The New Riddle of Induction

You are likely to be eaten by a grue

Preface: This is a riddle, not a paradox. You may seem to see the impossible, but there is a perfectly reasonable answer for everything.

The New Riddle of Induction is traditionally introduced by example; we quote Wikipedia:

For some arbitrary future time t, say January 1, 2031, for all green things observed prior to t, such as emeralds and well-watered grass, both the predicates green and grue apply. Likewise for all blue things observed prior to t, such as bluebirds or blue flowers, both the predicates blue and bleen apply. On January 2, 2031, however, emeralds and well-watered grass are bleen and bluebirds or blue flowers are grue. The predicates grue and bleen are not the kinds of predicates used in everyday life or in science, but they apply in just the same way as the predicates green and blue up until some future time t. From the perspective of observers before time t it is indeterminate which predicates are future projectible (green and blue or grue and bleen).

The riddle is: why should we expect things to stay blue, but not expect things to stay bleen? The obvious solution is indeed a correct one: we should not expect a time-dependent predicate such as bleen to apply consistently across time. However, trying to rigorously describe why that is so will require overcoming several interesting problems.

What is Induction?

Before we get to new induction, we must discuss what induction means. Induction is a word with dozens of meanings, we note a few important ones here.

• Mathematical induction refers to a method of proof. If a base case N=1 is true for a statement, and if the statement being true for N=m implies it is true for N=m+1, then it is true for any N=n.

• Membership induction, more often called initiation, involves a process where an individual joins a group. One might have an induction ceremony where a person is inducted into a Hall of Fame.

• Electromagnetic induction is the principle that a change in magnetic fields will generate an electric current, and vice versa. We defer the task of explanation to Mr. Feynman.

• Logical induction, more often called inductive reasoning, is the use of statistical evidence to support a claim. Also phrased as the statement that the future will resemble the past. We recommend reading the Stanford Encyclopedia of Philosophy article on Inductive Logic.

It is that claim that “the future will resemble the past” that is the old riddle of induction. The riddle is, why should you expect that?

Old Riddle of Induction

Why should we believe the future will resemble the past? A first retort is that it has worked in the past; that argument merely begs the question¹.

There is a mathematical argument here. If, for the past 98 days, you have had some unusual phenomenon occur, you can reasonably expect a 99% chance (that is, (98+1)/(98+2) chance) that it will occur the next day. No higher, no lower, 99%.

One thing that does not impact that 99% chance is the possibility that some adversary is governing whether or not the event happens. As a simple explanation, “it’s only your first time once”. In a more metaphorical sense, their ability to trick you by having the event carry on unexpectedly and their ability to trick you by having the event stop unexpectedly cancel each other out. Or rather, they cancel out when you assign the correct probability, which here is 99%.

There are other reasons to believe that the future will resemble the past. There may be an over-arching theory governing the behavior. Perhaps it is a seasonal phenomenon and you can measure when it will end months in advance. Perhaps it is that your daily newspaper has arrived; if you have not canceled your subscription and there is not an Act of God impacting the area there should be a substantially higher SLA than 99% regarding delivery.

The sun rises every morning because of what the Solar System theory describes. Our model above gives a 1 in 1.5 trillion (age of the earth in 24-hour days) chance that the sun will disappear tomorrow. If you believe השם created the world 6000 years ago, your estimate is closer to 1 in 2 million. Perhaps the risk from aliens noticing our illegal radio emissions is substantially higher than that. Perhaps there are no aliens, and nothing other than the collapse of physical reality will stop the sun from rising. We leave both discussion of aliens and discussion of the collapse of physical reality to others.

For less mundane events, those same existential risks will still apply, but there are other risks. The lights and the elevators and the cars outside will keep operating as normal, unless there is a power outage.

New Riddle of Induction

Now we have predicates with a change built-in to them. We are tempted to say that this is an artificial problem, and by excluding them the riddle is solved. This is twice not the case. Many useful predicates have time-dependent changes in them. And many of those predicates have the opposite behavior to the bleen and grue scenario; it is the time-dependent predicates that one should consider to believe.

As a (still artificial, yet easily demonstrable) example, we consider Bitcoin². If, at block 10000 or 350000 or 475000, you said that certain properties (SegWit) would apply only at or after block 481824, you would be correct, but your predicate would be very much a grue/bleen structure. This is not academic if you own and want to transact with Bitcoin.

Clearly, some predicates with time-sensitive clauses are different than others. Our question remains: how are they different, and how can we tell them apart? We must rely on some external system to explain why Segwit AFTER_AND_NOT_BEFORE 478120 is true, and for 484120 it is not. Surely a large number of archived email messages will be submitted in evidence for any legal decision on the topic. But we do find an explanation: a bunch of people who govern Bitcoin agreed that it would change at that time, and then it did. Our predicate “Bitcoin” behaves like bleen and grue, we cannot be surprised that other predicates that interact like it have the same nature. And other useful phrases, like “legal tender for all debts, public and private, according to the Government of the United States of America under the Constitution of 1787”, also have that nature.

There is a mathematical answer here as well³. There is only one way to define blue = “light at 440nm⁴”. Assuming you aren’t defining nanometers relative to the length of the Queen of England’s thumb, this is constant. Yet there are a trillion billion ways of defining bleen and grue; perhaps the wavelengths are slightly different, or the cutover times are slightly different, or there are two different cutovers. Clearly, the one simple definition is more likely than the median of the myriads of complex definitions.

And regarding the likelihood in aggregate: it is substantially less likely that our 440nm laser will suddenly emit light at 510nm than it is that aliens will arrive and destroy the sun. And nothing you might say about grue and bleen should convince anyone otherwise.

1. We use “beg the question” here properly: to make an argument that pre-supposes the truth of the conclusion. Sure, “the future will resemble the past” was true in the past, but why should you think it will still hold in the future?

2. As a disclaimer: this is not a recommendation to buy Bitcoin. Our recommendation is that Bitcoin is worth $7000-$10000 per coin.

3. This isn’t rigorously Kolmogorov complexity, but that’s what you should look for to find out more.

4. Technically, perhaps we should refer to a point on the CIELAB color space gamut rather than a wavelength of light. For the purpose of argument, it’s simplest to assume we are talking about the color of light sources rather than arbitrary objects.