In the opening passage of THOR, we hear Holmes apologizing: "I am getting into your involved habit, Watson, of telling a story backward." (THOR 1056) There is surely an element of implicit criticism in this apology, suggesting that "backwards" is not the proper way to tell a story. But is this criticism really justified? In SIGN, Holmes criticizes Watson’s account of STUD for being too much concerned with romantic details instead of concentrating on "the curious analytical reasoning from effects to causes" (SIGN 90). But in this mode of reasoning, as Holmes points out himself, the natural course of events is reversed: "the grand thing is to be able to reason backward" (STUD 83). So, if the Good Doctor occasionally did tell a story backwards, was he then not simply following the natural train of reasoning, just as Holmes wanted him to? And did he then not deserve praise instead of blame?

In this essay, I will examine Holmes’s reasoning in THOR, which I take to be another instance of the kind of backward reasoning that is so distinctive of the Holmesian method. This is the kind of reasoning that logicians nowadays call abduction, as opposed to ordinary logical inference, or deduction. In a deductive inference, the conclusion is a statement which follows logically from the given facts and which therefore, in a way, does not contain any new information. For example:

Premises: If it rains, the streets will be wet.

It is raining.

Conclusion: The streets will be wet.

If we know that the premises hold true, then we know (implicitly) that the conclusion holds true. By contrast, abduction means inferring (hypothetically) a statement from which the given facts follow – instead of the other way round – and which therefore provides an explanation of these facts. Thus:

Premises: If it rains, the streets will be wet.

The streets are wet.

Conclusion: It has been raining.

The conclusion in this case does not follow logically from the given facts (the streets may be wet because someone has sprayed water), but if it is true that it has been raining, then this explains the fact that the streets are wet. And explaining facts is the business of the detective (consulting or otherwise), as Holmes points out to Miss Dunbar in her time of despair: "If you will find the facts, perhaps others may find the explanation." (THOR 1066)

However, since abduction is not a logically valid mode of inference in the same way as deduction, it must be applied with great care. There is always a possibility of error, and there are always alternative explanations possible for any given set of facts. This is why it is not sufficient for the detective to be able to observe facts and draw inferences (abductive or deductive). He must also have the capacity to conceive the possible alternatives, a capacity that first and foremost requires an extensive knowledge of previous cases that present similar features. This is very clearly pointed out by Holmes when discussing the merits of his French colleague François le Villard: "He possesses two of the three qualities necessary for the ideal detective. He has the power of observation and that of deduction. He is only wanting in knowledge, and that may come in time." (SIGN 91) However, sometimes not even extensive knowledge is enough, because some cases present unique features, and then the detective must rely on his powers of imagination. Again, this is noted by Holmes in relation to the shortcomings of a fellow detective: "See the value of imagination [...] It is the one quality which Gregory lacks." (SILV 344)

Given the requisite knowledge and imagination, the reasoning detective can use abduction to infer possible explanations for the given facts. However, he must also have a method for distinguishing the correct explanation from the erroneous alternatives. This is where the famous process of elimination comes in: "Eliminate all other factors, and the one which remains must be the truth." (SIGN 92) This elimination is carried out through the collection of new facts, facts which support some explanations and contradict others, until finally only one explanation remains which accounts for all the facts. In the words of the Master himself: "That process [...] starts upon the supposition that when you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth. It may well be that several explanations remain, in which case one tries test after test until one or other of them has a convincing amount of support." (BLAN 1011)

Let us now turn to consider, in some detail, how the Holmesian method of backward reasoning is put to use in the particular case of THOR. Let us first make a brief review of the basic facts, as they appear at the beginning of the case (THOR 1056-1057):

These facts, together with Miss Dunbar’s lack of alibi (she even admitted that she had seen Mrs Gibson at the scene of the crime) and her possible motive (the prospect of marrying the Gold King), were enough for the coroner’s jury and the police-court to charge Miss Dunbar with the murder of Mrs Gibson.

I want to argue that the whole chain of Holmes’s reasoning in this case starts from the single problematic fact F2, a fact which, in Holmes’s view, is not properly explained by the official theory that miss Dunbar murdered Mrs Gibson. The passage where Holmes first becomes aware of this fact is worth quoting in full:

There is no doubt that, already at this point, Holmes is beginning to question the theory that miss Dunbar murdered Mrs Gibson because, although this theory would explain some of the given facts (F1 and F3), it is hardly compatible with the fact that the revolver was found on the floor of Miss Dunbar’s wardrobe (F2), something that Holmes’s knowledge of previous cases tells him very clearly. As he later explains to Dr Watson: "Where a crime is coolly premeditated, then the means of covering it are coolly premeditated also. I hope, therefore, that we are in the presence of a serious misconception." (THOR 1065)

