Last updated on 18 Sep 2017
Sometimes, as a philosopher, one forgets that not everyone has been forced to undergo a logic class. This is a problem, both because logic is taught as the second most boring subject after calculus, and because, like calculus, it is enormously relevant to everything we do. Most especially it is something that is relevant to scientists. Now, I do not want to imply that all scientists do not understand logic, or misuse it, but it is worthwhile occasionally revisiting the basics. Especially for the nature of classification and inference in science.
Last time I wrote about natural classification, I discussed the use of clades as a straight rule for induction. An induction, for those who do not recall their introductory philosophy of science, is an inference from a limited number of particular observations to a general conclusion: all the swans I have seen are white, so swans are white. Inductions can be wrong. Deductions move from the generalisation (“All swans are white”) to the particular case (“this is a swan, so it is white”). Deductions cannot be wrong if the premises (the generalisation itself, and the claim this is a swan) are true. Now, the most widely known philosophy of science, that of Karl Popper, is based upon a logical deduction – if the general claim (the “law”) says that all As are Bs, and this B is not an A, then the law is false. He called this “falsification”. It is based on what we call the modus tollens, and is bandied about all the time by philosophers and scientists alike. It seems to me that not everybody understands what is at issue here. So, a simple introduction follows below the fold.
One standard, and possibly the most widely used, form of inference or argument, is called modus ponens, a Latin term we inherited from the marvellous logicians of the middle ages. It works like this: suppose I have a conditional statement, which we sometimes call an “if-then” statement. If something is true then something else is true; IF A THEN B. [We represent this with an arrow in logic: A ? B.] This is a whole statement. It tells us something about the world it represents, that As always imply Bs. As a whole statement it is either true or false.
An argument is a series of statements that have a logical conclusion (cue Monty Python and the Argument Clinic sketch: “An argument is a series of propositions intended to establish a conclusion. It is not merely the automatic gainsaying of everything the other person says”). We usually lay this out like a sum:
A ? B (Premise 1)
A (Premise 2)
This is a modus ponens argument. If the conditional statement is true, and the second premise is true, then the conclusion must be true. Now, suppose the conditional is true, but we know that the conclusion, B, is false; what then?
A ? B (Premise 1)
¬ B (Premise 2)
¬ A (Conclusion)
Where “¬ ” means “NOT-“. [In basic or first order logic, we use simple operators like NOT, AND, and OR to make up our logical “equations” the way we use PLUS, MINUS, MULTIPLY and DIVIDE in arithmetic.] Here, the fact that the second part of the conditional statement, called the “consequent”, which we have shown as the letter B here, is false, suggests that the first part of the conditional (the “antecedent”) is false. Stay with me here, because this is the machinery of the point of this post. This is called modus tollens.
Modus ponens and modus tollens (we always italicise them, since they are used as Latin names or terms), are the foundations of inference. And moreover, modus tollens is the foundation for Karl Popper’s philosophy of science. Let me review this briefly.
Popper was concerned about what is known as the “problem of induction”: no matter how many observations we make of swans, there is always the possibility that the next one we observed (in Perth) might be black. So how can we justify the use of induction? This problem goes back to David Hume (who, by the way, did not ever use the term induction), but was raised most strongly by John Stuart Mill in the mid-19th century. It was a standard philosophical staple in the 1930s, when the Vienna Circle was arguing about the logic of science.
Popper noted that the problem of induction was insoluble, contrary to philosophers like Hempel and Carnap who thought we might be able to formalise it, and came up, instead, with the principle of falsification: if you can disprove a theory or generalisation by modus tollens, then it is dead and buried, but you can never prove the generalisation. Inductive inference did not, in Popper’s world, exist. This leads to a problem: how does science actually get the generalisations in the first place? Popper did not care: guess, conjecture, dream or cast dice. In a book the English title of which was The Logic of Scientific Discovery, Popper had no logic of scientific discovery.
Why is this relevant to natural classification? Largely, because systematists adopted Popper as their philosopher of science, despite his treating classification as something that could be dispensed with. Systematists recast their classifications as “hypotheses” and “models”, and treated their data as potential falsifiers of these models, rather than building up the classifications inductively. This sea change was dramatic. In the “traditional” systematics, the whole point of doing classification was to provide inductive generalisations (“All mammals have character A”) which then called for explanation (and lest anyone assert this is a pre-Darwinian or ancient metaphysics or epistemology, note that people were publishing this in the 1960s, as well as in the 1860s; that it was commonly understood before the arrival of numerical taxonomy and cladistics is sometimes forgotten, or deliberately ignored).
