Wednesday, 20 April 2016

Deduction, Induction, Abduction and Fallacy

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

Why did Sherlock Holmes say this? Because it was in the script, obvs.

This oft-repeated statement, delivered by some of the allegedly most logical characters in all of fiction (not least twice that I'm aware of by Mr Spock), and repeated in the real world by real people, is a quite beautifully meretricious utterance and seems, on the face of it, to be unassailable. 

Most people I encounter think that they have a reasonable grasp of how logic works, but there's a good reason that people study it for decades, namely that it can be difficult. In fact, in all honesty, I can't claim to have mastery of logic myself, but that might be because I understand it just well enough to know precisely how little I understand of it. There's good justification for this view, made concrete in a study by David Dunning and Justin Kruger, and published in a paper entitled Unskilled and Unaware of It: How Difficulties in Recognizing One's Own Incompetence Lead to Inflated Self-Assessments. Of course, it's entirely possible that I'm subject to this effect myself, and entirely unaware of my own incompetence in this area. I'll leave such conclusions, in the tradition of such greats as Newton, to the diligent reader.
 
Any readers who are, like me, avid fans of quiz shows, will have come across a statement in response to a question to the effect that 'My uncle will go mad if I get this wrong, because he's an expert in entomology, so I should know about the mating habits of fruit flies' or some such. Sadly, this is what passes for logic in the general populace and, in my not so very humble opinion, this is what's at the root of many of the societal problems we face today.
 
In this post, I want to pick apart some of the fundamentals of logic and, along the way, expose some of the most pernicious and ubiquitous fallacies I encounter in my sojourn both around the net and in meat-space. By the time I get to the end, I hope to have exposed the reader to enough of how logic really works that they might see the glaring flaws in that statement at the head of the post, and to show that the vast majority of what Sherlock Holmes engaged in was not actually deduction, but induction (or abduction, but we'll come to that). There will be a little repetition of points already made, but here I want to make them more robust.

So, what is logic?

At its most basic, logic deals with how we arrive at conclusions. It deals with what assumptions (premises) lie at the foundation of an argument and whether those assumptions can be taken as axiomatic (whether they're true), as well as whether the route from those premises to any conclusion derived therefrom is sequent, i.e. that said conclusions follow, using proper rules of inference, from the premises

Among other things, basic logic focusses on some fairly straightforward rules (there are logics known as 'explosive logics', in which these rules can be questioned, but they have little relevance for our purposes).

These rules can be stated in fairly simple terms:
1. No proposition can be both true and false simultaneously - the Law of Non-contradiction (LNC).
2. Nothing can be both what it is and not what it is - the Law of Identity.
3. For any proposition, either that proposition is true, or its negation is true - the Law of the Excluded Middle (LEM).

I will be coming back to these later in the post but, for the time being, this is the essence of the three laws of classical logic.

Reasoning falls into three main types:

1. Deductive:

Deductive reasoning is reasoning of the type in which, if the premises are true, and the route from premises to conclusion valid, the conclusion is guaranteed to be true. It cannot be otherwise.

The most common form of deductive argument one might encounter is the syllogism, which takes the form of at least two connected premises (P1, P2) followed by a conclusion (C). An example of a valid deductive syllogism follows:

P1: All men are mortal.
P2: Hitler was a man.
C: Hitler was mortal.

Note that I only assert this argument as valid. I make no assertions thus far about its soundness, which can be questioned (I will be doing this shortly). I should also point out that syllogisms are generally only used in the teaching of logic and/or by flim-flam artistes and religious apologists, while their use among logicians is, I'm assured by somebody better qualified than I, non-existent. They're useful for elucidating the core principles of logic, and not much beyond that, except to lend weight to shoddy arguments presented to the uninitiated.

Deduction is often presented as reasoning from the general to the specific but, while this can serve as a vaguely useful first approximation, it fails to capture what deduction is really about. What it's really about is the application of valid reasoning from true premises to a conclusion that cannot fail to be true. When you have all those elements in place, you don't merely have a valid argument, you have a sound argument (soundness dealing with both the truth of the premises and the valid application of rules of inference), and your conclusion is, in this instance, unassailable.

2. Inductive:

Inductive reasoning is reasoning of the type in which the premises, if true, stand as strong evidence that the conclusion is true. In contrast to a deductive argument, in which any conclusion logically following from true premises is necessarily true, an inductive argument is one in which a conclusion logically following from true premises is probably true, for a given value of 'probably'. An example follows:

P1: Every observation of a dropped pencil thus far resulted in the pencil falling to the floor.
C: Every dropped pencil falls to the floor.