Thus, in my view, the fact that the revolver was found in Miss Dunbar’s wardrobe suggests to Holmes from the very beginning that Miss Dunbar did not murder Mrs Gibson. Moreover, if Miss Dunbar did not murder Mrs Gibson, and did not put the revolver in the wardrobe, then it can be inferred (abductively) that someone else must have put the revolver there, for where there is an effect there must be a cause. And, to follow the causal chain backwards even further, the revolver was probably put there to incriminate Miss Dunbar, and it was probably put there by the real murderer. As Holmes later explains: "When once your point of view is changed, the very thing which was so damning becomes a clue to the truth. For example, there is the revolver. Miss Dunbar disclaims all knowledge of it. On our new theory she is speaking the truth when she says so. Therefore, it was placed in her wardrobe. Who placed it there? Someone who wished to incriminate her. Was not that person the actual criminal? You see how we come at once upon a most fruitful line of inquiry." (THOR 1065) This, I want to argue, is how far Holmes’s reasoning has proceeded before he has even seen his client for the first time (although he does not reveal it to Watson until much later). To be able to reason backwards is indeed "the grand thing"!

The next logical step in Holmes’s method is to start conceiving alternative explanations for the facts of the case (still using backward reasoning, or abduction). Let us see if we can follow in the Master’s footsteps and find out what the conceivable alternatives are at this point. There are, as Holmes notes, "three persons concerned" (THOR 1056) in the case: the American Gold King, his Brazilian wife, and the English governess. This looks very much like a classical ménage à trois, and it seems very probable, from the outset, that the person who killed Mrs Gibson is to be found within this group. Still, we must not confuse the impossible with the improbable, and our first alternative theory must therefore be:

It seems, however, that Holmes more or less excludes this possibility from the beginning (cf. the earlier quote about "three persons concerned"). With the benefit of hindsight, we can of course say that he is right in doing so, but on what grounds does he dismiss T1 as a possible explanation? I can only assume that this is where Holmes’s vast experience of previous cases comes into play, an experience that tells him that T1 is extremely unlikely in the circumstances at hand. Still, it is clear that Holmes is taking a definite risk in discarding T1 offhand. Even during the final demonstration on Thor Bridge Watson notes that "my friend under all his habitual coolness was in truth deeply agitated" and Holmes himself admits that "one can always conceive alternative explanations which would make our scent a false one". (THOR 1069)

For the sake of completeness, I will put down the official police theory as our second alternative:

T2. Miss Dunbar murdered Mrs Gibson.

As I have already explained, this theory is discarded by Holmes because it does not properly explain the fact that the revolver was found in Miss Dunbar’s wardrobe (F2). However, Holmes also realizes that, as long as no convincing alternative theory can be found, his reasoning is unlikely to impress a judge and jury and is therefore of limited practical value to Miss Dunbar and to his client. We must therefore go on to consider the next logical possibility:

T3. Miss Dunbar killed, but did not murder Mrs Gibson.

That Miss Dunbar killed Mrs Gibson accidentally or in self-defence is suggested to Holmes by Mr Gibson, but it is clear that Holmes has already considered this possibility: "That possibility had already occurred to me, [...] Indeed, it is the only obvious alternative to deliberate murder." (THOR 1062) The adjective "obvious" is important here; while T3 may be the only obvious alternative to T2, we have already seen that it is not the only possible alternative. And there is more:

T4. Mr Gibson killed Mrs Gibson.

This theory is suggested by Sergeant Coventry, but again it is clear that Holmes has been aware of this possibility from the start: "I have been considering that." (THOR 1062) This theory, however, is ruled out by the fact that Mr Gibson appears to have an alibi for the evening of the murder, although Holmes himself expresses himself more cautiously: "There is no evidence at all that Mr. Gibson had been out of doors since his return from town at five o'clock." (THOR 1064-1065) Furthermore, given the relation between Miss Dunbar and her employer, this theory is hardly compatible with the fact that the revolver was put in Miss Dunbar’s wardrobe in order to incriminate her.

Summing up, there are (at least) four alternative theories; two of them (T2 and T4) can be discarded because they do not fit all the facts, and one of them (T1) is at best extremely improbable. This, I want to argue, is the result of Holmes’s reasoning before he has even left Baker Street. There is, of course, yet another alternative, but it seems that this possibility has not yet occurred to Holmes, and that this is (part of) the reason why he later observes: "I fear, Watson, [...] that you will not improve any reputation which I may have acquired by adding the case of the Thor Bridge mystery to your annals. I have been sluggish in mind and wanting in that mixture of imagination and reality which is the basis of my art." (THOR 1070)

At this point, Holmes is clearly in need of more facts, facts that may contradict or give support to the alternative theories under consideration. Although Holmes’s first impulse is to go and see Miss Dunbar, there is some delay in obtaining the necessary permits and Holmes and Watson therefore ends up going to the Gibson estate instead. An interview with Sergeant Coventry and an examination of the scene of the crime brings several new facts to their attention (THOR 1062-1063):

