You Can "Prove" a Negative
Addressing a common claim and hopefully being helpful
Im a big fan of Flint Dibble and his attempts to improve knowledge communication from our academic institutions to the general public. Flint is an archaeologist and (notoriously) engaged in a public debate on Joe Rogan’s podcast with pseudo-archaeologist Graham Hancock about the existence of Atlantis.
In a recent podcast, talking about his experiences dealing with Graham, at 33:35 Flint says:
Im working on a book now about Atlantis, and so one of the things that I am trying to figure out how to explain adequately is how you do disprove a negative? And what I’ve realised is Philosophically you can’t disprove a negative, on the level of philosophy of science and things like that. On the other hand, scientifically in practise, we discard and disprove negatives all the time …
I slightly disagree with Flint here, though I don’t think all that he says is wrong. I’ve also frequently come across folk engaged in debate around Atheism and Theism making the claim “you can’t prove a negative”. So, I thought overall this could be a good opportunity to try to clarify my thoughts on this issue in a helpful way that applies across issues.
In the phrases “you can’t disprove a negative” and “you can’t prove a negative” what are the meanings of the relevant terms?
I don’t have any good data to interpret what most ordinary people think they mean when they say these things, however, in Philosophy, “a negative” is typically taken to be some claim which involves negation. Philosophers like to abstract claims, using algebra to focus on the logical form.
Take the claim “2+2=5” and call it `A`. A Philosopher would say that the claim “2+2=5 is not true” or, “NOT: 2+2=5” is `~A` — the little squiggle here means “not” or “negation”. Here we have a negative `~A`. Here is a short proof of ~A.
Assume 2+2=5 (A)
Subtract 2 from both sides: 2 = 3
This contradicts the definition of 2 and 3 as distinct natural numbers
Therefore, the assumption 2+2=5 must be false (~A)
We have proven a negative.
Can you disprove a negative?
Lets take the claim B as “2+2=4”, so ~B is to say that that isn’t true.
Define 2 as the successor of 1: 2 = S(1)
Define 4 as the successor of 3: 4 = S(3)
By the definition of addition: 2+2 = S(2+1)
Applying the definition again: S(2+1) = S(S(2+0))
2+0 = 2 by the identity property of 0
So we have: S(S(2)) = S(3) = 4
Therefore, 2+2 = 4.
There’s nothing about a proposition being a negation or not that makes it an “illegal” move to have it as your conclusion. The only relevant considerations in these arguments are whether or not the premises are true and the logical validity of the overall form of inference being used.
These cases don’t help us that much though. These are highly idealised Mathematical cases using deductive reasoning and propositional logic. That doesn’t really come close to modelling what happens in something like a philosophical dispute about Atheism, Science or Archaeology.
“Proof” has this strict, intimidating Mathematical sense, however its semantic range is much broader. “Proof” can be taken to mean something like “make a convincing argument for”, “give me good reason to believe” and so on. In these senses of the word, the types of claim being made, formalisms we might use to model the inference in arguments and epistemic norms we should follow can differ from the Mathematical, deductive example.
Things get more messy with these different types of inference. And to make things more complicated there are different types of claims that different forms of inference apply to; claims I shall classify as quantificational, modal, and probabilistic. These sorts of claims are more granular, allowing for finer detail about the scope of a claim and the type of belief a claim is held with. I shall turn to each of these in turn.
Quantificational claims include statements about classes of things. “Some cats are black” uses the term “some” and it is sufficient for a single thing to be both a cat and black for this to be true - in order to evidence or substantiate our knowledge here then we only need knowledge of a single example. “All cats are black” is about the entire class “cats”. In this case, the claim requires more knowledge to substantiate because it is about the entire class of cats, and one single counter-example would render the claim false. This type of logic is the type with which Aristotle was concerned and I’ve made a video about it below.
Im sympathetic to the intuition that might lie behind the idea that certain quantificational claims can’t be proven or disproven. It’s sort of like “that claim about a whole class is about a lot of stuff and there’s always room for error in your knowledge of that class”. I think that’s right, but again it’s more complicated.
The first observation in response to this line of thinking is that this affects (so-called) positive claims, just as much as negative ones. For example, consider “All X’s are Y’s”. In order to substantiate the positive claim, we would have to know, or have very good information about “All X’s”. In principle, this needn’t be an issue for some types of claim, such as where we get to define the relevant reference class. For example, we construct the reference class “triangles” to mean “shapes with three sides” and as such I can be sure that all triangles have three sides because anything that doesn’t satisfy that I wont consider to be a part of the class. Even though I do happen to think all our categories are constructed, I think that there are concepts such as “Raven” that are less rigorously and clearly constructed than that of “Triangle”. If I come across some new species as a biologist, I will consider “should this fall under the concept of Raven, do I need to modify my concept of Raven to include this new thing” and our concepts with their fuzzy boundaries shift and change over time.
Further, there are logical inferences between positive and negative claims. i.e.
All X’s are Y’s = NOT: Some X’s are NOT Y’s
Some X’s are NOT Y = NOT All X’s are Y’s
I am no fan of Christian Apologist William Lane Craig and have made many videos about things I disagree with him on. However, there are a rare few points we do agree on, and our ability to “disprove negatives” is one of them (though I don’t like the way he dunks on some random college kid). I think he articulates the point well in this video.
There are no Tyrannosaurus Rex’s on The Planet Earth
There are no Muslims in the United States Senate
Actually, after re-watching that, there’s a lot I disagree with. I don’t think he really gave good reasons to accept his claim that you can disprove a negative and simply restated conclusions that follow from that if you could (i.e. question begging) — but here’s to bridge building!
There are further types of claims and forms of inference that people can make. Im going to make some future posts in response to this discussing specifically statistical methods, Bayesianism and Inference to the Best Explanation. I think these have far more import to the sciences and archaeological disputes and I will discuss the idea of “negative claims” in these frameworks too.
For now, to make it digestible, I will leave it there. Suffice to say, I hope I left a stone in your shoe with respect to “proving”/”disproving” a negative. With language, and inference, things get sticky very quickly and there is no rule that can’t prove or disprove a thing called a negative.