I presented it in this manner for a very specific reason, namely to highlight the problem with this type of reasoning, elucidated fairly comprehensively in my last post as 'the problem of induction'.

There are, of course, circumstances in which letting go of a pencil will not result in it falling to the floor. In space, for example, the concepts 'drop' and 'floor' have little meaning, being corollaries to the concept 'down', which is meaningless in a weightless environment.  

It's also known that inductive reasoning is fallible. As a concrete example, let's imagine ourselves in a world in which Galileo had not made his famous observations of bodies falling from the tower at Pisa, or his experiments with inclined planes. It would be reasonable to inductively assume that heavier things fall faster and, indeed, this is what was widely thought prior to Galileo. Imagine then seeing the video of David Scott dropping a hammer and a feather on the moon (setting aside for the moment that, without the understanding Galileo provided, we might never have made it to the moon):

 

Induction is often presented as reasoning from the specific to the general but, while this can serve as a vaguely useful first approximation, it fails to capture what induction is really about. What it's really about is the application of valid reasoning from true premises to a conclusion that is probably true.

3. Abductive:

Abductive reasoning is heuristic reasoning (a heuristic is a tool of discovery), of the type in which an observation b can be used to infer a hypothesis a. In form, it precisely mirrors the deductive fallacy of affirming the consequent, not least because hypothesis a may not be the only possible explanation for observation b. Thus, abduction is never taken as conclusive, only as a method for choosing between competing hypotheses.

The most important feature of abductive reasoning is parsimony. This feature is popularly known as 'Occam's Razor', named for William of Ockham, a medieval monk, probably because, in one form or another, he is said to have used it fairly often (it's entirely possible that he never used it at all; the first naming of it as Occam's Razor appears in the 19th century, while the principle itself can only reliably be tracked to the 17th). This principle is often horribly misunderstood, so it's worth a little bit of unpacking. 

In most of my encounters of Occam's Razor, or, as I often refer to it, the shaving implement of the late, lamented cleric of Surrey, it's referred to as a principle of simplicity, or 'the simplest hypothesis is the best', but this is a gross oversimplification (see what I did there?) 

In its original Latin form, it's expressed as Non sunt multiplicanda entia sine necessitate, which translates directly into English as 'you shall not multiply entities beyond necessity'. This should nicely highlight the error in treating it as a principle of simplicity. What it should be referred to as is a principle of parsimony, or, in the vernacular, economy. In other words, the most economical hypothesis is generally the one we should go for. Note that it doesn't suggest that the most parsimonious hypothesis is correct, only that we should work with the most economical until it presents us with observational problems. Then, and only then, should we move on to less economical hypotheses.

Critically, we need to understand just what's meant by 'competing hypotheses', because this is another area in which the razor is understood poorly. To show how it really works, we need an example:

H1: The automobile operates by means of an internal combustion engine and a series of gears delivering kinetic energy via driveshafts to the wheels.

H3: The automobile operates by means of an internal combustion engine and a series of gears delivering kinetic energy via driveshafts to the wheels, and a group of pixies pushing from behind.

You'll note that the only distinction between these hypotheses is the pixies. In all other aspects, they're exactly the same. Only in such a scenario can hypotheses be said to be competing, for the purpose of the application of the shaving implement. In reality, we can't actually rule out the pixies in any rigorous sense, but we also can't rule out such absurdities as a Wimbledon champion who played all his matches with a spatula, or a star made entirely of teddy-bears.

We should also note that, in the context of this discussion, an assumption qualifies as an entity. Thus, Occam's Razor can be expressed in its most basic scientific terms as among competing hypotheses, the one with the fewest assumptions should be selected.

All three of the above are employed in science and, overall, the logic of scientific discovery works as follows:

1. Observe phenomenon.
2. Formulate hypothesis (abduction).
3. Compute the consequences of your hypothesis (deduction).
4. Compute a consequence that, if observed, will show your hypothesis to be incorrect (deduction).
5. Devise experiment (or observation) that will show one or other of the above.
6. Observe phenomenon:
7a. If 3 is observed, your hypothesis survives (induction).
7b. If 3 is not observed, your hypothesis is incorrect (deduction).
7c. If 4 is observed, your hypothesis is incorrect (deduction), and should be modified (at least) or discarded. 
8. Rinse and repeat.

