(Preamble: I apologize in advance for cluttering TSZ with these three posts. There are very few people on either side of the debate that actually care about the details of this “conservation of information” stuff, but these posts make good on some claims I made at UD.)
Laws of nature are universal in scope, hold with unfailing regularity, and receive support from a wide array of facts and observations. The Law of Conservation of Information (LCI) is such a law.
Dembski hasn’t proven that the LCI is universal, and in fact he claims that it can’t be proven, but he also claims that to date it has always been confirmed. He doesn’t say whether he as actually tried to find counterexamples, but the reality is that they are trivial to come up with. This post demonstrates one very simple counterexample.
First we need to clarify Dembski’s terminology. In his LCI math, a search is described by a probability distribution over a sample space Ω. In other words, a search is nothing more than an Ω-valued random variable. Execution of the search consists of a single query, which is simply a realization of the random variable. The search is deemed successful if the realized outcome resides in target T ⊆ Ω. (We must be careful to not read teleology into the terms search, query, and target, despite the terms’ connotations. Obviously, Dembski’s framework must not presuppose teleology if it is to be used to detect design.)
If a search’s parameters depend on the outcome of a preceding search, then the preceding search is a search for a search. It’s this hierarchy of two searches that is the subject of the LCI, which we can state as follows.
Given a search S, we define:
- q as the probability of S succeeding
- p2 as the probability that S would succeed if it were a uniform distribution
- p1 as the probability that a uniformly distributed search-for-a-search would yield a search at least as good as S
The LCI says that p1 ≤ p2/q.
In thinking of a counterexample to the LCI, we should remember that this two-level search hierarchy is nothing more than a chain of two random variables. (Dembski’s search hierarchy is like a Markov chain, except that each transition is from one state space to another, rather than within the same state space.) One of the simplest examples of a chain of random variables is a one-dimensional random walk. Think of a system that periodically changes state, with each state transition represented by a shift to the left or to the right on an state diagram. If we know at a certain point in time that it is in one of, say, three states, namely n-1 or n or n+1, then after the next transition it will be in n-2, n-1, n, n+1, or n+2, as in the following diagram:
Assume that the system is always equally likely to shift left as to shift right, and let the “target” be defined as the center node n. If the state at time t is, say, n-1, then the probability of success q is 1/2. Of the three original states, two (namely n-1 and n+1) yield this probability of success, so p1 is 2/3. Finally, p2 is 1/5 since the target consists of only one of the final five states. The LCI says that p1 ≤ p2/q. Plugging in our numbers for this example, we get 2/3 ≤ (1/5)/(1/2), which is clearly false.
Of course, the LCI does hold under certain conditions. To show that the LCI to biological evolution, Dembski needs to show that his mathematical model of evolution meets those conditions. This model would necessarily include the higher-level search that gave rise to the evolutionary process. As will be shown in the next post, the good news for Dembski is that any process can be modeled such that it obeys the LCI. The bad news is that any process can also be modeled such that it violates the LCI.