But let us return to how logic works, and look at a common fallacy used in science. I’m going to argue that it is a fallacy, but that it is sometimes a justified fallacy. It is called the Fallacy of Affirming the Consequent, and it works like this. Recall our argument form of modus ponens: If A, then B, A, therefore B. A common mistake, made by children and adults alike, is to argue this way: If A, then B (our conditional), B (the consequent of the conditional), therefore A. Here’s an example:
P1. If the theory of evolution is true, then we should see lots of convergence (that is, organisms should “solve” the same problems the same way)
P2. We do see lots of convergence
C. Evolutionary theory is true
Why is this a fallacy? Because convergence could arise in many ways: two of the most obvious are the creationist claim (God made them the same way because He liked that “solution”) and the Lamarckian claim (species go through a predetermined sequence of stages or grades of evolution, so we will observe similarities of function and similarities of form). In fact, there are an infinitely large number of possible explanations, so finding convergences do not bolster the generalisation of evolutionary theory. This is why, in my view, convergence (called “analogy” or “homoplasy” in systematics) is not informative about evolutionary history, and does not form part of classification; something I have talked about before.
Now, this means that we cannot use convergence as evidence for evolution. But we do. This is in fact commonly the case in science: we find things are as they were expected to be according to the theory and so we say the theory is “confirmed”, in defiance of all logical strictures. The Fallacy of Affirming the Consequence is the inversion of modus ponens (a similar fallacy is the fallacy of inverting modus tollens, called the Fallacy of Denying the Antecedent). We can show this as a table:
Valid forms Fallacious forms Modus ponens
If A then B
Affirming the consequent
If A then B
If A then B
Denying the antecedent
If A then B
Why is it that science relies upon affirming the consequent? It has to do with a rather complex topic of Confirmation Theory, and so typically philosophers of science will appeal to Bayesian logic:
a hypothesis is confirmed just to the extent that the likelihood that the hypothesis was true given the evidence is greater than the likelihood that the evidence would occur otherwise, very ( very ) roughly [here’s why] a hypothesis is confirmed to some extent just in case the probability that it is true is greater in light of the evidence than otherwise. Because science needs to affirm the consequent in a world where we can never make simple deductive inferences from known true laws, we needed to have a logic that allows us to commit this formal fallacy with justification. But this has ridden roughshod over some thorny ground, in my opinion, ignoring the fact that we do other things than simply address theories, hypotheses, conjectures and models in science.
Classification is somewhat a forbidden topic in philosophy of science as practiced by the post-Vienna analytic tradition. It’s what librarians do, for convenience, or psychologcal tendencies cause. It’s anthropomorphic and anthropogenic. Subjectivism, conventionalism, psychologism and conceptualism are terms thrown about in the debate. But classification, I think, can explain why we affirm the consequent. It has to do with limiting the scope of the argument.
Suppose I have a bag of marbles of various colours, and I want to make inferences about what colours there are (because the black ones are more valuable and I am about to trade my bag for a bicycle with Tommy, but I don’t want to give away the black ones and I don’t want to count them in front of Tommy, which would undercut my bargaining position for arcane reasons any ten year old boy will recognise). Now if there were an indefinitely large population of marbles, my hypothesis that black ones are rare might not be either confirmed or disconfirmed, no matter how many bags of marbles I had previously observed. I might have been in the area of a shop that happened to sell the brand that had few black marbles, whereas in general, only yellow ones are rare and the black ones are worthless. But suppose instead I am trying to make inferences in only my neighbourhood. There are many fewer marbles there, and the frequency of black marbles is more likely to represent that population, and hence the worth of the colours.
An inductively-based classification is more representative of a limited population, so in that case, the unbounded ignorance forced upon us by indefinitely large possible cases is trimmed away substantially. We can affirm the consequent here if the scope is small, because observations and actual frequencies converge. More than simple logic rules here; we aren’t so ignorant to begin with. Because we have classified our domain and iteratively refined it, we can confirm our hypotheses. It’s defeasible (that is, we might be wrong in a given case), but it’s the best we can do under uncertainty about the natural world. And it doesn’t need to be subjective or any of those other Bad Words. Yes, classification is a human cognitive activity, but all science is, and if we’re waiting for a God’s Eye View we may be waiting some time.
Classification restricts the inferential domain the arguments work in, and so the absence of a consequent does undercut the antecedent of the law or generality, if the scope is countable. I can confirm my claim there are no elephants in my left pocket by observing all the things in my left pocket (this goes to the claim we cannot prove a negative, which I do with respect to elephants in my pockets all the time). By setting the domain up (“my actual left pocket”) the formal point of the fallacy, which applies when the domain is all possible worlds, becomes unnecessary.
So sometimes the fallacy of Affirming the Consequent is a fallacy, and sometimes it isn’t, depending on what is a live and viable question. Since we do not know that ahead of time, we have to make inferences from where we begin.