&R. A familiar practical problem in science. Curve-fitting problem. Suppose you have x, y points. (Might be: x = years of education, y = lifetime income; or: x = fat level in diet; y = heart attack risk....).
For any set of points, there is more than one curve that will fit the set (either exactly or, as in the figure, approximately). How should you extrapolate? What should you expect for the presently unobserved case when the value of x=5?
Replies. Simplicity? Certainly used in practice. Why should a simpler curve be preferred? Role of other background assumptions? (These can override or support simplicity - eg., you might believe the increase cannot continue linearly forever).
Nelson Goodman: this problem is everywhere, and simplicity does not help (perhaps not much, perhaps not at all). Fact, Fiction, and Forecast, 1955.
His own set-up. Immediate goal: to show that there can be no "formal" theory of what makes some inductions good and some bad.
Deductive logic: goodness of an argument is entirely a matter of its form, not its particular content.
All Fs are G. Premises a is an F . a is G. Conclusion
The argument is good no matter what you put in for "F" and "G" (as long as you put in the same thing consistently). Goodman: induction does not work like that.
All the emeralds observed prior to the year 2019 have been G. All emeralds are G Don't worry about the fact that the conclusion should be qualified; this is not a deductive argument. You can add 'probably' or whatever you like.
But now:
An object is grue if and only if it was first observed before the year 2019 and is green, or if it was not first observed before 2019 and is blue.
* Note that grue objects do not change color in some mysterious way. Lots of ordinary objects are grue.
The argument above seems fine when G=green but bad when G=grue. But the form of the argument is the same. The same data (the same pile of emeralds) seems to point us towards two contradictory conclusions, depending on how we describe the emeralds (as green or as grue)
That is stage 1 of Goodman's problem. Good and bad inductions can have the same form.
Note that "grue" works perfectly well in deductive arguments. Induction works differently. But how?
Can we hang onto the idea that as we seem more and more cases that fit a generalization, the support for that generalization increases?
Can we exclude inductions using the term "grue" because it is defined in a way that makes reference to time? Goodman: no.
Another new word: an object is bleen if and only if it was first observed before the year 2019 and is blue, or if it was not first observed before 2019 and is green.
We can use the English words "green" and "blue" to define "grue" and "bleen."
If we do so we must build a reference to time into the definitions of "grue" and "bleen." But suppose we spoke a language that was like English except that "grue" and "bleen" were basic, familiar terms and "green" and "blue" were not. Then we would need to define "green" in a way that involves a reference to time. (A green object is one that was first observed before 2019 and is grue, or was not first observed before 2019 and is bleen).
Goodman's own response to all this: Goodness of inductive arguments is language-relative. Simplicity, also, is language-relative.
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More Back to the comparison with the curve-fitting problem. Can cast that problem as a form of Goodman's problem.
All the many combinations of fat level in diet and heart attack rate I have seen have fit the relationship y = mx +c, so I expect future cases to fit that pattern.
All the many combinations of fat level in diet and heart attack rate I have seen have fit the relationship y = ax^2 + bx + c, so I expect future cases to fit that pattern.
In the curve-fitting case, we start with a sort of "raw" description of the data (x and y values) and we look for a relationship between them. Goodman: even if that works in the curve-fitting case, it does not help generally. We have to start out with some categorization of the cases we observe, and that categorization (green, grue) will push towards some extrapolations and away from others. (You could say: "all the emeralds I have seen have been grue, so I expect the next one to be green." But why extrapolate to a different property from the one you have seen to fit your data?)
How about simplicity? This seemed to help with curve-fitting. Goodman: simplicity is language-relative. If you speak ordinary English, a description in terms of "green" will look simple. If you speak a different language, a different description (perhaps one in terms of "grue") will look simpler.
(Does this also undermine what we said about the curve-fitting case? That is a tricky problem. A straight line is objectively simpler than lots of other functions. But whether you get a straight line on a graph depends on how you categorize the observations. E.g., logarithms can turn straight lines bent, and vice versa.)
Main replies: (1) Both the grue-emerald and green-emerald inductions are OK as arguments. We can make a choice between the two conclusions on other grounds, though. For example, the hypothesis that "all emeralds are grue" has a low prior probability (a low initial credibility). So it is still not something you should believe. (End of course - we return to ideas like this.)
(2) Goodness of inductive arguments is language-relative.
(3) The role of "natural kinds." The category grue is not "natural.
(Psychiatric categories as an interesting problem case.) I don't accept any of these.
(4) I think the right answer is related to the right answer to the ravens problem. The relevance of an observation (including instances of generalizations) depends, in many cases, on the procedures being followed and other aspects of the background. (More exactly, there is part of the grue problem that is related to the ravens problem and another part that involves simplicity or something related.) All this is wide open.
See Chapter 14 of T&R, or "Induction, Samples, and Kinds" on my website. See also the book edited by D. Stalker called Grue! Frank Jackson's 1975 paper"Grue" is good.
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