For the most part, a really good scientist should be working hard to break his hypothesis, or show it to be incorrect. This isn't just because it's good science, it's also because, if the person proposing the hypothesis hasn't spotted all of the opportunities to falsify his hypothesis, he's likely to be embarrassed by somebody who has. We saw an example of that only 18 months or so ago, with the BICEP2 results showing B-mode polarisation in the cosmic microwave background, announced in a blaze of glory, then scuppered when another team spotted the fatal flaw, namely that the BICEP team hadn't corrected for contributions from interstellar dust, a known potential source of B-mode polarisation. Once the result was corrected for this, the polarising signal faded to statistical insignificance.
 
For most of the rest of this post, I want to focus on deduction, and especially its pitfalls, as these are the areas in which logic fails in the real world.

There's a principle known to computer nerds as 'GIGO', which stands for 'garbage in, garbage out'. It's an important idea in logic, and one that we should spend a little time on. Let's go back to our original syllogism:

P1: All men are mortal.
P2: Hitler was a man.
C: Hitler was mortal. 

Are the premises in the above syllogism true? Dunno, but there's sufficient information in place to suggest that they might be suspect. Are all men mortal? It certainly seems so, but this premise runs hard up against the problem of induction (unless, of course, one defines a man as a mortal human male). It may be that somebody, somewhere, has solved the problem of mortality. It's known that a fair portion of ageing is rooted in errors in the copying of cells, and that at least some of this can be traced back to the fact that our own requirements (along with those of the vast majority of life on the planet, plants excepted (for reasons that should be clear)) include the consumption of oxygen (one of the most reactive, and thus corrosive, elements in the cosmos). Perhaps the malign influence of oxygen (or other factors) on cellular replication and ageing has been, or will be, solved by somebody, unbeknownst to the rest of us. It has exactly the same status held by the assertion that all swans are white, prior to discovering a black swan. It's probably true but, unless and until we discover an immortal human, we have to be careful with this premise because, barring omniscience, we might be wrong.


Was Hitler a man? It certainly seems so on the face of it but, given recent advances in the understanding of gender identity, it's reasonable to be sufficiently suspicious to avoid taking this premise as axiomatic.
 
Note that even the event of the premises in the above syllogism being false doesn't render the conclusion itself false, it merely gives us cause to assert that the argument hasn't demonstrated its truth.

What tools can we bring to bear to determine if an argument makes its case? The first port of call must be to see whether any fallacies have been committed on the way.

A fallacy is an error in reasoning, and it can take many forms, although they broadly fall into two categories, formal and informal. A formal fallacy is an error in the structure of an argument. All such errors take the form of the non sequitur (does not follow). These are generally easy enough to spot once you grasp the basics of logical structure, not least because it's usually easy to render the premises into some propositional form. With our syllogism above, we can render the argument as follows:
  
P => Q, P
Q

This looks like a slightly different form of argument, but it's exactly the same. It reads P implies Q, P, therefore Q. P = man, Q = mortal. Thus, man implies mortal, man, therefore mortal. This form is known as modus ponens (the way that affirms by affirming), which is an argument form and a valid rule of inference. It says nothing about the truth of the premises, only the validity of the structure of the argument. The beauty of this sort of approach is that one can render any argument in this manner, which helps in trying to pin down whether any formal fallacies have been committed, because it strips away the content of the argument and leaves only the structure, clearly exposed and open to easy analysis.

We can see more about how it works by playing with the order a bit to make it invalid:
  
P => Q, Q
P

In this form, our argument now says:

P1. All men are mortal.
P2. Hitler was mortal.
C. Therefore, Hitler was a man.

This form is invalid, and commits the fallacy known as 'affirming the consequent'. It's fairly easy to see where this goes wrong, not least because men aren't the only things that are mortal, and this conclusion disregards all of them. Note that, as stated earlier, this is exactly the same form as abductive hypothesis selection, so what gives? Why is this an error here and not in an abductive argument?

The distinction is really simple; in this case, we're employing a deductive argument, which means that we're looking for a conclusion that cannot fail to be true. In an abductive argument, we're merely looking for a hypothesis that's supported by the facts. Thus, affirming the consequent is a deductive fallacy only.  

The very observant reader may notice the similarity of the invalid form of this to something discussed in earlier post, namely the modus tollens (the way that denies by denying):

P => Q, ¬Q
¬P

Again, this is a valid argument form and a rule of inference, and this is how deduction enters the logic of scientific discovery. This is our 7b above. It says Proposition P implies Q, not Q, therefore not P. The modus tollens also has a negative form:

P => ¬Q, Q
¬P

Or Proposition P implies not Q, Q, therefore not P. In scientific reasoning, this form is employed in affirming the null hypothesis, our 7c.

