(intro music) My name is Marc Lange. I teach at the University of
North Carolina at Chapel Hill, and today I want to talk to you about
the paradox of confirmation. It's also known as the
"paradox of the ravens," because the philosopher Karl Hempel,
who discovered the paradox, first presented it in terms of
an example involving ravens. The paradox concerns confirmation, that
is, the way that hypotheses in science and in everyday life are supported
by our observations. As we all know from detective stories,
a detective gathers evidence for or against various hypotheses about who
committed some dastardly crime. Typically, none of the individual pieces of evidence available to the detective is enough all by itself
to prove which suspect did or did not commit the crime. Instead, a piece of evidence
might count to some degree in favor of the hypothesis
that the butler is guilty. The evidence is then said
to confirm the hypothesis. It might confirm the hypothesis
strongly or only to a slight degree. On the other hand, the
piece of evidence might, to some degree, count against
the truth of the hypothesis. In that case, the evidence is said
to disconfirm the hypothesis. Again, the disconfirmation
might be strong or weak. The final possibility is that
the evidence is neutral, neither confirming nor disconfirming
the hypothesis to any degree. The paradox of confirmation
is concerned with the question "what does it take for some piece of
evidence to confirm a hypothesis, "rather than to disconfirm it
or to be neutral regarding it?" The paradox of confirmation begins with three very plausible ideas,
and derives from them a very implausible-looking
conclusion about confirmation. Let's start with the first of these
three plausible-looking ideas, which I'll call "instance confirmation." Suppose that we're testing a hypothesis like "all lightning bolts are
electrical discharges," or "all human beings have
forty-six chromosomes," or "all ravens are black." Each of these hypotheses is general, in that each takes the
form "all Fs are G," for some F and some G. Instance confirmation says that if we're
testing a hypothesis of this form, and we discover a
particular F to be a G, then this evidence counts,
at least to some degree, in favor of the hypothesis. I told you this was going to be
a plausible-sounding idea. Isn't it plausible? The second idea is called
the "equivalence condition." Suppose we have two hypotheses that say
exactly the same thing about the world. in other words, they are equivalent, in the sense that they must either
both be true or both be false. For one of them to be true and the
other false would be a contradiction For instance, suppose that one hypothesis is that all diamonds are made entirely
of carbon, and the other hypothesis is that carbon is what all diamonds
are made entirely out of. These two hypotheses are equivalent. What the equivalence condition says is that if two hypotheses
are equivalent, then any evidence confirming one of
them also confirms the other. this should strike you
as a very plausible idea. Let's focus on our favorite hypothesis:
that all ravens are black. The third idea is that this hypothesis
is equivalent to another hypothesis. That other hypothesis is a very clumsy
way of saying that all ravens are black. Here it is: that anything that
is non-black is non-raven. Let me try a different way of explaining
the equivalence of these two hypotheses, just to make sure that
we're all together on this. The hypothesis that all Ravens are black
amounts to a hypothesis ruling out one possibility: a raven that isn't black. What about the hypothesis that all
non-black things are non-ravens? It also amounts to a hypothesis
ruling out one possibility: a non-black thing that isn't a non-raven. In other words, a non-black
thing that's a raven. So both hypotheses are equivalent
to the same hypothesis: that there are no non-black Ravens. Since the two hypotheses are
equivalent to the same hypothesis, they must be equivalent
to each other. Okay, at last, we are ready for
the paradox of confirmation. Take the hypothesis that all
non-black things are non-ravens. That's a general hypothesis. It takes the form "all Fs are G." So we can apply the instance
confirmation idea to it. it would be confirmed by the
discovery of an F that's a G. For instance, take the red
chair that I'm sitting on. I am very perceptive, and I've
noticed that it's a non-black thing, and also that it's not a raven. So the hypothesis that all
non-black things are non-ravens is confirmed at, least a bit, by
my observation of my chair. That's what instance confirmation says. Now let's apply the equivalence condition. It tells us that any observation
confirming the hypothesis that all non-black things are non-ravens automatically confirms any
equivalent hypothesis. And we've got an equivalent
hypothesis in mind: that all ravens are black. That was our third plausible idea. So my observation of my chair confirms
that all non-black things are non-ravens, and thereby confirms the equivalent
hypothesis that all ravens are black. Now that conclusion about confirmation
sounds mighty implausible, that I could confirm a hypothesis about
ravens simply by looking around my room and noticing that my chair, not to
mention my desk and my coffee table, that each of them is
non-black and also not a raven. I can do ornithology while remaining
in the comfort of my room. So here is the challenge that you face. either one of those three ideas must be
false, in a way that explains how we could have arrived at are false
conclusion by using that idea, or the conclusion must not in fact
follow from those three ideas, or the conclusion must be true,
even though it appears to be false. Those are your only options. I leave it to you to think about
which of them is true. Subtitles by the Amara.org community
I think the issue of the paradox has to do with the innate fuzziness of inductive arguments and a flawed understanding of what confirmation actually entails with respect to it. If one wanted to say, with absolute certainty, that all ravens are black with no appeal to definitions or characteristics of ravens (which would be a deductive argument) that statement must be confirmed by evidence. However, the existence of any raven that is not black, or any non-black thing that is not a non-raven, even if there is only one such thing, automatically overthrows the hypothesis completely and totally. Therefore, in order to have absolute certainty that your hypothesis is true from an inductive argument, you must either go through the set of all ravens and show that they are black (proving the given hypothesis), or you must go through the set of all non-black things and show that they are all non-ravens (thereby proving the equivalent hypothesis).
