You Can Prove Negatives

A quick note: someone whipped out this old bit of folk-logic that “you can’t prove a negative” at me earlier today. This statement shows up an awful lot in all sorts of debates, but despite its folklore position as some sort of rule of elementary logic, no logician ever has actually proposed it.

And that’s because it is pretty clearly not true, after all, it contains within itself its own handy refutation:  “you can’t prove a negative” is itself a negative statement, so if you can’t prove a negative, you can’t prove that “you can’t prove a negative.”

But of course there are plenty of examples of negative statements that people can prove. One of the real elementary laws of logic is that any proposition P is identical to the negation of its negation, that is, to not-not-P. So if you can prove a positive statement, then you also prove a negative statement which is equivalent to it. (If Descartes could prove that he existed with cogito, ergo sum, he could also thus prove that he wasn’t nonexistent.)

Austrian Economics: Some Questions

Some proponents of Austrian economics (particularly those of the more severe Rothbardian and Hoppe-ian (Hans-Hermann Hoppe) bent) argue that the advantage Austrian economics has over other forms of economics is that, since Austrian economics is logically deduced from a priori axioms, its truth is known even in the absence of empirical evidence.

Now, if indeed all of Austrian economics is deducible from a set of a priori axioms (which I doubt, since I know of Austrian economists, like Ludwig Lachmann, who use empirical methods), I have the following questions:

  1. I’ve never seen an Austrian economist manage to provide a complete list of all the axioms of Austrian economics. Typically they point to only one, the axiom of human action, but of course one assumption is incapable of proving anything. (For example, if I assume that the sky is green, I can’t prove anything not tautologous to “the sky is green” without introducing another assumption.) Without such a list, how is it possible to argue that Austrian economics is all deducible from these axioms?
  2. Relatedly, I’ve never actually seen a formal logical proof provided for any claim of Austrian economics. I’ve seen arguments, I’ve seen arguments that use deductive logic, but there’s a reason logicians (and philosophers and mathematicians) use formal logical proofs: they make it easy(er) to demonstrate that no hidden assumptions slipped in, and that no fallacious reasoning took place. Particularly, they make it possible to ensure that all theorems proven follow from the axioms and other theorems proven from the axioms.
  3. Moreover, provided some such exact and finite list of axioms existed, why are there no attempts to prove consistency? Any axiomatic system that wishes to produce theorems (true statements) distinct from false statements must first demonstrate that its assumptions are consistent, that is, they do not in some way contradict each other. I’ve never seen this done for Austrian economics, no doubt due to the lack of such a clear and precise list, but if somehow the axioms aren’t consistent (and this isn’t always obvious!) then they could be used to prove literally any statement.

Notice, I’m saying “I’ve never seen” instead of “does not exist.” I’ll admit, I’d be very surprised to find out that such things do exist, because I have read (most of) Human Action and a number of other Austrian books and have never encountered any mention of them. But I’m more than welcome to be corrected (also why the above are questions, not statements.)

My point being: in the absence of any one of the above three (list of axioms, logical proof, consistency), any claim that Austrian economics produces logically necessary truths is simply baseless and premature. That’s not say it is wrong — Cantor’s set theory was largely correct in its conclusions, despite the inherent instability of its construction (notably, the contradiction derived form considering the set of all sets.) Nonetheless, Cantor’s system was not capable of making any broad claims about necessary truths: after all, his theory was contradictory. Equally, without positively demonstrating 1, 2, and 3 above, there is a possibility that Austrian economics is self-contradictory, in which case it requires substantial architectural revision if it wishes to be respected as an axiomatic, logical system.