You Can’t Prove An Empirical Claim Apriori

This is a post on a rather technical question, which is nonetheless very important in my view. I often need the argument, which is actually quite simple, and that’s why I wanted to write it up in a separate post as a reference. One application would be in my discussion of the Malthusian argument (see my overview of the posts in this series here). If you have understood the main point, you will find many — even very many — instances where people get this wrong. As far as I can tell, I have learned the argument from reading David Stove, an Australian philosopher.

— — —

As an aside: David Stove is so underappreciated that I would like to plug him here. This is not a blanket endorsement of all his opinions, which are often in my view wrong. But at least even when they are wrong they are so in an interesting and thought-provoking way. Stove was a conservative, but in a tradition that I would trace back to an old Liberalism. He can be very blunt and judgmental. In addition, he did not shy away from arguing controversial positions, which I think caused him some problems in an academic environment.

While I would object to many of Stove’s arguments, there are also quite a few that I have learned from him:

• He convinced me that something with Karl Popper’s philosophy is deeply wrong and that he was an intellectual trailblazer for postmodernism, which first surprised me, but then convinced me. The fundamental error comes from David Hume (Stove otherwise held Hume in high regard, and rightly so).
• I also had a naive view of the Enlightenment. Here again David Stove showed me what’s wrong with it. See, for example, his “On Enlightenment.” Basically, it is the expectation that all problems can be solved just by shining some light on them. Everything is actually very simple once you think about it, all you need is some calculus and just do it. This leads to a massive overestimation of what can be achieved with education. I will write more about this. And while I am already plugging: Isaiah Berlin’s “The Roots of Romanticism” is also a “one of the 100 you should read before you die” book that makes the point.
• Few people have changed my mind on major points, but even fewer on several of them. So I was willing to read also what David Stove had to say about Darwinism in his “Darwinian Fairytales.” To understand the background: My father was a great admirer of Charles Darwin, and that was what I grew up with, so this was a challenge for me. David Stove was an atheist (and so am I). This is not about “creationism” or “intelligent design.” — And David Stove once more convinced me that something is wrong here. I don’t think his argument is good enough, and that’s why I try to improve and expand on it. However, it is still pretty good, enough to kick off an intellectual journey for me. One part of Stove’s argument is a critique of the Thomas Malthus’ theory of population. This part is excellent, but again, I think you can make it even stronger. I try to do this in my series of posts, which are a spin-off from a book project: “Synopsis: What’s Wrong with the Malthusian Argument?” I am massively indebted to David Stove here, not the least for hammering the question in: Even if it seems self-evident, is it really true?

David Stove is fun to read, at least in as much as that can be the case with sometimes complicated philosophical arguments. And he can also cut to the chase like no other. Someone has compared this to watching Fred Astaire dance: It goes so fast, you can’t keep up with how he does it. And that is true. I was often disappointed that an argument was over so fast when I expected much more. It then took me some time to let it sink in and understand that the argument was really this simple. — I have perhaps even somewhat taken on Stove’s tone: If something is ludicrous, call it out, even if it comes from someone who is held in awe as a great thinker.

— — —

Now, to make the argument, I have to explain the relevant terms. It is not complicated, you only have to pay close attention because it is easy to lose track. My terminology may be imperfect for experts and I might get some finer points not exactly right. But I hope that I don’t totally mess it up. Here goes:

A statement is empirical if it can be either true or false. Both cases can happen, and it is not possible to exclude one of them. An example for an empirical statement would be: “It will rain tomorrow.” Perhaps there is some necessary reason why this assertion could only be true or false, eg. because you refer to the moon where it cannot rain. But prima facie, it can be either way under normal circumstances: It could rain tomorrow or it could not.

A statement is tautological if it can only be either true or false. You could also say that the statement is necessarily true or necessarily false. An example would be: “It will rain tomorrow or it will not rain tomorrow.” This may look similar to the first statement, which was empirical, but it is not. Either it rains, then the first part is true, and hence the whole statement, or it does not rain, then the second part is true, and hence also the whole statement. It can only be true, no matter what happens. In a way, you can obtain a tautological statement from empirical statements here. Technically, this is “A or not-A,” where A is “it rains.” It does not really matter what A is to find out whether “A or not-A” is true, it is always so.

