Why the "Problem of Induction" really isn't a problem. (And why theists don't even get it right)

From Logic

What is Inductive Logic?


Gregory Lopez and Chris Smith

We can define any type of logic as a formal a priori system (axiomatic) that is usually employed in reasoning. In general, if we feed in true propositions, and follow the rules of the particular system, the logic will crank out true conclusions.

We can define 'induction" as a thought process that involves moving from particular observations of real world phenomena to general rules about all similar types of phenomena (a posteriori). We hold that these rules that we generate are probably, but not certainly, true, because such claims are not tautologies.

Inductive logic therefore, is a formal system that can be distinguished from deductive logic in that the premises we feed into these arguments are not categories or definitions or equalities, but observations of the real world - a posteriori world. Inductive logic therefore, is the reasoning we do every day while working in the real world - i.e. the probabilities that we deal with while making judgments about the world. We can think of it as learning from experience and applying our prior experiences to new, but similar, situations. History

Inductive logic is basically a form of probability. While human beings have used intuitive forms of inductive reasoning all throughout history, probably theory was first formalized in 1654 by the mathematicians Pascal and Fermat - during their correspondence over the game of dice! In their attempts to understand the game, they created a set of frequencies - or possibilities that described the likelihood for particular rolls of the dice. In doing this, they accidentally set down the basics of probability theory.

It was only a short time later, in 1748, that someone noticed a problem in probability theory - that it included the presumption that the future would be just like the past, yet this assumption could not in of itself provide a sufficient condition for justifying induction, seeing as there is no valid logical connection between a collection of past experiences and what will be the case in the future. Hume's Inquiry Concerning Human Understanding" is noted, even today, for pointing out this problem - the "problem of induction". However, few realize that a solution to the problem appeared only a few years later: In 1763, Thomas Bayes's presented a theorm that unaware to him, could be used to provide a logical connection between the past and the future in order to account for induction. More recently, Kolmogorv (1933) axiomized probability theory, which means that he gave probability theory an axiomatic foundation. Induction, therefore, while a probabilistic enterprise, is founded on a deduced system:

The three axioms of formalized probability theory:

1. The probability of any proposition falls between 1 and 0.

2. Certian propositions have a probability of 1

3. When there is no overlap, P(P or Q) = P(P) + P(Q)

and the definition of conditional probability:

P(P/Q) = P(P & Q)/P(Q)

If you accept these axioms, you must accept Bayes Theorem. It follows logically from the axioms.

These are the key points to the history of induction as far as the formal origins and formal supports for induction. I will cover these points in more detail below. But first, let's look at the different types of inductive logic. Types of Inductive Logic

Let's do a brief review of some kinds of Inductive Logic

Argument from analogy . This occurs when we compare two phenomena based on traits that they share. For example, we might hold that Object 'A' shares the traits w, x and y, with with object 'B,' therefore, object A might also share other qualities of object B.

Statistical syllogism. This inductive logic is similar to the argument from analogy. The form of the logic follows: X% of "A" are "B", so the probability of "A' being "B" is X%

Example: 3% of smokers eventually contract lung cancer. John Doe is a smoker, therefore, he has a 3% chance of contracting lung cancer.

Generalization from sample to population The best example of this inductive logic would be a poll. Polls rely on random samples that are representative of a group by virture of their random selection (i.e. the fact that every person had the same chance of being chosen for the sample).

On my website, I will also discuss John Stuart Mill's Method of Causality. For now, let's return to the aformentioned "problem of induction" and take a deeper look both at the problem of induction, and some solutions for this problem.

The problem of induction

You've probably heard about Hume's famous 'problem of induction"

How do we know that the future will be like the past?

Or... more comedically

How do we know that the future will continue to be as it always has been?!

Consider the following example: we observe two billiard balls interact. From this, we observe that they appear to obey a physical law that could be presented in the formula: F=ma - Force = Mass X acceleration. From this observation, we then generate a general law of force. However, the problem then arises: how can we hold that this law will really apply to all similar situations in the future? How can we justify that this will always be the case?