Of course, it is much much easier to find evidence that supports the equivalent hypothesis that all non-black things are non-ravens, non-black and non-raven things being very easy to come by. But this ease is offset by the fact that there are so many remaining members of the set of non-black things which in turn makes any evidence you have gathered in support of this hypothesis to be very very weak indeed. The heuristics involved in checking millions of ravens versus a relatively infinite number of non-black things (a set so large that even its size is a nebulous concept) should be enough to show exactly how difficult the equivalent hypothesis is to prove with any degree of certainty
All in all, I appreciate the video very much. Thanks for sharing!
The solution seems simple; there's many orders of magnitude more non-raven things (N) than ravens (Nr << N), so observing a non-raven only confirms the proposition a tiny bit (going like 1/N) and effectively appears like nothing at all to us.
On the other hand observing a raven is more significant, because there's not so many ravens in the universe.
You could formalize this in terms of probability distributions and information.
Consider the world, or universe to have a finite number of "things". All of those things are is an set - the set of all things. Next, you have a hypothesis that all are ravens are black. You could prove the hypothesis by examining the entire set, and finding no non-black ravens. Picking one item out of the set, say a red chair, reduces the number of things to examine, thus making the hypothesis more likely to be true. Thus it has given a very small "confirmation". Really the problem is the term "confirmation". It should be evidence.
Easier to grasp that a red chair is an infinitesimally small piece of evidence that all ravens are black by the mere fact of not being a non-black raven.
Instance confirmation is a form of abductive reasoning, and cannot serve to confirm a deductive argument. Deductive claims can only be confirmed once all possible relevant evidence is obtained. Without that, deductive claims become matter of statistical confidence rather than of certainty.
Summary: In this video, Marc Lange (UNC-Chapel Hill) introduces the paradox of confirmation, one that arises from instance confirmation, the equivalence condition, and common inference rules of logic.
Thanks for watching!
Wi-Phi
Maybe I'm simple, but that just seems trivially correct and boring.
Imagine you have four face-down cards:
Q♣ K♥ 8♦ Q♠
...and the statement is "all queens in the set are black."
Flipping over the first queen, king and the eight lends some evidence to that statement because you've failed to discover a red queen. There's fewer cards left to disprove your hypothesis.
The closer you are to running out of "things" to look at, without discovering any pink ravens, the more credible that statement becomes. The only reason it's pointless is because there's too many "things" for that to make any significant impact. Define a finite set of "things" and what you're doing is obviously right. Where's the paradox?
I mean, if your sample space is a red sofa, blue minivan and a black raven...
The paradox of ravens is, why do people whose academic training should be able to identify a posteriori, inductive classifications when they see them never seem to be able to do so? The definition of "raven" is constructed inductively, and whether or not a non-black thing that otherwise behaves as a raven is a raven is a decision, not a deduction.
If you don't believe me, try this on for size: all planets in our solar system orbit the Sun in approximately the same plane. That was true until 1930 and after 2006 but was false in the interim. Nothing significant changed about our knowledge of Pluto's orbit, rather it was a reformulation of the definition based on additional evidence gathered about other things, in other words through inductive reasoning. The same applies to everything else in the universe. Can we move on?
Here's a better thought.
Suppose I said all unicorns fart rainbows. Therefore all things that don't fart rainbows aren't unicorns. So now every object I see "proves" that all unicorns fart rainbows? Does this imply that unicorns exist?
Hmm. The video linked in the media preview is wrong.