At a glance, the Raven Paradox is disturbing, but seemingly insignificant. The average reader correctly assumes that a simple but non-obvious flaw exists in the argument, and moves along. A deeper look at the puzzle, however, reveals that the Raven Paradox is not just a philosophical test-of-wits, but has implications for the way we think about observation and proof.

The Raven Paradox is a set of four plausible ideas which, together, are contradictory. The premises are as follows:

I. The empirical hypothesis “All ravens are black” is equivalent to “All non-black things are non-ravens.” To use the classic example, the statement “All ravens are black” implies that if something isn’t black, it’s not a raven.

II. Examples of black ravens confirm “All ravens are black.” Though a mere instance of a black raven doesn’t prove that all ravens are black, it certainly lends credence to (in philosophical terms, confirms) the idea that all ravens are black.

III. Confirmers of one formulation of a hypothesis are confirmers of any logical equivalent formulation. This, too, seems obvious: The phrasing of an idea shouldn’t effect how true it is; it is the concept that is the important.

IV. Non-black, non-ravens do not confirm “All ravens are black.” Simply put, a white tennis shoe does not confirm that all ravens are black: white tennis shoes have nothing to do with black ravens.

The combination of these “obvious truths” leads to an obvious contradiction. By premises I, II and III, the hypothesis “All ravens are black” should be supported by evidence of white tennis shoes — something that is preposterous, according to premise IV.

But closer inspection shows that perhaps this is not as preposterous as we may at first believe. Consider a world in which there are thirty black things and only one white. In this case, it is less conclusive to show that one of the black things is a raven than it would have been to show that the only non-black thing was a tennis shoe, that is, not a raven. Clearly, premise IV is false.

In fact, evidence of non-black, non-ravens lend greater confirmation that “All ravens are black” whenever there are more ravens than non-ravens. This point, along with the fact that counting either every black raven or every non-black non-raven can confirm “All ravens are black,” suggests an interesting fact about the nature of observation. It seems reasonable that the degree to which a hypothesis has been proved can be viewed as equal to the conditional probability that the hypothesis is true given the evidence, or P(H|E). (See A Brief Introduction to Probability Theory.) Analyzing the quandary with the example of ravens and blackness, we see that the degree of support increases as a higher portion of black ravens and non-black non-ravens are discovered.

Looking at a Venn diagram for ravens as a subset of black things, we can see that so long as no counterexamples are found, the degree of proof increases linearly as a higher portion of confirming events are discovered: the area of black ravens observed in B, for example, or the area of non-black non-ravens found in non-B. Note: the formulation of the hypothesis guarantees that n ravens É n black things and n non-black things É n non-ravens.

The Raven Paradox is resolved, then, as premise IV is incorrect. In fact, both II and IV could be updated in terms of the degree of support. Premise II, for example, could more accurately read “Instances of black ravens confirm ‘All ravens are black’ in that they add a value of (1/n)* to the degree of support, where n is the number of ravens.” Likewise, premise IV could be replaced with “Non-black, Non-ravens confirm ‘All ravens are black’ in that they add a value of (1/m)* to the degree of support, where m is the number of non-black things.” The contradiction disappears, as does the paradox.

So why the confusion about premise IV? The confusion results because in our world, there are many more non-black things than there are ravens, and evidence of a black raven is thus more confirming than evidence of a non-black non-raven (for example, a white tennis shoe). The primary usefulness of the degree of support method is in situations where “caps” on the number of Ss, Ps, non-Ss and non-Ps are known: The system derives from examining the number of possible universes our evidence could represent, and finding in how many of those our hypothesis is correct.

This probabilistic interpretation not only solves the Raven Paradox, but also makes substantial inroads in solving Hume’s Problem of Induction. The Problem of Induction lies in the idea that there are no necessary connections, that future events will not necessarily resemble the past. Though somewhat counterintuitive, the problem seems rational if looked at this way: All hypotheses, no matter how well “proved,” are generalizations based on prior observed events. And as Hume points out, past events are but a small subset of event-space, and observed past events an even smaller subset. Therefore, why should we expect anything but observed, past events to confirm to this rule? As our evidence has been subject to conditions absent from all future events, we have no way to generalize regarding them — the event-space is far to large to prove anything.

This problem is nicely illustrated by the degree-of-support model, but we must first add another level of sophistication. In order to apply the model to the Problem of Induction, we must realize that the size of our event-space is dynamic: as time progresses, event-space grows. Consequentially, more events will exist in each S, P, non-S, and non-P as time goes on. We know from our earlier example that in any circumstance where a finite amount of observations have been made out of a possibly infinite span of events, P(H|E) approaches 0 as event-space approaches infinity. This is the circumstance Hume points to: In reality, all our empirical observations are trapped in event-space between time 0 and time t. To inductively prove a hypothesis for all time, P(H|E) would equal a finite number (n of observed SPs) divided by an infinite one (n of Ss), and thus P(H|E) would be infinitesimal. This problem can be avoided, however, if we limit the scope of our hypothesis to include only the next predicted event. That is to say, the P(H|E) for the event-space between time 0 and time t+ , where t+ is the time that the next S is encountered. This results in a P(H|E) that is equal to a division of a finite number (n of observed SPs) by an only slightly smaller number (n of Ss + 1). While hypotheses inductively determined from observation are weak in the long run, in the short run they can be very reliable. Thus is Hume’s Problem of Induction satisfied. Or is it?

To be fair to Hume, time was not the only condition that caused a problem with Induction—there is also a problem with observation. Only events that have occurred in the past can truly be observed. This implies that we are trying to prove “All Xs are Ys” when really we can only confirm that “All OXs are Ys” (that is all observed Xs). Luckily, this too has a fairly simple solution, though perhaps one not as reassuring.

Let us define two sets: One, the set of all situations that have been or will ever be observed, which we’ll define to be set “So.” Two, the set of all situations that have never been nor will ever be observed. We’ll name this “Su.” The solution to this second aspect of the Problem of Induction lies here: Our universe, the only universe we’ll ever exist in or need worry about, consists entirely of So.

It is important to note that by “observed” we do not denote some ephemeral phenomena, but a time-independent categorization: So and Su are fixed throughout eternity and do not change with time, they are as concrete and certain as the fact of existence. We live in the world of the observed. Anything unobserved is unproveable: lack of observation is the great equalizer, we fail equally to observe the obscure and the trivial, the deep and the random, the coincidentally false and the necessarily untrue. We never see, hear, touch, feel, taste or otherwise perceive anything that we do not observe. The set of Su is the set of all things we have no way of testing, proving, affecting or being affected by: in other words, the set of all things we cannot determine and about which we should not care.

Given this, Hume’s second condition — the condition of being observed — becomes a trivial concern. From a positivist viewpoint, P(“All OXs are Ys”|E) means merely P(H&So | E&So), which is analytically equivalent to P(H|E). This can be seen by expanding P(“All OXs are Ys”|E): (the n of known OXYs) / (the n of OXs).

By analyzing the Raven Paradox, we discover that a trivial mental game can provide insight into the workings of observation. Furthermore, the degree of support method which was here applied to the paradox can also be used, in conjunction with a positivist approach to observation, to solve Hume’s Problem of Induction.

Footnote:
A quick mental test reveals that this is an oversimplification. The formula for the probability of a hypothesis of the form “All Xs are Ys” given the evidence observed is
P(H|E) = 1 - (n Xs-n XYs)(n Ys-n XYs).