If we argue that "we can know this, because the balls have always acted this way in the past" we are not really answering the question for the question asks how how we know that the balls will act this way in the future. Of course, we can then insist that the future will be just like the past, but this is the very question under consideration! We might then insist that there is a uniformity of nature that allows us to deduce our conclusion. But, how do we know that nature is uniform? Because in the past it always seemed so? Again, we are simply assuming what we seek to prove.

So, it turns out that this defense is circular... we assume what we seek to justify in the first place, that the past will be like the future. So this argument fails to provide a justification for induction.

But this in itself is not the whole story, in fact, if we stop here, we get the story all wrong. You see, the 'uniformity of nature' is in fact a necessary condition for induction but it could never be a sufficient justification of inductive inference anyway. The actual problem of induction is more than this: it is the claim that there is no valid logical "connection" between a collection of past experiences and what will be the case in the future. The classic "white swans" example serves: the fact that every swan you've seen in the past was white means simply that: every swan you've seen has been white. There is no logical "therefore" to bridge the connection "all the swans I've seen are white" to "all swans are white" or "the next swan I encounter will be white".

So, yes induction presupposes the uniformity of nature, but while this is a necessary condition for induction, the UN is not sufficient to justify inductive inferences epistemologically. So, any attempt to solve the problem by shoring up the 'uniformity of nature' will never work to begin with. When the next swan turns out to be black, it shows your statement "all swans are white" had no actual "knowledge" content. What you've done is presupposed nature to be uniform, but not in fact justified any particular inductive inference you may wish to make.

So,solving the 'problem' of induction is more than just trying to find a way out of the 'circle' of uniformity of nature/justifying induction. There is a problem that needs a solution. Interestingly, many critics seem to believe that the story ends here - that there simply is a problem, and that all solutions are merely circular. But this is untrue. There are responses to the problem.

Since it was Hume who first uncovered this problem, let's begin by looking at his response:

David Hume's Response: This assumption is a 'habit'

Hume's answer was that we had little choice but to assume that the future will be like the past..... in other words, it was a habit born of necessity - we'd starve without it! And, given that there was nothing contradictory, logically impossible or irrational to holding to the assumption, this utility of induction was seen to support the assumption on a pragmatic basis. This is a key point lost upon many people: there is nothing illogical or irrational about assuming that induction works, nor are there any rational grounds for holding that 'induction is untrustworthy'. The fact that I cannot be absolutely certain that the sun will rise tomorrow does not give me any justification in holding that it will not rise tomorrow! This error is called the fallacy of arguing from inductive uncertainty.

But merely holding that an assumption is 'not irrational' is not a satisfying enough answer for many. Hume himself stated: "As an agent I am satisfied but as a philosopher I am still curious." So let's continue our search for an answer to the problem.

What is the Basis for Inductive Logic? - An examination of Probability Theory

Curiously, the axiomatic foundations for inductive logic only tell us how a probability behaves, not what it is. So let's begin our examination by first defining what we actually mean by saying the word "probability".

Three common definitions:

Classical - the classical definition describes probability as a set of possible occurrences where all possibilities are 'equally likely' - but a problem arises from this definition. For example, how do you define "possibility" in a univocal manner? Is an outcome 50/50 (either it happens or it does not) or is an outcome actually 1/10, 1/100? In many cases there are possible reasons for each choice. So let's look at another definition.

Frequentist - the 'frequency' is the probability for a given event, that is determined as you approach an infinite number of trials. For example, as with the central limit theorm, you could learn what a probability might be for the roll of a 7 on a pair of dice, after rolling them for a large number of trials. This is the most popular definition, including in science and medicine. This view is backed up by axiomatic deduced probability theory (based on infinite trials (like coin flips)) the law of large numbers. The frequency converges to the probability when we reach infinity. But there are problems here as well: does the limit actually exist? Do we ever really know a probability, since we can't do things infinitely? Also, this method gives us very counterintuitive interpretations. For example, consider a 95% confidence interval - often this is read to mean that 1 out of every 20 such studies is in error. In actuality, what this means is that if the experiment were repeated infinitely, you'd get the real mean 95% of the time. This is hardly what people think when they read a poll.

