First posted on August 10, 2000. Updated July 30, 2001.
This article is not a detailed discussion of the multitude of questions related to the anthropic principle. There is a vast literature devoted to such discussions, where the anthropic principle has been analyzed in great detail and from various viewpoints. This article offers a very simple and to a certain extent superficial discussion of the anthropic principle in Bayesian probabilistic terms. While it simplifies the problem under discussion, it is offered with an intention to demonstrate that certain interpretations of anthropic principle can easily be refuted based on quite simple probabilistic grounds.
The term anthropic principle apparently started gaining popularity after 1973, when an English physicist, Brandon Carter introduced it at a gathering of scientists on the occasion of the 500th anniversary of Copernicus's birth. Carter noted that the values of physical constants must be within a very narrow range in order to enable the existence of life, and that the observed values of those constants happen indeed to be in that narrow range. In other words, the universe appears to be "fine-tuned" for the existence of life. Were any of the physical constants slightly different, life would be impossible.
Since then, the anthropic principle, as Carter's observation was labeled, has become a favorite subject of discussion by many adherents of the Bible's inerrancy, who view it as a convincing proof that the universe, as we see it, and intelligent life in particular, could not emerge by chance but must have been created for a purpose and according to a detailed design by a supernatural agent.
For example, Hugh Ross has extensively written about the anthropic principle, listing multiple examples of physical constants whose values seem to be very finely tuned for the existence of life (as in his article ). Professor Nathan Aviezer also argued  in favor of the above supernatural interpretation of the anthropic principle. Similar discussions can be found in many other publications by the proponents of the supernatural creation of the universe and life, such as books by Fred Heeren  or Patrick Glynn , a paper by Walter L. Bradley  and many other publications.
Of course, those adherents of such view, who approach it from a Christian perspective, interpret it differently from those who are believing Jews. For the former, the anthropic principle points to Jesus Christ as the creator of the universe, whereas for the latter Jesus has nothing to do with the creation of the world by Yahve (Jehovah). It is easy to see that if we accept the thesis that anthropic principle indeed proves that the universe was created by a supernatural agent, it still leaves many variations of that explanation equally possible. Without contradicting the mentioned general interpretation of the anthropic principle, one can equally guess that the creator of the world was either the biblical Yahve or Quetzalcoatl, or Krishna, or Jupiter, or a band of smaller gods all working in cahoots.
However, in this article I will try to show that the above interpretation is unsubstantiated in general, regardless of the particular choice of a candidate for the role of the supernatural Creator of the world. I will try to show that if the universe is indeed fine-tuned for life this does not logically point to the supernatural creation of the world and of life, such an interpretation being just one of many possible, equally arbitrary assumptions.
First, let us note that after Carter's introduction of the concept of the anthropic principle, a number of versions of the latter have been offered. One such version is often referred to as the Weak Anthropic Principle (WAP). For example, here is how the famous physicist Stephen Hawking defines the WAP in his popular book A Brief History of Time : "The weak anthropic principle states that in a universe that is large or infinite in space and/or in time, the conditions necessary for the development of intelligent life will be met only in certain regions that are limited in space and time. The intelligent beings in these regions should therefore not be surprised if they observe that their locality in the universe satisfies the conditions that are necessary for their existence." From that definition we see that the WAP does not require the assumption of a supernatural agent being responsible for the creation of the universe and of life.
Another version of the anthropic principle is often referred to as the Strong Anthropic Principle (SAP). Here is how Hawking defines the latter: "According to this theory, there are either many different universes, or many different regions of a single universe, each with its own initial configuration and, perhaps, with its own set of laws of science. In most of these universes the conditions would not be right for the development of complicated organisms; only in the few universes that are like ours would intelligent life develop and ask the question, 'Why is the universe the way we see it?' The answer is simple: if it had been different, we would not be here." We see that the SAP itself does not necessarily imply the supernatural creation of the universe either.
One more version of the anthropic principle was offered  under the name of the Participatory Anthropic Principle. This version made use of the concepts of quantum mechanics in their so-called "Copenhagen interpretation." This version implies that the very existence of the universe can only be understood within the framework of an intelligent mind observing the universe.
Then, there is also the so called Final Anthropic principle  which goes even further, positing that the very existence of the universe is due to the human mind observing it. Reviewing all described versions of the anthropic principle, Martin Gardner  suggested an all-embracing derogatory term "CRAP, the Completely Ridiculous Anthropic principle."
