When you have eliminated the impossible, whatever remains, however improbable, must be the truth.

Fig. 1 - Sir Arthur Conan Doyle.

Long held to be an inviolable rule of logic, this is actually a sentence often placed in the mouth of fictional detective Sherlock Holmes by his creator Sir Arthur Conan Doyle (Fig. 1). At its core is the straightforward process of elimination. And lately it has become the watch-cry of conspiracists who quite obviously want us to believe in something improbable, and quite often attempt to elicit that belief by appearing to "eliminate the impossible."

The statement is true enough under the right consequences. In deductive logic we know not to reject a proposition for which a suitable proof has been constructed, simply on the basis that it seems absurd. A deductive case, if inferred according to sound rules of inference from premises known to be correct, is airtight.

In mathematical reasoning the Holmesian Maxim is crucial. An integer, for example, must either be positive, negative, or zero. If we wish to prove some given value is negative and a direct proof is problematic, but we can show that it is neither positive nor zero, then we will have proved rigorously by indirection that it must therefore be negative. This would hold even if a prior informal assessment of the problem suggests otherwise. Very often we must rely on proving what something is not, and this is a powerful tool in categorical logic.

However, in inductive reasoning the Holmesian Maxim does not provide a helpful strategy.

Recall that inductive reasoning starts with specific observations and attempts to collate them to form a single, more general conclusion. This is what happens in a court of law. Bits of evidence are presented which, when considered collectively, help the court arrive at a general declaration of guilt or innocence. Inductive cases are never airtight the way deductive or mathematical proofs must be. Induction does not assure correctness, and as such relies on a certain allowance for error, known as "inductive skepticism".

Induction governs historical investigations, which covers the explanation of all phenomena and the characterization of all events whether recent or ancient. The same principles that govern an investigation of how the Romans breached the fortress of Masada also govern the identification of a UFO sighted yesterday: one begins with observations of artifacts and accounts and then forms a picture of what really happened. The historian has no delusion of being able to do so conclusively or correctly. He merely strives to draw a conclusion which among its peers is the most likely to explain the given evidence material.

Therein lies the difficulty. In the mathematical example above, we can perfectly enumerate the available possibilities: positive, negative, and zero. Categorical reasoning by definition presents us with such clearly partitioned categories. But this esoteric purity is not available in inductive investigations. There we are confronted with the reality that an empirical observation might quite literally have been produced by almost anything. And we are also faced with the possibility that the one true answer may lie beyond our ability to imagine it, and therefore even to consider or test it.

Let us say that again: the true antecedent to any observation may not occur to us during our investigation. The indirect method of reasoning, expressed in the Holmesian Maxim, requires that we specifically enumerate all the possibilities that may apply to the observation -- whether or not we favor them. And it requires us to know conclusively that we have considered all of them. In an artificial context such as mathematics, blessed with rigid definitions, this assurance of completeness is easy to come by. Not so in the "messy" real world.

Even if we could establish, by some good fortune, the assurity of having enumerated all the possibilities in preparation for examining them, we would still have the task of soundly disproving them. And in a historical inquiry that means an empirical proof with limited available evidence -- in other words, a limited ability to draw unassailable conclusions. Bear strongly in mind that an indirect proof requires conclusive disproof of the competing hypotheses. If the argument can show only that some competitor is merely improbable and not fully impossible, then it fails the stipulation of the Holmesian Maxim. The choice is then between two improbables, not the assertion of an improbability over an impossibility. And then the relative merits of each become important and one must supply direct proof for the desired proposition.

Historical inquiries are limited to the evidence at hand. The nature of empirical investigations does not allow us to develop crucial evidence on a whim. And so it may happen that the bit of evidence necessary to substantiate or eliminate a hypothesis is simply not available and not likely to be forthcoming. In the absence of key evidence we cannot rule either way on the hypothesis, and that conspicuously disallows us from saying it is impossible. And this means we cannot use an indirect proof.

