The Inductive Detective

I was watching an old episode of Sherlock Holmes last night – from the classic  Granada TV series featuring Jeremy Brett’s brilliant (and splendidly camp) portrayal of the eponymous detective. One of the  things that fascinates me about these and other detective stories is how often they use the word “deduction” to describe the logical methods involved in solving a crime.

As a matter of fact, what Holmes generally uses is not really deduction at all, but inference (a process which is predominantly inductive).

In deductive reasoning, one tries to tease out the logical consequences of a premise; the resulting conclusions are, generally speaking, more specific than the premise. “If these are the general rules, what are the consequences for this particular situation?” is the kind of question one can answer using deduction.

The kind of reasoning of reasoning Holmes employs, however, is essentially opposite to this. The  question being answered is of the form: “From a particular set of observations, what can we infer about the more general circumstances that relating to them?”. The following example from a Study in Scarlet is exactly of this type:

From a drop of water a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other.

The word “possibility” makes it clear that no certainty is attached to the actual existence of either the Atlantic or Niagara, but the implication is that observations of (and perhaps experiments on) a single water drop could allow one to infer sufficient of the general properties of water in order to use them to deduce the possible existence of other phenomena. The fundamental process is inductive rather than deductive, although deductions do play a role once general rules have been established.

In the example quoted there is  an inductive step between the water drop and the general physical and chemical properties of water and then a deductive step that shows that these laws could describe the Atlantic Ocean. Deduction involves going from theoretical axioms to observations whereas induction  is the reverse process.

I’m probably labouring this distinction, but the main point of doing so is that a great deal of science is fundamentally inferential and, as a consequence, it entails dealing with inferences (or guesses or conjectures) that are inherently uncertain as to their application to real facts. Dealing with these uncertain aspects requires a more general kind of logic than the  simple Boolean form employed in deductive reasoning. This side of the scientific method is sadly neglected in most approaches to science education.

In physics, the attitude is usually to establish the rules (“the laws of physics”) as axioms (though perhaps giving some experimental justification). Students are then taught to solve problems which generally involve working out particular consequences of these laws. This is all deductive. I’ve got nothing against this as it is what a great deal of theoretical research in physics is actually like, it forms an essential part of the training of an physicist.

However, one of the aims of physics – especially fundamental physics – is to try to establish what the laws of nature actually are from observations of particular outcomes. It would be simplistic to say that this was entirely inductive in character. Sometimes deduction plays an important role in scientific discoveries. For example,  Albert Einstein deduced his Special Theory of Relativity from a postulate that the speed of light was constant for all observers in uniform relative motion. However, the motivation for this entire chain of reasoning arose from previous studies of eletromagnetism which involved a complicated interplay between experiment and theory that eventually led to Maxwell’s equations. Deduction and induction are both involved at some level in a kind of dialectical relationship.

The synthesis of the two approaches requires an evaluation of the evidence the data provides concerning the different theories. This evidence is rarely conclusive, so  a wider range of logical possibilities than “true” or “false” needs to be accommodated. Fortunately, there is a quantitative and logically rigorous way of doing this. It is called Bayesian probability. In this way of reasoning,  the probability (a number between 0 and 1 attached to a hypothesis, model, or anything that can be described as a logical proposition of some sort) represents the extent to which a given set of data supports the given hypothesis.  The calculus of probabilities only reduces to Boolean algebra when the probabilities of all hypothesese involved are either unity (certainly true) or zero (certainly false). In between “true” and “false” there are varying degrees of “uncertain” represented by a number between 0 and 1, i.e. the probability.

Overlooking the importance of inductive reasoning has led to numerous pathological developments that have hindered the growth of science. One example is the widespread and remarkably naive devotion that many scientists have towards the philosophy of the anti-inductivist Karl Popper; his doctrine of falsifiability has led to an unhealthy neglect of  an essential fact of probabilistic reasoning, namely that data can make theories more probable. More generally, the rise of the empiricist philosophical tradition that stems from David Hume (another anti-inductivist) spawned the frequentist conception of probability, with its regrettable legacy of confusion and irrationality.

My own field of cosmology provides the largest-scale illustration of this process in action. Theorists make postulates about the contents of the Universe and the laws that describe it and try to calculate what measurable consequences their ideas might have. Observers make measurements as best they can, but these are inevitably restricted in number and accuracy by technical considerations. Over the years, theoretical cosmologists deductively explored the possible ways Einstein’s General Theory of Relativity could be applied to the cosmos at large. Eventually a family of theoretical models was constructed, each of which could, in principle, describe a universe with the same basic properties as ours. But determining which, if any, of these models applied to the real thing required more detailed data.  For example, observations of the properties of individual galaxies led to the inferred presence of cosmologically important quantities of  dark matter. Inference also played a key role in establishing the existence of dark energy as a major part of the overall energy budget of the Universe. The result is now that we have now arrived at a standard model of cosmology which accounts pretty well for most relevant data.

