When debating important topics, whether with your friends, online with strangers, or in a public debate with spectators, there is always a high emphasis placed on proving your case. People ask why you believe what you do, ask you to show your evidence, ask you to give them your arguments. There is nothing wrong with an inclination to ask people for reasons for what they believe – this is healthy. But there is a very real danger of letting this instinct run amuck. This danger goes under the name of *arguing from ignorance*. Here, we’d like to have a brief discussion of what this logical fallacy is and why it is fallacious, as well as pointing out some examples of how this fallacy can play out in conversation.

**What is an Argument from Ignorance?**

The key word to understand is perhaps the word proof. This word has at least two commonly-used meanings. In mathematics, logic, and (sometimes) philosophy, the word *proof* denotes something *irrefutable*. Proofs in this context are required to be so persuasive that rejecting the proof amounts to rejecting one of the foundational rules of logic. Outside of these very narrow fields (and even occasionally within some of these fields), this super-charged kind of proof doesn’t exist. Most of the time, when we use the word *proof* we really mean *evidence-based reasoning*. This is much, much broader and is the normal mode of reasoning in the physical sciences, politics, historical studies, and pretty much anything else you can think of.

With an understanding of how we use the word proof, we can talk about *arguments from ignorance*. An argument from ignorance looks something like: * because you can’t prove X, X must be false* or

**.**

*because you can’t disprove X, it must be true***Why is this Fallacious?**

There are a great many things wrong with arguments from ignorance. In order to treat the issue with the right level of finesse, I need to distinguish between the two kinds of provability. The most obvious difficulty comes up when you try to apply the strong type of provability to a realm like experimental science, history, or politics, fields in which that kind of provability is completely outside the picture. This issue is hardly worth commenting on because of how obviously bad it is – it would be on par with taking issue with a political party platform because it does not prove any mathematical theorems. Such an objection would be completely off base. But we can do even better. If we try to use the correct kind of proof in the correct domains of inquiry, the fallacy still shows up.

*Mathematics has Debunked Arguments from Ignorance*

The previous discussion should convince the thinking person that arguments from ignorance are fallacious. But what is even more surprising, in the realm of mathematics arguments from ignorance have been *proven* to be fallacious – in the strong sense of the word prove, the kind of proof that is *irrefutable*. This is one of the most unexpected developments in the entire history of mathematics, and only came about in very recent times.

Now, what is this development? These are the two “incompleteness theorems” of mathematician and logician Kurt Gödel. What these are is summarized well by the Stanford Encyclopedia of Philosophy [1]:

*“Gödel’s two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent).”*

The second incompleteness theorem is fascinating in its own right, but doesn’t really concern us here. We are concerned with the first incompleteness theorem. Let’s slow down and explain the philosophical jargon. A *formal system* is just the body of true things that can be proven mechanically using only logic and a finite number of *axioms *which are taken to be self-evident truths within the formal system. The system being *consistent* means that there are no statements that are both true and false within that system – since that is inherently absurd. When it talks about a certain amount of arithmetic, this is essentially referring of the ability of this logical system to accurately describe the basic properties of addition, subtraction, and counting (the exact level of detail doesn’t matter much for our purposes here). The conclusion of the theorem is that, if all the previous assumptions work, there is a statement *G* within that formal system that is neither provable nor disprovable. We can extrapolate this to mean that *there is a statement about whole numbers that,* *even though it is true, cannot ever be proven to be true*. This is a disproof of the idea of arguing from ignorance. Arguments from ignorance say that because you can’t prove something, it must be false. But Gödel’s first incompleteness theorem uses the laws of logic to deduce that there are statements about whole numbers that are true but cannot be proven. This is a direct refutation of the entire concept of arguing from ignorance.

**What About that Other Kind of “Proof”?**

But what about if by “prove” we mean “provide reasonable evidence for”? Surely, a claim about which we can’t find any evidence must be false. Again, no. I could produce endless examples, but just one should do.

Suppose you go back in time a few thousand years, back to a time when nobody knew yet that the earth is a sphere and not flat. Imagine then that you get into a debate with an ancient human about the shape of the earth. Being so far back in time, civilization is not yet developed enough to produce the evidence that we need to show that the earth is round. If the method of argument from ignorance is valid, then the ancient man can conclude that since you cannot show him evidence that the earth is round, therefore the earth must be flat. But of course, the earth is not flat. Ergo, the method of argument that concluded that the earth is flat is flawed, ergo the method of argument from ignorance is flawed.

You could do the same thing with any scientific theory, or a great number of other truths we take for granted.

Conclusion

I think this ought to be a clear enough presentation that you can’t dismiss something as definitely false on the basis that nobody has proved it to be true. Of course, we should be skeptical of claims about which no evidence has been produced. But we equally must be skeptical that we have adequately looked for evidence in an unbiased way. For me, it would take years of serious searching to come anywhere near concluding that there is no evidence at all for some position, and in fact I think there is evidence for many positions that I disagree with – including the position that God does not exist. Now, I don’t think the evidence there is very convincing, not nearly as convincing as the evidence in favor of the truth that God does exist, but I’m quite willing to grant that the atheist does have a nonzero amount of evidence. I think that is the proper position to take, and I think humanity would benefit greatly from such an approach in the hotly debated issues of our time.

**Citations**

[1] https://plato.stanford.edu/entries/goedel-incompleteness/