Each of these facts can be shown to play a role in Holmes's reasoning. The first two (F4 and F5) I will set aside for later. The third one (F6) is important because it contradicts one of the theories still entertained by Holmes at this point, namely the theory that Miss Dunbar killed Mrs Gibson accidentally or in self-defence as the result of a struggle (T3 above). The fact that there was no trace of a struggle on the scene of the crime – together with the fact that T3 was hard to combine with the assumption that the revolver was put in Miss Dunbar's wardrobe by the person who killed Mrs Gibson in order to incriminate the governess – was probably enough for Holmes to finally discard this theory. Usually the refutation of a theory in the light of new facts is a good thing because it means that at least one alternative has been eliminated. But in this particular case, it meant that all the alternatives had been exhausted and that Holmes was left with no theory at all! In addition, there was the most curious fact of them all (F7): Why was Mrs Gibson still holding Miss Dunbar's note in her hand when she was killed?

All in all, this meant that there were simply too many facts to account for, and none of the theories considered so far (T1-T4) could explain them all. In Holmes’s own words: "Well, Watson, we have come on a good many facts, some of them new ones, and yet I seem some way from my conclusion." (THOR 1064)

Nevertheless, Holmes made one positive discovery at the scene of the crime: there was a small and apparently recently-made chip on the parapet, some fifteen feet from the place where the body was found. I will refer to this as another unexplained fact (F8), although at the time of its discovery it might just as well have been coincidental to the case.

Holmes now had to go and see Miss Dunbar: "We have several very vital questions to ask that young lady" (THOR 1065). The purpose of this interview was again to collect more information that could either lead to a reinterpretation of some of the problematic facts or suggest new theories that could better explain the facts. And the interview with Miss Dunbar revealed the following important facts (THOR 1066-1067):

After being presented with these facts, Holmes at last starts to perceive the true explanation of the extraordinary events at Thor Bridge:

Let us pause for a minute and ask ourselves whether Dr Watson is really justified in speaking of "the supreme manifestation of [Holmes's] genius" in the passage just quoted. Shouldn’t Holmes have known the truth much earlier? I have already quoted the Great Detective himself saying that his reputation would not be improved by the publication of this case, and I think it has to be admitted (albeit reluctantly) that this is a rather fair assessment of the situation.

First of all, although the case may have presented some singular features, the way in which Mrs Gibson met her death was certainly not unique. A remarkably similar case (even involving the chip on the parapet) had been reported by Dr Hans Gross, Professor of Criminology at the University of Prague, in System der Kriminalistik in 1893 (CC-THOR, note 133, page 57-58). Surely Holmes, who prided himself with an exact knowledge of the history of crime, must have known of this case and ought to have seen the similarity. (This is undoubtedly why he later admits to Watson: "I confess that the chip in the stonework was a sufficient clue to suggest the true solution, and that I blame myself for not having attained it sooner." THOR 1070) Secondly, even without knowledge of the previous case, it should have been possible to infer the correct solution on purely logical grounds. Let us consider once again the alternative theories T1-T4:

T2. Miss Dunbar murdered Mrs Gibson.

T3. Miss Dunbar killed, but did not murder Mrs Gibson.

T4. Mr Gibson killed Mrs Gibson.

If all of these theories have been eliminated (as they had been, at least provisionally), then it is clear that one and only one logical alternative remains:

T5. Mrs Gibson killed Mrs Gibson.

And once this alternative is conceived, it should be obvious that it yields by far the best explanation of the given facts:

F1, F4, F5, F6 and F8 are all explained by the way in which Mrs Gibson inflicted death upon herself, as later demonstrated by Holmes on Thor Bridge using Dr Watson’s revolver. F2, F3, F7, F10 and F11 are explained by Mrs Gibson's desire to involve Miss Dunbar "in a fate which was worse far than any sudden death could be" (THOR 1070), which in turn is explained by F9. It can even be argued that, although Holmes’s dramatic demonstration on Thor Bridge may have been necessary in order to convince a judge and jury, it was completely superfluous from a strictly logical point of view, because "when you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth" (BLAN 1011).

At last, then, Holmes had arrived at the theory which could explain the single problematic fact from which the whole reasoning process originated, as well as all the other facts encountered along the way. As always, although perhaps somewhat slower than usual, he had done this by a method of backward reasoning, supported by the gathering of facts to eliminate false alternatives. Finally, on the relation between reasoning and story-telling, I think we may safely conclude that even if the publication of THOR did not improve Holmes’s reputation as a master of reasoning, it must surely have contributed to the "booming" of Dr Watson in the other domain. I am not sure whether this is an example of forward or backward story-telling, but I am quite sure that we wouldn’t want the Good Doctor to tell his stories in any other way!