Nothing is certain, of course, and this model may well turn out to be flawed in important ways. All the best detective stories have twists in which the favoured theory turns out to be wrong. But although the puzzle isn’t exactly solved, we’ve got good reasons for thinking we’re nearer to at least some of the answers than we were 20 years ago.

I think Sherlock Holmes would have approved.

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15 Responses to “The Inductive Detective”

  1. Anton Garrett Says:

    A friend of mine used to run the Sherlock Holmes Society of London and was kind enough to give me membership after I gave them a talk on Professor Moriarty’s science – Moriarty is recorded as having written a treatise called The Dynamics of an Asteroid. Many people in the society considered Jeremy Brett the best screen Holmes.

    Karl Popper’s notion of falsifiability is pernicious. Whoever thought it was good to see your theory proved WRONG? Probability theory, done properly (ie the objective Bayesian view), IS the same thing as inductive logic (altough Popper would disagree). His problem is that by rejecting induction he was rejecting the only way to compare competing theories in the light of inevitably noisy data – given the data, which theory (eg, Newton vs Einstein) is more *probable*?

    Of course, the REALLY hard part is not hypothesis testing, of which this is simply an Olympian example, or even getting the philosophy right, but hypothesis creation – coming up with a theory that is novel and consistent with all previous physics while being capable of explaining new stuff. For that we remain dependent on the minds of giants like Newton, Maxwell, Schroedinger, Dirac, Einstein.


    • telescoper Says:

      You can imagine the fun we had at Nottingham when a colleague in the Physics department called Phil Moriarty was promoted to Professor…

    • ps. Jeremy Brett’s life was rather tragic – after his wife, Joan Sullivan, died he struggled with a depressive illness while filming these series, and he eventually died from complications caused by his medication. The combination of lethargy and manic energy that he brought to the role of Sherlock Holmes was something he knew all about from his own illness.

  2. Peter,

    I’m not sure you’re right to call Hume an “anti-inductivist”. Indeed, he stressed the importance of inductive reasoning in our everyday lives: the Wikipedia page on Hume to which you link includes the quote that “Nature, by an absolute and uncontroulable necessity has determin’d us to judge as well as to breathe and feel”. His main concern was just to make clear the distinction, as you did, between deductive and inductive reasoning and then to stress the difference between the basis on which we apply each – logical necessity versus psychological/social propensity.


    • Bob,

      Hume concluded that inductive reasoning was not a rational means of inference, but one to do with custom and habit. It was that distinction that spawned the irrationalist tradition that led to Popper and ultimately reached its dead end in Feyerabend. I contend that there’s not much in the work of these philosophers that can’t be traced back to Hume.


  3. Peter,

    I don’t doubt any of that chain of influence, but I don’t think Hume saw anything wrong with it being custom and habit that is the basis for our use of inductive reasoning – hence, my feeling that calling Hume “anti-inductive” is misleading: inductive reasoning certainly had its place for Hume, and an important one, just a different one from deductive reasoning.


  4. Anton Garrett Says:


    Have a look at David Stove’s writings to see that Hume – whom Stove otherwise admired – WAS anti-inductivist. This work is Stove at his least polemical. (His polemics are not everybody’s cup of tea although I enjoy them.)


  5. [...] look at a star by glances.. I’ve blogged before about my love of classic detective stories and about the intriguing historical connections between astronomy and forensic science. However, I [...]

  6. [...] For comments on induction versus deduction in another context, see here. [...]

  7. [...] and related this to the processes involved in solving puzzles. That piece makes a number of points I’ve made before, so I thought I’d update and recycle my previous post on that theme. The main reason for [...]

  8. mike tabacco Says:

    Thank u for this! I came to same conclusion as your ‘thesis’ and was looking for some outside agreement…’d think Sherlock writers would be more accurate with their use of deduv vs inductiv..

  9. […] interplay between analysis and synthesis (and between deductive and inductive reasoning) involved not only in detective stories but also – I would contend – in the scientific method generally. I  agree with Poe […]


    PLEASE, and with apologies as I fail to relate this to the arguments on inductive reasoning, but…I’m unable to find video for Jeremy Brett in “A Study in Scarlet” anywhere and it’s not even listed in the titles of the Granada boxed set. I believe this was the first and introductory episode, a “special”, but again am unable to locate it anywhere on the interwebs or other equally confusing places. Please if you would, Induct me into the mystery of how you were watching this and where I can get a copy of the same?

    • telescoper Says:

      Apologies for the confusion. It wasn’t a Study in Scarlet I was watching – but the quote later comes from that. I don’t think Jeremy Brett ever made “A Study in Scarlet” actually. It’s not in my Box set but and this story is one that has been adapted for film and TV much less frequently than the other stories.

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