While one can be skeptical in regard to the mentioned esoteric interpretations of Carter's observation of the values of physical constants, the very fact of their being seemingly "fine-tuned" for the existence of life has been rather commonly accepted. The question is, though, what is the meaning of that apparent hospitality of the universe for the existence of life. Depending on the preferred answer to that question, we can see a clear demarcation between two opposite interpretations of the mentioned "fine-tuning." One interpretation perceives in the fine-tuning of the physical constants an indication of the creation of the universe by a supernatural agent (God or gods) for a purpose and according to a plan. The opposite interpretation looks for a natural explanation of the fact that the universe seems to be fine-tuned for life.
I suggest to denote all versions of the anthropic principle that imply only a natural origin of life in our universe, as the Natural Anthropic Principle (NAP) and all those versions that imply a supernatural creator or creators, as the Supernatural Anthropic Principle (SNAP).
My further discourse will be of probabilistic nature. I will estimate what the probability is that the fine-tuning of the universe for life indicates a supernatural creation of the universe vs. the probability that the universe had no supernatural creator.
A few words about the terminology are in order. The adherents of SNAP often discuss the problem in terms of "what if the physical constants were different." For example, if the value of the strong force were slightly less than it is in our universe, many atomic nuclei would not exist, so that the only element in such a universe would be hydrogen, and, hence there would be no life. On the other hand, in a universe where the strong forces were slightly stronger than in our universe, all hydrogen would be replaced with helium at an early stage of that universe's formation, so the stars as we know them would not exist, and therefore there would be no life. Whereas some theories have been suggested in which the actual existence of alternative universes have indeed been postulated, in many discussions hypothetical alternative universes with different values of physical constants are imagined only for the sake of discussion. Most often the adherents of SNAP do not assert that indeed such alternative universes exist or even could exist. Likewise, in my probabilistic discourse, I will use similar references to "alternative universes," not implying that such alternative universes really exist or even may exist.
Before turning to the probabilistic presentation of the anthropic principle, let me make a brief excursion into some seminal concepts of the probability theory.
Let p(A&B) be the probability that two events A and B both take place. The probability theory tells us that if A and B are independent events, then
p(A&B) = p(A)×p(B)..........(1)
This equation is often viewed as the definition of the events' independence.
However, if A and B are not independent, i.e., if the occurrence of A changes the probability of B, then we have instead of (1):
p(A&B) = p(A)×p(B|A)..........(2)
where p(A) is the probability of event A, and p(B|A) is the conditional probability of event B provided event A takes place. This relation is pretty self-evident and sometimes is viewed as the definition of the conditional probability p(B|A).
Likewise, we can write
Since, obviously, A and B are arbitrary notations, which may be swapped, we can write:
which, comparing (2) and (3), yields:
From (5) we get a universal relation
Our equation (6) could have simply been presented as a consequence of Bayes's theorem, well known in the probability theory . Alternatively, Bayes's theorem can be easily derived from equation (6). For this discussion, we will limit ourselves to the particular, simplified form of Bayes's theorem presented as equation (6).
It is easy now to succinctly represent any version of the anthropic principle in probabilistic terms. For example, for NAP and SNAP it can be done as follows:
Let p(FT) be the probability that a certain universe is fine-tuned for life. Let p(FT|L) be the conditional probability that a universe is fine-tuned for life provided life exists in that universe. Let p(L) be the probability that life exists in a universe, and p(L|FT) the conditional probability that life exists in a universe provided that that universe is fine-tuned for life. The equation (6) of probability theory becomes:
p(FT|L)/p(FT) = p(L|FT)/p(L)..........(7)
The Natural Anthropic Principle then boils down to the statement that
NAP: p(FT|L)» p(FT)..........(8)
The meaning of (8) – the probabilistic representation of the Natural Anthropic Principle - is the assertion that the existence of life in a universe makes it much more probable that the universe in question must be fine-tuned for life (as compared with a universe where there is no life).
From (7) then follows that also
The meaning of (9) is that if a universe is fine-tuned for life, this considerably enhances the probability that life will exist in that universe (as compared with a universe which is not fine-tuned for life). This is a reasonable assumption (some readers may even view it as obvious). If (9) holds, (8) holds as well. Hence, NAP seems to be a reasonable and logically faultless assumption.
2) SNAP (Supernatural Anthropic Principle). This interpretation of the anthropic principle is based on the assertion that the existence of life in our universe is best explained by assuming that the universe was intentionally "fine-tuned" by a supernatural intelligent agent. In probabilistic terms, this position can be expressed by the following inequality:
where p(S) is the probability of a supernatural creation of the universe and p(S|FT) is the probability of a supernatural creation provided the universe in question is fine-tuned for life. Expression (10a) states that the fine-tuning of our universe makes much more probable its supernatural origin as compared with a universe which is not "fine-tuned" for life.