You can also move from empirical statements to other empirical statements. Take for example: “If it rains tomorrow, then the road will be wet.” The road could be wet or not, so this is also an empirical claim. And you can also get tautological statements from other tautological statements. That’s what you have all the time in mathematics. An example would be: If it rains tomorrow or it does not rain tomorrow, then it will not rain tomorrow or it will rain tomorrow.” That is pretty trivial, I only exchanged the order of the statements. On an abstract level this is: “If A or not-A, then not-A or A.”

— — —

But here is the main point: It is not possible to obtain an empirical statement only from tautological statements by a logical derivation. The reason is about this if you work through the three cases:

→ Empirical to empirical: You plug something in that can either be true or false, and you get something else that can be either true or false. You start with two possibilities and end up with two.

→ Empirical to tautological: You plug something in that can either be true or false, and get something that can only be true or false, but not sometimes this, sometimes that. You start with two possibilities and reduce them to one. This was the example with “A or not-A” where A was “it will rain tomorrow.” Note though, that the empirical claim here is superfluous. But then the conclusion works.

→ Tautological to tautological: You plug something in that can only be true or false, but not sometimes this, sometimes that, and you get something of the same type. Here you move from one possibility to one possibility.

But if you tried to get from tautological to empirical, you would plug something in that can only be one thing, and you would end up with something that can be either way: two possibilities. Somehow you would have created another possibility out of nothing. You can only arrive at an empirical statement if you plug some empirical statement in somewhere. There is nothing wrong with also using tautological statements, they don’t hurt. But tautological statements alone cannot do the trick.

Logicians have to grant me a lot of sloppiness here. My explanation is certainly wanting. However, the argument stands as far as I can tell even if I might be unable to make it.

— — —

I had a statement of the type “A or not-A” above, which is tautologically true. There are not many ways to get such statements, and they are all equally banal, though perhaps more complicated. What you have in mathematics is similar, but slightly different. You start with axioms, statements that you assume to be true. There is no way of proving them, that is what you start from as a premise. And then you look what else you can obtain only with tautological transformations from the axioms.

Axioms play a similar role here as tautological statements because they are true by assumption, and not false, only one of the two possibilities. In addition you also have outright tautological statements that are necessarily true and can serve as ways to conclude something from the axioms. If you start from axioms, what you derive can be interesting. For example, you could find that there are only five types of Platonic solids, which is not obvious from the start. But in a way, the conclusions are already baked into the axioms, and all the derivations just transform them to other statements that are tautologically true under the assumption that the axioms are true.

If you have seen some real mathematics, that is not how it looks. But that is so because the axioms are deep in the background. You already start with conclusions that have been derived from them. Hence it is not visible that the arguments are only a sequel to a sequel to a remake of a sequel after a few seasons. Still, the bottom line remains the same: You can only obtain tautological statements in this way that can only be true or false, but never a something that can be either true or false, an empirical statement. That’s what gives such results their strength that they can only be true out of necessity.

In other contexts, axioms could also be truisms that appear impossible to deny. If you grant them, and that can be very plausible, you have axioms as in mathematics. If that is all you build your argument on, then the conclusion is still the same: You cannot obtain empirical statements from them. This is so for very fundamental reasons and has nothing to do with the specific arguments you deploy. To obtain empirical claims, you have to plug in at least one empirical claim somewhere, or it won’t work.

— — —

Up to this point, this seems like a rather academic point that I am making, and it is. Still it has many applications. What you have to look for are arguments of this type:

Premises: There are truisms A, B, C, and further tautological statements.

Proof: A long derivation with tautological arguments that cannot produce an empirical statement with two out of one.

Conclusion: That is why the empirical statement X is true.

No matter what happens in the middle, no matter how many times the author claims that steps in the derivation are obvious, no matter how many times they stomp their foot: The argument is incorrect. It cannot work as a matter of principle. Something has to go wrong somewhere. It may be difficult to find the mistake, but at least one has to be there that makes the derivation incorrect.

There are many possibilities: The author might use some hidden assumptions that are not explicit and that are empirical. In that case, the argument might work and you could fix it. However, if that is not the case, the proof still has to be wrong as it stands.

A similar case is that the author uses equivocations: A statement can be understood in two ways: As a truism and as an empirical statement. It cannot be both, so while they may look the same, they are different. Equating them is a rhetorical trick, not an argument.

The author may then start out with the interpretation that is a truism, and underhandedly exchanges it for the empirical statement during the argument. That makes it then possible to obtain an empirical conclusion. However, the author still treats it as a proof apriori that needs no empirical statements. If cornered, they will then revert back to the interpretation as a truism.

