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Saturday, September 10, 2011

Induction and the Grue Paradox

The last critique of science I intend to address before moving on to whatever you guys vote for is the so-called "problem" of induction.  But before I can defend the inductive method used in science, I must first clarify what this method is and how it works in science.  To this effect, I'm going to formulate what I will call the Generalized Inductive Hypothesis (GIH), which will of course require some notation.

P(X) = probability of X
P(X|Y) = probability of X, assuming that Y is true.

GIH: Given two hypotheses H1 and H2 and an observation O, if P(O|H1) > P(O|H2), then the ratio P(H1)/P(H2) is increased by observation O.

To put this in less technical terms, the basic claim is that observations support hypotheses that predict them and count against hypotheses that don't.  This is the essential ingrediant in the scientific method.  When a hypothesis makes a prediction, and that prediction is later observed, the hypothesis is gains support.  But if a hypothesis says something shouldn't happen, yet that something is observed, the hypotheses loses support.

This generalized principle can be used to reach all sorts of conclusions.  The obesrvation of a black raven, for instance, lends support to the hypothesis "All ravens are black" while diminishing the hypothesis "Several ravens are white."  Since each observation lends support, a legnthy succesion of observed black ravens can lead us to conclude that the claim "All ravens are black" is highly probable. 

This type of induction is performed over a population, in that we take observations of some ravens in order to draw a conclusion about all ravens.  We could also perform induction over other sets.  For instance, we can use induction with past experience to draw conclusions about the future, or we can use induction with experiences from one place to draw conclusions about another place.

Here are some notes about the GIH that will refute most of the objections raised against induction:
1. There is no "principle of uniformity."  The idea that the future resembles the past can be concluded from applying induction to past experience.  It is not a prerequisite for the application of induction.  Induction comes not from a uniformity principle but from basic probability considerations (as I will demonstrate next week).
2. Any attempt to assign particular values to any probabilities involved will have to comply with standard constraints from probability theory.  The most notable of these is normalization
3. The GIH never concludes anything like "H1 is true."  The conclusion involves probabilities, so observing that induction sometimes leads us astray is simply the observation that when P(H1) = 0.85, H1 is sometimes false.
4. P(O|H#) may depend on the manner in which O is obtained.  For instance, the probability of drawing a red object from a bag may be different if you select a random object from the bag than if you select a random cube from the bag.

Note 1 is the most important.  Many of the issues raised by philosophers (such as the Grue Paradox tackled later in this article) rely on the idea that induction requires a "uniformity principle."  Philosophers seem to think that induction requires us to assume that the future resembles the past.  This has never made any sense to me.  In fact, the reson predictive power is useful is because the future does not resemble the past in many ways.  While it is true that science treats many things as constant in time, the constancy is a conclusion of induction, not the basis.  This is why it is possible for scientists to consider ideas like "What if the speed of light isn't actually constant?" or "What if the continents don't sit still?" or "What if mass isn't actually conserved?  What if it can be transformed into energy?"

While it is obvious that a time-dependent variation in the speed of light would cause current physics to become inadequate, it would not be the end of the scientific method.  The constant c would simply be changed to a time-dependent function c(t), and we would reformulate our theories and perform more tests and so on.  When science says that something is constant in time, this is a conclusion reached through the GIH.  It is not an assumption on which all scientific induction is based.  Observing that the laws of physics change in time would merely indicate that several quantities physicists currently treat as constants are actually time-dependent functions.  It would wreck havoc with current physics, but it would do nothing to discredit the GIH.

As an example of philosophical objections gone wrong, consider the Grue Paradox (Stanford Encyclopedia of Philosophy's, section 3.2).  In this "paradox," we consider repeated observations of emeralds.  Emerald 1 is observed to be green.  As is emerald 2, 3, 4, 5, and so on up to, say, a million emeralds.  The GIH is then applied to conclude that all emeralds are probably green, and thus we can expect the next emerald we observe to be green.  But the philosophers asked, why do we not instead conclude that all emeralds are grue?  Grue is defined as follows: An object if grue if it is green and observed before some arbitrary time t, or blue otherwise.  So long as we choose t to be in the future, each observed emerald is grue.  So why do we use induction to conclude that all emeralds are green instead of concluding that all emeralds are grue?

To resolve this apparent paradox, we must first point out that grue is parameterized by a time.  Grue with t = October 1st, 1986 is observably different than grue with t = yesterday.  Goodman, who developed the Grue paradox, objects to this observation by noting that we could define another property bleen, which means blue if observed before some time t and green otherwise.  Then we could define green as grue if observed before t, blegg otherwise.  As such, Goodman claims that any objection to grue on the basis of including a time can be applied to green.  But this is not so.  Unlike with grue, there is no observable difference between Goodman's obscure definition of green with t = October 1st, 1986 and the same definition with t = yesterday.  While we can define green in a manner that references a time, such a reference is unimportant because the meaning of green, even with this definition, doesn't change when we change the referenced time.  On the other hand, the time reference in the definition of grue is important.  In phsyics-speek, green, even with Goodman's obscure definition, is time-invarient while grue is not.

So now we see there is a meaningful time-dependence in the grue property which does not exist in the green property.  As such, all further references of the grue property will include the time, such as grue(yesterday).  Combined with the observation that induction can be applied over time, we proceed as follows.  Our observations indicate that emeralds are not grue(last year), nor grue(two years ago), nor grue(three years ago), etc.  We apply the GIH to conclude that "emeralds are grue(any time)" is much less likely then its negation.

Notice how this same kind of argument could not be used to conclude that emeralds probably aren't green even if we defined green in the seemingly time-dependent way proposed by Goodman..  This is precisely because of the fact that Goodman's obscure definition of green is actually time-independent, even though there appears to be time-dependence in the definition.  Thus we can use the GIH to conclude that "all emeralds are green" is much more likely than "all emeralds are grue."

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