Why Sherlock Holmes' reasoning is illogical

The Sherlock Holmes character has often been assumed to be logical in his analysis of situations. Here are a couple of examples of his reasoning from two of Conan Doyle's stories.

'In solving a problem of this sort, the grand thing is to be able to reason backward. That is a very useful accomplishment, and a very easy one, but people do not practice it much. In the everyday affairs of life, it is more useful to reason forward, and so the other comes to be neglected. There are fifty who can reason synthetically for one who can reason analytically'.

'I confess’, said I, ‘that I do not quite follow you'.

'I hardly expected that you would. Let me see if I can make it clearer. Most people, if you describe a train of events to them, will tell you what the result would be. They can put those events together in their minds, and argue from them that something will come to pass. There are few people, however, who, if you told them a result, would be able to evolve from their own inner consciousness what the steps were which led up to that result. This power is what I mean when I talk of reasoning backward, or analytically’.

(Doyle, 1887, pp. 83-84, A Study in Scarlet)

So, what Sherlock Holmes means by people who reason 'synthetically’ is that they can work out that B is the case whenever A is the case, i.e., modus ponens. Sherlock, however, argues here that he reasons 'analytically' and ‘reasons backward' so that given B, then A should follow. As we have seen, this is affirming the consequent. It is a form of reasoning known as abduction (see Figure 4.2). An example of what can be derived using this form of logic is shown in the next example.

'I can see nothing’, said I, handing [the hat] back to my friend.

'On the contrary, Watson, you can see everything. You fail, however, to reason from what you see. You are too timid in drawing your inferences’.

'Then, pray tell me what it is that you can infer from this hat?'

He picked it up and gazed at it in the peculiar introspective fashion which was characteristic of him. 'It is perhaps less suggestive than it might have been’, he remarked, 'and yet there are a few inferences which are very distinct, and a few others which represent at least a strong balance of probability. That the man was highly intellectual is of course obvious upon the face of it, and also that he was fairly well-to-do within the last three years, although he has now fallen upon evil days. He had foresight, but has less now than formerly, pointing to a moral retrogression, which, when taken with the decline of his fortunes, seems to indicate some evil influence, probably drink, at work upon him. This may account also for the obvious fact that his wife has ceased to love him'.

'My dear Holmes!'

'He has, however, retained some degree of self-respect', he continued, disregarding my remonstrance. 'He is a man who leads a sedentary life, goes out little, is out of training entirely, is middle-aged, has grizzled hair which he has had cut within the last few days, and which he anoints with lime-cream. These are the more patent facts which are to be deduced from his hat. Also, by the way, that it is extremely improbable that he has gas laid on in his house’.

(Doyle, 1892, p. 246, The Blue Carbuncle)

So, where does Sherlock get these conclusions from? Here are a few of the features of the hat that he uses: The hat is expensive but now out of date and hasn't been replaced (so the owner has ‘fallen upon evil days’). Sweat stains show he is out of training, several tallow drips from a candle suggest he does not have gas. The hat has not been brushed, therefore he does not now have a wife to either exhort him to brush it or to brush it for him. As with the case with ‘if it rains then I will get wet. I am wet’, there may well be other explanations for the details that Sherlock used to derive his conclusions. Maybe the man had stolen the hat, for instance. However, an abductive inference is an explanatory hypothesis and the likelihood of it being accurate depends on the quality of the explanation given (Lombrozo, 2012; Thagard, 1989). Important words in the last excerpt include ‘probability", ‘seems to indicate', ‘probably’, and ‘improbable’, which attests to the view that Conan Doyle was well aware that abductive reasoning can only be probabilistic. I am obviously using abductive reasoning here, though.

The three types of reasoning discussed are shown dia-grammatically in Figure 4.2.