Disproof often requires the proof of a negative proposition. And proving a negative is almost always difficult. For example, to disprove the hypothesis

The shadow in that photograph was cast by the sun.

The shadow in that photograph was not cast by the sun.

The practical impossibility of assuring a complete set of competing hypotheses, together with the limitations on disproving them constitute an imposing barrier to successful inductive proofs constructed according to the Holmesian blueprint.

So why is this method so common among conspiracists?

First, it creates the illusion of support for a proposition that has no direct evidence at all in favor of it. Since most conspiracy-related propositions are pure conjecture, a direct proof is not possible. The conspiracist would have nothing to write at all if not for the practice of indirect proof.

Second, an indirect proof carries a semblance of rigor. If the author is unable to fully enumerate the competing hypotheses then it's unlikely the sympathetic reader will be able to; and thus he won't necessarily notice the absence of a serious competitor that the author has failed to consider. Unless the reader is predisposed to dig deeper than the author, he will likely consider the case complete and decisive.

This circumstance is especially effective on specialized topics such as science. Most readers are not experts in all technical matters. And so competing hypotheses that arise from obscure -- but nevertheless valid -- scientific principles are likely to be missed by both author and reader. And when a critic brings up these obscure principles in his objection to the conspiracy theory, he can often be accused of trying to muddy what would otherwise be a "straightforward" case.

In this devious way, the conspiracist pares down the set of possibilities to something which appeals to the "common sense" intuition of the reader. The reader is spoon-fed just enough information to validate an overly simplistic view of the problem. And this simpleton view then generates what appears to be a small set of competing hypotheses, which the author can usually convince the reader is complete. Then after disqualifying each of this small set of straw men as an explicator, the author proclaims his desired proposition vindicated by process of elimination.

Third, the Holmesian Maxim supplies language to predispose the reader to accept a conclusion he might otherwise reject as absurd. The astute conspiracist author realizes that his controversial proposition will encounter skepticism. By introducing his attempt at indirect proof with the Holmesian Maxim, the author imparts a degree of intellectual comfort to the reader who can then accept the proposition against his better judgment. The reader believes himself to have remained rational if he accepts a preposterous conclusion that nevertheless must be true by the process of elimination.

While conspiracists can easily create indirect inductive proofs that seem rigorous even when applied to baseless propositions, they seldom acknowledge the ease with which such indirect proofs can be refuted. The two impassable obstacles in an indirect inductive proof -- assurance of completeness and strength of elimination -- give predictable rise to the two basic methods of refutation.

Any plausible competing hypothesis that the author does not consider in his indirect proof, is sufficient refutation of the proof. It does not matter whether the competitor's proponent is able to prove the competing hypothesis in the specific case. It matters only whether the author is able to disprove it in the specific case. The author has the burden of proof to "eliminate the impossible". The critic's burden of proof is for mere plausibility -- that it is "not impossible". So saith the Holmesian Maxim.

Since each competitor must be conclusively eliminated, the strength of each eliminative proof must be aggressively tested. As noted above, the eliminations are, by nature, difficult proofs to construct to sufficient rigor. And the lack of empirical evidence may eliminate the testability altogether, in which case impossibility may not be assumed. But very often the simplism of a putative elimination is its own undoing; it may serve only to suggest that a hypothesis is improbable, not that it is truly impossible. And as stated above, this reduces the argument to an evaluation of relative probability among improbable hypotheses.

To compare one hypothesis to another on the basis of its relative probability is the process of direct inductive proof. One must examine the merits of the desired hypothesis, not the conspicuous lack of merit in all its competitors. And because an indirect inductive proof invariably reduces, upon scrutiny, to a direct proof, the smart proponent adopts a direct proof strategy at the outset. And knowing that a purely conjectural hypothesis cannot prevail according to a direct proof, the smart proponent avoids advancing a conjectural hypothesis altogether. And this leaves the Holmesian Maxim safely where it belongs -- away from the inductive case.