As I said, that is a trick. I would not say that this is necessarily done in bad faith. An author may delude themselves into thinking they only argue with truism and tautologies. That they believe this, then also supplies a sense of self-evidence. Only a fool could doubt this. Slam-dunk. Deluding yourself here is a good start for successfully deluding also others. If done in good faith, though with a muddled head, it can be even more convincing because you do not sense the sophistry here.

You can fix the argument by sticking with the empirical interpretation. But then the self-evidence is gone. All you can argue is something like this: “If this empirical statement is true, then X follows.” But then you start from an empirical claim that could be either true or false, which is hence open to debate and no longer something that only a fool could doubt.

Basically, you have to address the empirical content of your argument somewhere. It is not possible to escape this and make it look as if you could argue apriori only from on the basis of truisms and tautological statements.

— — —

If that is still too abstract, let me flesh it out for the Malthusian argument that I am particularly interested in (here is an overview of all my posts on the subject that I keep updated: “Synopsis: What’s Wrong with the Malthusian Argument?”)

Thomas Malthus in his “Essay on the Principle of Population,” first published in 1798 argues like this in the first edition:

Premises: There are two “postulata,” ie. truisms: (1) Humans need a minimum amount of food to survive over the longer run. (2) The “passion between the sexes” is always “nearly” in the “present state.” (All these terms are never exactly defined.)

Proof: A convoluted sequence of arguments with many assertions that the derivation is so self-evident that Malthus cannot imagine how anyone could doubt it.

Conclusions:

→ Population growth is exponential “when unchecked” (an ill-defined term in his derivation that is hard to pin down).

→ Subsistence, which is mostly the food supply, grows linearly.

→ “It is an obvious truth, which has been taken notice of by many writers, that population must always be kept down to the level of the means of subsistence; […]” (Cf. P.3. Although this is supposed to be the conclusion from his argument, Malthus bluntly introduces it as an obvious truth right in the preface.)

However, all these conclusions are obviously empirical: A population could not grow exponentially “when unchecked,” subsistence not linearly, and a population might not always be “kept down to the level of the means of subsistence” (a statement that at closer inspection is an equivocation between an empirical statement and a banal truism: there can never be more people than there is food for over the longer run). There are plenty of examples where it is not so.

The meaning of “when unchecked” remains obscure in Malthus’ essay. He himself concedes that the case has never happened, so it is hard to tell whether his claim about exponential growth is true or false in concrete cases because there are none. As far as I can tell, Malthus’ conclusion rests on the assumption that “when unchecked” means that a population grows exponentially, which is hidden in his scant remarks. Of course, in this case, it is tautologically true that the conclusion follows. It is basically “if A holds, then A holds,” or what is called “begging the question,” you just assume the conclusion.

However, Malthus then tacitly jumps to the claim that human populations in general will grow exponentially when there is enough food. And that is clearly false in many cases, ie. it is an empirical statement, and Malthus knew this. He spends a lot of time on explaining contradictory evidence away, which he can only do by introducing empirical claims via his “checks.” But then this is a different statement from the one he has “proved” by begging the question. It is not self-evident, it is false.

That is already an analysis of where Malthus’ argument breaks down. But that’s not even necessary. All you have to do is point out that the conclusion is empirical, and the supposed premises are all truisms and tautological statements, so it cannot work.

— — —

This type of fallacious reasoning is very popular, and I will provide more examples in further posts. Writers of the Enlightenment are especially prone to fall for this fallacy. The reason is that they view mathematics as how it should be done, but miss that mathematical proofs cannot work for empirical statements. The temptation here is that you think you can get self-evidence without doing the empirical part. Only a fool could doubt the conclusion!

Other examples that I immediately have to think of are all the “praxeological” economists starting with Ludwig von Mises who think they can reach empirical claims apriori. I will dissect such an example from Murray Rothbard in a future post (if you can’t wait, here is my argument in German). Hans-Hermann Hoppe is a sheer goldmine in this regard, but it is hard to isolate this specific fallacy from the general sophistry of his style. What is supremely ironic here with these writers is that they are totally hostile to using mathematical methods in economics as a tool, but fashion their argument on mathematical proofs although this is inappropriate for empirical claims.

— — —

To sum up: Watch out for the premises and the conclusion. If the former are all truisms and the latter is empirical, you can safely conclude that the argument in the middle is wrong. It is not even necessary to find out where it breaks down, it has to break down somewhere. Hidden assumptions that are empirical and especially equivocations are usually the crux.