Are You Sure That's the Best Explanation?

Jonah Bossewitch

PHI 313

Professor Rosen

November 24, 1993

When analyzing human reasoning there are two tasks that are crucial to distinguish from one another. There is the empirical question of how people actually reason and there is the philosophical question of whether or not that method of reasoning produces justified conclusions. The discussion that follows deals with the latter question and in particular explores the justification of ideal forms of reasoning. Whether the human mind actually reasons using these ideal methods is a separate matter to be considered at another time.

The term reasoning is meant to describe the process of inferring non-obvious information from some given information. In considering the justification of various sorts of reasoning we take as a given that we can trust our senses and that there exists an external reality. Even after these hairy skeptical challenges have been answered, the skeptic may still be able to argue that we are not justified in believing many of the conclusions that we often attempt to infer from our sense data.

The one form of reasoning that even a (rational) skeptic cannot refute is deductive reasoning. Given a body of observed data, deductive reasoning with no implicit assumptions will not add any new information to the system. All that deductive reasoning has the power to do is make information that is implicit in the premises explicit in the conclusion. For example, in the standard example of deductive reasoning that goes "All A are B, x is A" the conclusion "x is B" is not telling us anything we did not previously know. All that deduction does is make "hidden" information apparent. If this were the only form of valid reasoning then conclusions that contribute new information to the system would be unjustifiable (without adding implicit premises). Abstractions such as scientific hypothesis and laws, generalizations from an observed sample to an unobserved sample, and predictions about the future based on the past all fall into this category. Any conclusion that makes a claim concerning objects or events outside of those enumerated in the premises is adding information that cannot be deductively inferred from just the initial premises. To be more specific, we are talking about an instance where I observe a regularity and want to infer that that regularity always holds. This type of inference is non-deductive, and depending on how it is formulated, an example of inductive, or abductive reasoning.

There are two types of problems in trying to justify non-deductive conclusions. First, we need to justify the information in the conclusion that was not implicit in the premises. I will not discuss this problem here but one way it has been approached is by trying to find a justifiable premise, such that, if included, would make the argument deductively valid. The second type of problem that Goodman raised is present even if the first problem has been solved. Goodman raised this problem with respect to the form of non-deductive reasoning known as induction. An inductive argument is of the syntactic form "All observed A's are B's" therefore "All A's (including those not observed) are B's." Goodman's new problem of induction focuses not on the conclusion, but on our justification of the premise to the inductive argument.

The problem that Goodman raises is subtle and is best illustrated by analyzing the premise "All observed A's are B's." When we claim that all observed A are B we really mean that all of the A's that we have observed have a particular quality B in common. We would then like to conclude that all A's have this quality B.

To understand Goodman's problem it helps examine what it means for something to have a quality. When I say that dolphins are slippery, or have the quality of slipperiness, do I mean that they actually possess some intangible essence of slipperiness that they store somewhere (like under a fin)? It is plausible that to possess a quality means nothing more than to be a member of the set of all of the things that also possess that quality (footnote?). So the dolphin isn't in possession of slipperiness, it is just a member of the set of all of the things that also possess slipperiness. When an object satisfies a quality predicate, (i.e. it is true that a dolphin is slippery) it is helpful not to think of the object as having that quality, but rather as being a member of the set all objects that also have that quality. I will call such a set a quality set.

In light of this analysis of qualities the new problem of induction becomes clearer. Given a set of observed A's, there are an infinite number of quality sets that A can belong to. If I observe a parking lot full of cars, there are many different qualities that all of the cars I observe share. All observed cars have four wheels, all observed cars have antennas and have fenders, all of the observed cars have two doors or four doors etc. The problem with inductive inferences is that there are an infinite number of "quality sets" that observed A's belongs to, and some of them diverge for unobserved A's. In other words while all of the observed A are in both sets C and D, some of the unobserved A may be in C and not in D or in D and not in C.

Goodman's illustrates this point by defining some new quality sets that are nothing more than combinations of quality sets that we are familiar with. He defines two terms that together specify all of the elements of the sets green and blue. The term grue means the set of all things that are {{green and observed before time t} or {blue and not observed before time t}}, and the term bleen means the set of all things that are {{blue and observed before time t} or {green and not observed before time t}}. If all of the emeralds I have observed have been a member of the quality set green, it is just as logically valid to claim that all of the observed emeralds are green as it is to say that all of the observed emeralds are grue. This is true since all of the observed emeralds belong both to the set of things that are green as well as to the set of things that are grue. The question is what should I predict about the emeralds that will be observed after time t? If all emeralds belong to the set of things that are green, then following the rule of induction I should predict that emeralds observed after time t will be green. However, if all emeralds belong to the set of things that are grue, I should predict that emeralds observed after time t will be blue. How could I be justified in believing one prediction over the other? Following the rules of induction results in believing two contradictory conclusions.

Someone might object to this problem on the grounds that the qualities green and blue are more "primitive" than grue and bleen since they are complete sets not specified by the unions and intersections of other sets. However, the reason that Goodman's example is so devastating is because syntactically the terms grue and bleen are just as primitive as green and blue. If we start out considering grue and bleen to be complete sets, then green and blue can be defined in terms of the intersections and unions of the grue and bleen sets. Something belongs to the set of things that are green if it is {{grue and observed before time t} or {bleen and not observed before time t}} with blue defined in a similar fashion. Similarly, one might object to the arbitrary conjunction of the qualities of color and having been observed, however we can easily define two new qualities, obseen and obslue, that denote the sets of all things that are green and have been observed before time t and blue and observed after time t. Now, grue can be defined as {obseen or obslue} and there are no conjunctions to object to. Given the perfect syntactic symmetry between green and grue, how are we justified in categorizing something as green as opposed to grue? Is it just a matter of chance that we started using the term green first? Are our quality sets truly arbitrary?