Finally, we can't apply this method to singular cases. 'One case probabilities' are "nonsense" to the frequentist. How do we work out the probability of the meteor that hit the earth to kill the dinosaurs? Pshaw, who cares? We can't repeat this experiment infinitely! We can't repeat it once! We see the same problem with creationist arguments for our universe that attempt to assign a probability to the universe.

Subjective probability - Here, probability is held to be the degree of belief in an event, fact, or proposition. Look at the benefits of this model. 1) We can more carefully assign a probability to a given situation. 2) We can apply this to method 'one case events'. 3) This manner of defining probability gives us very natural and intuitive interpretations of events that fits with our use of the word "probably", circumventing the problems of frequentism.

MOST IMPORTANTLY: Allows us to rationally adjust our beliefs "inductively" by use of probability theory, which is a mathematically deduced theory, so we can latch on our beliefs onto a deductive axiomatic system. Here then, for many, is the solution to Hume's "problem" - induction is no longer merely "not irrational', but instead, can be seen as resting upon a firm deductive foundation.

How does it work?

How do you get a 'number' or probability, for subjective probability? Let's use the concept of wagering.... What would you consider to be a fair bet for a particular outcome? Is X more probable then getting Y heads in a row in your view? In brief, this is how the method works.

Subjective probability and frequency are linked by the "Principal Principle" (David Lewis) or Ian Hacking's "Frequency Principle" (his book cover appears at top). Subjective probablity is justified by a reductio argument: if your subjective probabilities don't match the frequency, and you know nothing else, you have no grounds for your belief.

A question may arise: How can we reason anything if probability's subjective? Well, it is true that you can just choose any starting ground you desire, HOWEVER, your choice must follow laws of probability, or else you're susceptible to 'Dutch Book Arguments' - what this means is that if your degrees of belief don't follow the laws of probability, you are being inconsistent and incoherent. You can choose to believe what you want, but at the risk of being incoherent. The beauty of this method is that a starting point is not necessarily very important: given differing starting probabilities, based on different subjective evaluations, two very different people who are shown enough of the same evidence will have their probabilities converge to the same value (a LAW OF LARGE NUMBERS) by probability theory - beliefs will converge to a similar value!

Being a subjectivist who wants to use probability as a basis of induction leads us to focus on a certain way of doing things using, Bayes' Theorm

BAYES' THEOREM

The simplest form of Bayes' Theorem:


where:

H is is the hypothesis. This is a falsifiable claim you have about some phenomena in the real world

E is the evidence This it the reason or justification you have for holding to the hypothesis. It is your grounds.

P(E|H) is called the likelihood : it is also the probability of E given H. In other words, it is the probability that the evidence would occur if the hypothesis were true.

P(H) is called the prior, or prior probability of H. It is the probability of the hypothesis being true without taking additional evidence into consideration. In other words, it is an unconditional probability. When I call something, "the prior" without qualification, I mean this probability.

P(E) is called the prior , or prior probability of the evidence E. It is the probability of E occurring regardless of H being true. This probability can be broken down further into the partition , as explained below.

The denominator of Eqn. 1 can be broken down as:


where H is the compliment of H, AKA not-H, and S is the sum over all independent hypotheses. This is sometimes called the partition. The top form is used when one is only considering whether a hypothesis H is true or false. The bottom form is more general, and holds for several independent hypotheses.

Plugging these into Eqn. 1 yields either:


which is useful when considering one hypothesis, being either true or false - this denominator of the right side of the equation multiplies the probability of the hypothesis being true against the probability of the hypothesis being false.

or it yeilds:


which is useful when considering how some evidence supports several independent hypotheses.

Personal tools