There are many experts weighing in on both sides of this issue. Here are a couple of citations from those who feel you can prove a negative, and my thoughts at the end.
Yes, you can prove a negative:
Google AI Overview 1
Logical equivalence
You can rewrite many negative claims as logically equivalent positive claims. For example, “No person taller than 5’6″ was at the party” is logically equivalent to “Everyone at the party was a 5’6″ or shorter”.
Negative proof
You can prove a negative by showing that the opposite is not true. For example, you can prove that there is no Loch Ness monster by draining the loch.
Scientific proof
You can prove a “universal negative” in the scientific sense. For example, proving Einstein’s theory of relativity is true also proves that there aren’t any spaceships anywhere in the universe accelerating to faster than the speed of light.
Inductive reasoning
You can use inductive reasoning to establish the plausibility of a negative claim based on observed evidence. For example, you can prove that there is no Bigfoot by searching the world for Bigfoot and finding no credible evidence of its existence.
The phrase “you cannot prove a negative” is a principle of folk logic, not actual logic. It’s also a negative claim that would not be true if it could be proven true.
I changed an example above for something more generic, keeping the meaning and context intact.
A principle of folk logic is that one can’t prove a negative. Dr. Nelson L. Price, a Georgia minister, writes on his website that ‘one of the laws of logic is that you can’t prove a negative.’ Julian Noble, a physicist at the University of Virginia, agrees, writing in his ‘Electric Blanket of Doom’ talk that ‘we can’t prove a negative proposition.’ University of California at Berkeley Professor of Epidemiology Patricia Buffler asserts that ‘The reality is that we can never prove the negative, we can never prove the lack of effect, we can never prove that something is safe.’ A quick search on Google or Lexis-Nexis will give a mountain of similar examples.
But there is one big, fat problem with all this. Among professional logicians, guess how many think that you can’t prove a negative? That’s right: zero. Yes, Virginia, you can prove a negative, and it’s easy, too. For one thing, a real, actual law of logic is a negative, namely the law of non-contradiction. This law states that that a proposition cannot be both true and not true. Nothing is both true and false. Furthermore, you can prove this law. It can be formally derived from the empty set using probably valid rules of inference.
[…]
Some people seem to think that you can’t prove a specific sort of negative claim, namely that a thing does not exist. So it is impossible to prove that Santa Claus, unicorns, the Loch Ness Monster, God, pink elephants, WMD in Iraq, and Bigfoot don’t exist. Of course, this rather depends on what one has in mind by ‘prove.’ Can you construct a valid deductive argument with all true premises that yields the conclusion that there are no unicorns? Sure. Here’s one, using the valid inference procedure of modus tollens:
- If unicorns had existed, then there is evidence in the fossil record.
- There is no evidence of unicorns in the fossil record.
- Therefore, unicorns never existed.
[…]
If we’re going to dismiss inductive arguments because they produce conclusions that are probable but not definite, then we are in deep doo-doo. Despite its fallibility, induction is vital in every aspect of our lives, from the mundane to the most sophisticated science. Without induction we know basically nothing about the world apart from our own immediate perceptions. So we’d better keep induction, warts and all, and use it to form negative beliefs as well as positive ones. You can prove a negative — at least as much as you can prove anything at all.
THINKING TOOLS: YOU CAN PROVE A NEGATIVE2
Steven D. Hales
Thinking Tools is a regular feature that introduces tips and pointers on thinking clearly and rigorousl
My thoughts
- Logical equivalence
- Provided the sample size is 100% then this would appear to prove a negative.
- Law of non-contradiction
- While using a set of rules of inference is good enough ‘proof’ for some, it is not the same as scientifically proven empirical3 evidence.
- Double negative
- This seems to prove a a negative for the same reason as logical equivalence.
- Negative proof
- I don’t care for this example; it seems closely related to inductive reasoning. Additionally, a better example might be to prove there is no monster currently in Loch Ness; which then is a negative of the Logical equivalence. In this case it could ‘prove’ a negative. A closer statement to scientific proof one might say ‘We found no monster in Loch Ness’.
- Scientific proof
- I don’t believe this would qualify as scientific proof. This requires one to assume that everything is known about the universe and Special Relativity. This rather bold assumption is then used as the base for a conclusion.
- Valid deductive argument
- Conclusion is based on assumptions.
- Inductive reasoning
- Conclusion is based on assumptions.
I think the interesting statement in the above article is that it “depends on what one has in mind by ‘prove.’ ” If one accepts conclusions based on probable assumptions then they would, of course, have a lower burden of proof. Using these reasoned assumptions and implied conclusions, one could perhaps ‘prove’ a negative. But if you wish to present a reasoned discussion based on provable facts based on empirical evidence, then other than a logical equivalence, proving a negative still might be something elusive.
- Gemini AI questioned “can you prove a negative” ↩︎
- THINKING TOOLS: YOU CAN PROVE A NEGATIVE by Steven D. Hales ↩︎
- Empirical: em·pir·i·cal /imˈpirək(ə)l/ adjective
based on, concerned with, or verifiable by observation or experience rather than theory or pure logic.
“they provided considerable empirical evidence to support their argument” ↩︎