Many people have the gut instinct that something strange is going on here and that there really is a difference between the qualities green and grue. The problem is that a formal rule of logic only distinguishes sentences based on the grammatical structure of the sentence, and not the meaning of the sentence. However, in contrast to formal logic, human beings have the ability to distinguish between sentences that are syntactically equivalent. We do this by interpreting the sentence and considering it's meaning. It is clear that to make induction work we must in some way limit the terms that we allow in the premise "All observed A are B" of our inductive argument. We want a rule that will allow us to induce from the observation "all observed A are green", but will not allow us to induce anything from the observation "all observed A are grue." As we have seen, any attempt to syntactically restrict the terms A and B (for instance not allowing a quality to be defined as a disjunction of other qualities) will fail due to the syntactic symmetry between quality sets. From a purely syntactical point of view the qualities green and grue are identical. However, perhaps a limitation the terms based on the meaning of the terms will allow us to formulate an induction that does not result in contradictory conclusions.

This sort of limitation would take the form "if all observed A are B, and B is C, all A are B" where C is the semantic restriction we are imposing on B. For example we might attempt to construct a semantic restriction based on the observation that in Goodman's example the qualities grue and bleen are not observable qualities. In order to tell if something is grue or bleen I need to know when it was first observed, which is not something that my senses can tell me. Does restricting B to only those qualities that are observable to the senses make inductive inferences valid?

While this sort of restriction has the flavor of the restriction needed, it does not solve Goodman's problem. As a counter-example suppose we define a new grue that means {{green and weighs less than 1 pound} or {blue and weighs more than a pound}}. Both color and weight are observable qualities and yet we are still stuck not knowing whether to predict that an emerald over a pound will be green (if all emeralds are green) or if it will be blue (if all emeralds are grue). So restricting B to observable qualities does not prevent us from drawing contradictory conclusions.

Finding the appropriate semantic limitation on B has proven to be quite difficult. Goodman has suggested that B must be limited to a "natural" quality. The trouble begins when we begin to wonder what makes one quality more natural than another. The fact that are familiar with it? Is there such a thing as a quality being objectively natural? If not, can we truly justify conclusions based on premises whose truth is subjective?

It is partially in response to this question that Harman proposed a method of non-deductive reasoning other than induction. Harman calls this method of reasoning "inference to the best explanation," or abduction. He argues that not only is this form of reasoning necessary to explain instances of non-deductive reasoning that are not induction, but that induction itself is an instance of abduction. Using abduction we would reason as follows: All observed A's are B's. Therefore all A's are B's if and only if the hypothesis that all A's are B's is better than the hypothesis that all A's are not B's. In this form of reasoning we place no restrictions on either the syntax or the semantics of A or B in the premise. The way that contradictions are avoided is by placing a semantic restriction on the valid abductive conclusions that can be drawn from the infinite number of premises. We limit the valid conclusions to only those that may be inferred from the best hypothesis.

At first, Harman's reformulation of non-deductive reasoning seems to accomplish nothing but a recasting of our old problem in a new vocabulary. We have traded the problem of defining a natural quality for the problem of defining a best hypothesis. However, it is important to recognize the fundamental difference between inductive and abductive reasoning. While induction allows us to add information by generalizing existing patterns, abduction allows us to add even more information by inventing a best hypothesis that does not contain any of the terms in the premises. For example if I observe that chemicals combine in whole number ratios, then induction will only allow me to infer that all chemicals combine in whole number ratios. Only abduction will allow me to hypothesize the existence of atoms to explain this phenomena. Similarly, a mechanic fixing a car may notice that some of the do not lights work. If he has seen many cases like this before he may be able to induce that none of the lights on the car work. However it is abduction that allows him to conclude that they do not work because the electrical system is shorted out. Since all inductions can also be recast as abduction, if we can justify abduction there is not necessarily a need to justify inductive inferences. It is true that by trading induction for abduction we have merely traded the problem of defining natural to defining best. However, justifying abduction justifies examples of reasoning that induction can never support.

The question of what makes one hypothesis better than another one still plagues scientists, philosophers, historians, linguists and many others. One tool that many of these individuals use in helping to resolve the matter in particular cased is the idea of confirmation. Competing hypothesis must continue to be consistent not only with the initial data, but with all of the new data that comes in. Of course the skeptic may still argue that while we may have discovered a way of deciding the best hypothesis among those that we have considered, what if there is a hypothesis that we have not considered that is even better than the ones that we have. Furthermore, as Van Frassen points out, our criterion for best may not reflect the truth of the matter. Given two hypothesis that both explain all the data, how do we decide which one is better. The one we thought of first? The simpler, more aesthetic one? Without an objective way of determining which hypothesis is better, at most we sat that our hypothesis is empirically adequate, and not that it is actually true.

A final important point concerns whether I am justified in believing a conclusion if I used an unjustifiable method of reasoning to arrive at the conclusion, but the conclusion is in principle justifiable. If we discover that human beings manage to always arrive at justifiable conclusions but do so by a process that depends on the last time that they ate humus. What does that imply about our justification in believing our own beliefs. Sure somebody can formulate their reasoning deductively or abductivly, but am I justified in believing my own beliefs until I have explicitly done so. Are we justified in believing something that is in theory justifiable before we have bothered to justify it? Does a method's being justifiable require anything more than that it produce correct results?