Other formal fallacies include 'denying the antecedent':

P => Q, ¬P
¬Q
  
The problem with this is fairly simple to spot. In this form, we can see that the argument implies the inverse from the original statement. While P might imply Q (assuming the truth of the premise), P may not be the only thing that implies Q. Let's look at our syllogism again, but cast it in this form:

P1. All men are mortal.
P2. Hitler wasn't a man.
C. Therefore, Hitler wasn't mortal.

It looks a bit odd to say that Hitler wasn't a man, until you realise that, in this case, I'm talking about the next-door neighbour's cat, also called Hitler (I'm making this up, in case you're wondering; my neighbour's cat is called Goebbels).

Again, the issue here is that men aren't the only things that are mortal, thus the conclusion doesn't follow from the premises, even if the premises are true.

There are quite a few formal fallacies so, rather than detail them all here, I'll link to some resources at the bottom of the post.

What about informal fallacies? These can often be more difficult to spot until one gets to grips with the most ubiquitous fallacies, as they can be subtle beasties, playing on our intuitions (see appeal to intuition below). Most are fallacies of relevance, which is to say that their reasoning is based on appeals to things that aren't relevant. Common examples include:

Post hoc ergo propter hoc (after, therefore because of). This is an extremely common fallacy, not only in argument and debate, but in the world at large. We've all encountered some form of this fallacy in our interactions with people. It takes the form of something along the lines of 'you were the last person to use my computer and now it doesn't work properly, therefore you broke it' (I've actually had this exact accusation levelled at me after repairing a faulty PC). The fallacy should be fairly obvious, since the current fault might have precisely nothing to do with the previous one or its repair, yet we all fall prey to this kind of reasoning. 

Einstein gives the best example of why this is wrong, with a particular example from special relativity. 


As you can see from the diagram (borrowed from Wikipedia), whether something happens simultaneous to, before or after  another event is dependent on one's own frame of reference. With v=0 (stationary observer), the line passes points A, B and C simultaneously. With v=0.3c (moving towards at 30% of lightspeed), the line passes C, then B, then A. If we take the passing of the line as being when lamps at points A, B and C light up, this observer could take the lighting of A to be caused by the lighting of B, which in turn is caused by C.

Argumentum ad verecundiam (appeal to authority). This can be a tricky one, not least because of what constitutes a valid authority. This fallacy is committed when one asserts that 'X person asserts this as true, therefore it's true', where X is some person set up as an authority for the purpose of the argument. It's a form of the genetic fallacy (see below). In reality, even appealing to somebody with relevant expertise cannot determine the truth of a conclusion, not least because even those who speak with relevant expertise are expected to demonstrate that their arguments are sound. Remember Feynman's admonition regarding the key to science:

"If it disagrees with experiment, it’s WRONG. In that simple statement is the key to science. It doesn’t make any difference how beautiful your guess is, it doesn’t matter how smart you are who made the guess, or what his name is… If it disagrees with experiment, it’s wrong. That’s all there is to it."
To give a concrete example, Einstein thought that quantum mechanics was deterministic and contained local hidden variables. Nobody would question whether Einstein was competent to speak on matters of theoretical physics, yet he was wrong about this, as subsequently demonstrated.

Most examples don't actually rely on valid authorities, though. Often, people are set up as authorities while having no expertise of relevance, simply because of their public stature. A lovely example is the very common citation of Richard Dawkins by creationists, when he says that DNA is precisely akin to digital code. Dawkins is wrong about this, and isn't a valid authority on what constitutes a code*.

Argumentum ad ignorantiam (argument from ignorance). This is committed when a conclusion is accepted as true simply because it cannot be proven to be false. This fallacy is also known as 'god of the gaps'.

Appeal to incredulity. This fallacy takes the form 'I can't imagine how this could be true, therefore it's false'. Somebody not conversant in the operating principles of quantum mechanics would be hard-pressed to imagine how something could be in two places at once. Concluding that it couldn't from this commits the fallacy, not least because something can be (and is) in two places at once, or we wouldn't be here to talk about it.

Appeal to intuition. This is a pernicious little beastie, not least because our intuitions are incredibly useful to us, so it's very difficult to address our reliance on it. The above examples should show us the problem with it, though, because quantum mechanics and relativity are extremely counter-intuitive, yet they seem to be pretty accurate descriptions of the phenomena we observe in their respective domains.