According to the proponents of SNAP, life would be impossible without the "fine-tuning." Therefore, the conditional probability p(S|FT) in the expression (10a) may be replaced by p(S|L) which is the conditional probability that our "fine-tuned" universe was created by a supernatural agent, provided life exists in that universe (as compared with a universe where there is no life).
Then, in probabilistic terms, SNAP can be rendered by the following expression:
p(S|L)/p(S) » 1..........(10b)
where p(S) is the probability that a universe was created supernaturally, and p(S|L) is the conditional probability that a universe was supernaturally created provided life exists in that universe.
The relation (6) is for this case as follows:
p(S|L)/p(S) = p(L|S)/p(L)..........(11)
where p(L|S) is the conditional probability that life exists in a universe provided that universe was created supernaturally. To satisfy (10) and (11) we must assume that
The meaning of (12) is that if we know that a universe was created supernaturally, this knowledge increases the probability p(L|S) of life's existence in such a universe compared to the probability p(L) of life's existence in the absence of knowledge of supernatural creation. This means the assumption that the supernatural creator of a universe must also have necessarily wished to have created life in it. This is an arbitrary assumption, since we have no knowledge of what a supernatural creator of a universe may have wished or not wished to do. Hence SNAP, i.e. expressions (10a) and (10b) also are arbitrary assumptions. Hence the Supernatural Anthropic Principle is logically unsubstantiated and is an arbitrary assumption.
The conclusion that the universe which is fine-tuned for life implies a supernatural creator is an example of "circular reasoning." In order to conclude that p(S|L)»p(S) which is a succinct representation of the SNAP, we must first assume that p(L|S)»p(L), i.e. assume a priori the existence of a supernatural agent who wished and planned to create life. But the latter assertion is exactly what was to be proven by the entire discourse.
Of course, establishing the arbitrariness of an assumption does not mean that such an assumption is necessarily wrong. It does mean, though, that one may not assert that such an assumption is correct. At best, the question about the correctness of such an assumption remains open until some convincing proofs of its being either correct or not are found. As the matter stands now, no such proofs have been suggested, so the assertion that the values of physical constants point to the supernatural origin of the universe and life remains an unsubstantiated assumption, reflecting religious preferences rather than factual evidence.
The above simple probabilistic discourse has shown that the supernatural interpretation of the anthropic principle, so popular among proponents of creationism, both of explicit and implicit kinds, is logically unsubstantiated. A further investigation of the logic of that proposition such, for example, as that by Michael Ikeda and Bill Jefferys (see The Anthropic Principle Does Not Support Supernaturalism) of which I did not yet know when first writing the above discussion, and which is based on the full form of Bayes's theorem, has provided an even stronger refutation of the hypothesis of a supernatural creation of life. That discourse seems to have shown that the more "fine-tuned" for life a universe is, the less likely is its supernatural origin.
The above discussion can be applied, with a slight modification, to the discussion of a logical procedure known as "abduction." William A. Dembski and Stephen C. Meyer provide  an explanation of the abduction procedure which differs in some respects from that concept as it is normally used in both philosophy of science and in artificial intelligence. Since this discussion is about Dembski & Meyer's use of that procedure in their attempt to prove the divine source of the big bang, I will not delve into the differences between their interpretation of abduction and that which is more commonly adopted but will rather discuss their paper adopting their definition of abduction.
Deduction is a process wherein a certain event A is believed to have actually happened. If some other event B is an inevitable consequence of A (we say in that case that A entails B), we infer from the occurrence of A that B necessarily must occur as well. Dembski and Meyer  present the deduction procedure in the following form:
Data: A is given and plainly true.
Logic: But if A is true than B is a matter of course.
Conclusion: Hence, B must be true as well.
On the other hand, according to Dembski and Meyer, abduction is a way of inference in certain respects opposite to deduction. Here is how Dembski and Meyer present the abduction procedure:
Data: The surprising fact A is observed
Logic: But if B were true, then A would be a matter of course
Conclusion: Hence, there is a reason to suspect that B is true.
It seems obvious that the inference in the first case is very strong and the conclusion is as certain as it possibly can be. However, in the second case, the inference is only tentative -- it only indicates the possibility of B being true. Indeed, while if B were true, A would be a matter of course, it does not mean that there are no other C, D, E, etc, which would entail A as well, and therefore the occurrence of A (the consequent) does not define the antecedent as being necessarily B, but only indicates that B is one of the possible antecedents.
