Re: What is the law of maximal specificity, relating to scientific explanation?

The short answer is that the law of maximal specificity requires that any information that could possibly be pertinent to a fact be included in an Inductive-Statisical (IS) explanation of that fact. It is needed to solve the problem that there may be explanations in the IS framework that say a fact is true as well as explanations that say a fact is false. The downside is that an enormous about of pertinent information might be needed to create an explanation.

The long answer requires us to examine more deeply the question: what constitutes a scientific explanation? It would be nice if all the explanations of facts in science were logical deductions from initial premises and scientific laws. This is the Deductive Nomological (DN) framework proposed by Carl Hempel and Paul Oppenheim. Of course, real science doesn't work that way. When a physicist reads a dial or measures a length, small errors are made. A physicist might record a measurement as 5 cm plus or minus 0.1 cm, indicating that the true value is probably near 5 cm, but there is a little variation.

In fact, there are no absolute certainties in scientific explanation, simply facts that have so much evidence for them that any rational person believes the fact is true. We say, for instance, modern physics has a very accurate description of gravity with General Relativity. However, there is not absolute certainty because all of the measurements involved in testing the theory are not exact, leaving the door open for a better theory to be constructed in the future.

To capture this spirit of explanations which only give "high probability" of the fact being true, Hempel proposed the Inductive-Statistical (IS) model for explanation (while Hempel's full paper is not available on the web, http://www.bun.kyoto- u.ac.jp/~suchii/stat.expl.html gives a brief description of the history involved). Roughly, instead of logical deduction using natural laws, an IS explanation uses statistical laws together with observations to show that the fact to be explained is true "with very high probability". A probability law looks something like, P(E|F) = r, which means that the probability of the event E occuring given that we have predicate F is equal to r. For example, my predicate might be "people from Latvia", the event might be "person is taller than 2 m.", and P(person is taller than 2 m.|person is from Latvia) = .05 is a probability law.

Unfortunately, this leads to a spectacular problem, so that Hempel was forced to add the law of maximal specificity to save the IS approach. Here is a toy example. Suppose I wish to determine whether the color of a star is "blue" or "red" (a spectroscopy problem). I have two machines for the task, machine A and machine B. Machine A is much faster than machine B, so I use A for 9900 observations, and machine B for 100 observations. Suppose that the true color is blue, and while machine A is very accurate, say P(blue|machine A, star is blue) = 0.99). Say that of the 9900 trials, 9800 come out blue. Sadly, machine B is broken and so P(blue|machine B, star is blue) = 0.05. Suppose on machine B's 100 trials, only 4 came out blue.

So out of 10000 trials, 9804 show up blue. This is overwhelming evidence that the color of the star is blue. But if I just look at the machine B results, 96 out of 100 trials say the color is red, giving overwhelming evidence that the color is red. So if IS works, I have a scientific explanation that the star is both blue and not blue, which is very bad for IS.

Hempel fixed this problem by introducting the law of maximal specificity. Here is a rough description of what is going on. We say that a predicate F is statistically relevant to fact E if there is some probability law P(E|F) = r. A predicate M is a maximally specific predicate if first of all M is logical equivalent to a union of predicates all of which are statisticallly relevant to E (e.g., M may be people in Europe, and the statistically relevent class may be people in France, people in Germany, etc.), M doesn't entail either E or not E, and if a statistically relevant predicate is conjoined to M, either no information is gained (the conjuction is logically equivalent to M) or the conjuction entails E or not E. In other words, a maximally specific predicate contains all the information that could be relevant to the problem (in our example, we information about the machine used) and putting in extra information either doesn't help us at all, or makes A (or not A) a logical certainty.

Now we can give an outline of Hempel's revised requirement of maximal specificity which states that in a statistical explanation for fact E that uses predicate F and law P(E|F) = r, there must be another predicate M either stronger than F or maximally specific such that there is a probability law P(E|M) = r.

In other words, conditional on F, the probability of E is just r, and adding extra information (in the form of M) does not change the probability. In our example of the machines, adding extra information about the machine definitely changes the probabilities involved, and so no longer constitutes a scientific explanation in the IS model.

The problem with maximal specificity is that the information requirement becomes quite large. This is a strong statement that rules out the ambiguity problem where an explanation for E and not E both exist, but it does it by requiring any possible information that could perturb the probabilities involved, making it unwieldy in practice.

Mark