Genetic fallacy. This is actually a quite broad class of fallacies, in which a conclusion is accepted or rejected based only on the source of the argument. A common subset of this is the argumentum ad hominem (argument to the man). Often misunderstood as being any instance in which an attack on the arguer takes place, the fallacy is only committed when the argument is rejected on the basis of whatever flaw is perceived in the arguer. An attack on the arguer would be ad hominem, as opposed to argumentum ad hominem. To clarify this, consider the following two examples.

1. Your argument is invalid, you moron, and here's why... (goes on to explain flaws in argument).

2. You're a moron, so your argument is wrong.

The former, while strictly containing an ad hominem, is not fallacious, thus does not commit the argumentum ad hominem. In the latter example, however, the argument is dismissed purely on the basis that the arguer is a moron (or, at least, perceived as such). This fallacy can best be criticised in the simple adage that even a broken clock is right twice a day.

Note that, as stated above, the argumentum ad verecundiam (appeal to authority) is also a subset of the genetic fallacy.

Petitio principii (begging the question). This is one of the most common fallacies you're likely to come across. It's committed when the conclusion of an argument is contained within the premises. It's generally easy to spot but, occasionally, the conclusion is 'smuggled' into the premises in a form not easy to spot. It will usually be in the form of a premise whose truth relies on the truth of the conclusion (hence begging the question) or vice versa. Note that a question-begging argument is always valid and, if the premises are true, then it's also sound (it couldn't really be other in cases where the conclusion is contained in the premises because, if the premise is true, and contains the conclusion, the conclusion must be true). Gary Curtis gives the following example:

P1. Murder is morally wrong.
P2. All abortions are murders. 
C. Therefore, abortion is morally wrong.

This is clearly a valid argument, and it doesn't appear that the conclusion is in the premises, so what's the problem?

It becomes clearer when you strip away the loaded term 'murder'. The first step is simply to remove the first premise, which lends nothing significant to the argument, and then to replace the word 'murders' with something more neutral. Then we're left with:

P. All abortions are wrongful killing
C. Therefore, abortion is morally wrong.

In content, this argument is identical to the original, yet now the circularity is crystal clear.

I highly recommend reading Gary's full exposition of this, which can be found HERE.

Onus probandi (shifting the burden of proof). This fallacy is ubiquitous in some circles, but it's also very easy to spot. It is committed wherever somebody denies their responsibility in supporting their argument, insisting instead that you must prove it false. It often goes hand-in-hand with the appeal to incredulity.

Inductive fallacy. This is a broad class of fallacies, including such fallacies as the hasty generalisation, in which a conclusion is erected based on weakly supporting premises or an insufficient sample set. It's an expression of ignoring the problem of induction. A good example is the first premise of the famous Kalam Cosmological argument, which is stated as 'everything that begins to exist has a cause for its existence'. Much like the example given above about all men being mortal, this premise relies on us never having found an exception (it's thus also closely related to the appeal to ignorance).

Fallacy of bare assertion. This one's fairly obvious. It relies on accepting a premise as true when it has never been demonstrated to be true. Most premise-related fallacies have this lurking in them somewhere. Any inductive fallacy, for example, asserts that its premises are universally true, thus committing this fallacy.

One thing to be very careful of is a subset of the fallacy of bare assertion, namely the assertion of fallacy. Simply asserting the commission of a fallacy does nothing to address or debunk an argument. This is a far-too-common practice, and it's just as fallacious as any of the other fallacies committed here.

Fallacist's fallacy, or appeal to fallacy. This fallacy is committed when a conclusion is asserted as being untrue based on the fact that a fallacy has been committed in arriving at it. Consider the following:

P1. All cars are blue.
P2. Jensen Button is German.
C. Therefore, diamond is an elemental form of carbon.

Neither premise is true nor are they connected, the conclusion doesn't follow from the premises. Pretty much everything about this argument is fallacious, yet the conclusion is a true statement.
  
Again, there are many examples of informal fallacies in deductive logic. I'll link to some excellent resources below.

So, what about the statement at the head of the post, which I term 'the Sherlock Holmes fallacy'? Let's remind ourselves of it:

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

Hopefully, I've given enough information in this post that seeing the flaws in this statement will be a fairly simple matter. I'll leave its solution to the interested reader. I'll be happy to expose it on request. 

Thanks for reading.
Some useful resources on logical fallacies:

The Fallacy Files (less comprehensive, but slightly more accessible.

*DNA is not a code, but our treatment of it is. We assign letters to the various chemical bases and build a language from them, but the map is not the terrain. DNA is a code in precisely the same way that London is a map.