Despite the relative weakness of inference by a way of abduction (in Dembski-Meyer sense), it is often resorted to as the only available path to a conclusion. However, if and when abduction has been utilized, one has to be aware of the tentative character of the conclusion which, if it is expected to be accepted as true, requires additional supporting evidence.
One of the examples of an unsubstantiated utilization of abduction is the inference of the supernatural origin of the big bang. In particular, such an inference is offered in the article  by Dembski and Meyer. In his book "The Design Inference"  Dembski turns several times to Bayes's theorem, showing familiarity with that version of a probabilistic analysis. However, in his paper  with Meyer, Dembski seems to have forgotten about Bayes's theorem, which is quite relevant to the discussion of the abduction-type inference. Actually, Bayes's theorem formulates the abduction inference (in Dembski-Meyer's sense) in probabilistic terms.
Let p(S) be again the probability that there is a supernatural agent responsible for the creation of the universe. Let p(S|BB) be the probability of the existence of that supernatural agent provided the big bang has indeed occurred. Let, further, p(BB) be the probability of the big bang's occurrence, and p(BB|S) be the conditional probability of the big bang's occurrence provided there is a supernatural agent responsible for the creation of the universe. Then our particular, simplified version of Bayes's theorem - equation (6) - takes the form:
The abduction inference, according to Dembski-Meyer's schema, if presented using the above version of Bayes's theorem, can be as follows:
Since, as they maintain, there exists no good explanation entailing any natural cause of the big bang, then in their view the hypothesis of a supernatural agent being the source of the big bang is a better explanation. Hence, if the big bang occurred, this enhances the probability of a supernatural agent being its source:
Note that (14) renders Dembski/Meyer's conclusion of their abduction argument in probabilistic terms.
In accordance with (13), if (14) holds, then also
In plain words, equation (15) means the assertion that the existence of a supernatural creator substantially enhances the probability of the occurrence of the big bang. This equation renders the "logic" step of Dembski/Meyer's abduction inference in probabilistic form. (The "data" step of their abduction argument is the assumption that the big bang had actually occurred.) Hence, according to Dembski-Meyer's abduction inference, if we assume that the big bang had actually occurred, there is a good reason to assume that a supernatural agent does exist, which is expressed by inequality (14).
The fallacy of that inference is in the inequality (15) - p(BB|S)» p(BB). This assumption asserts that the probability of the actual occurrence of the big bang is substantially enhanced if there is a supernatural creator of the universe. In terms of the abduction inference, it means that if a supernatural agent responsible for the creation of the universe does exist, the big bang is a matter of course. Obviously this is an arbitrary assumption because we have no knowledge about how a supernatural agent might act. If (15) is an arbitrary assumption, so is the conclusion - inequality (14).
Of course, again, the arbitrariness of the above assumption does not prove that it is wrong. However, it shows the lack of substantiation for Dembski-Meyer's hypothesis of a supernatural origin of the big bang, which is only one of many possible explanations. It has no advantage over any other explanation, including those denying supernatural creation.
 Hugh Ross, Big Bang Model Redefined by Fire, in coll. Mere Creation, ed. W. Dembski, InterVarsity Press, 1998.
 Nathan Aviezer, The Anthropic Principle, Jewish Action, Spring 1999.
 Fred Heeren, Show Me God, Day Star Publications, 2000.
 Patrick Glynn, God. The Evidence, Forum Publishers, 1999.
 Walter L. Bradley, The "Just So Universe: The Fine-Tuning of Constants and Conditions in the Cosmos, in coll. Signs of Intelligence, eds. W. Dembski and M. Kushiner, Brazos Press, 2001.
 Stephen Hawking, A Brief History of Time, Bantam Books, 1988.
 John D. Barrow and Frank J. Tipler, The Anthropic Cosmological Principle, Oxford University Press, 1986.
 Martin Gardner, WAP, SAP, PAP AND FAP, The New York Review of Books, May 8, 1986.
 C. Howson and P. Urbach, Scientific Reasoning: The Bayesian Approach, LaSalle. Ill. Open Court, 1993.
 William A. Dembski and Stephen C. Meyer, Fruitful Interchange or Political Chitchat? In coll. Science and Evidence for Design in the Universe, eds. M.J. Behe, W.A. Dembski and S.C. Meyer, Ignatius Press, 2000.
 William A. Dembski, The Design Inference, Cambridge University Press, 1999.