Is it possible to prove a negative?

there does not exists symbol
there does not exists symbol

Note: words in italic are technical terms with clear definitions in logic, which I’m going to omit explaining. Words in bold are substitutes for logical symbols with clear definitions and functions, which I’m going to omit explaining.

There are different kinds of impossibilities. One is physical. Another one is logical. Logical impossibilities are impossible, no matter what, no matter where, no matter the circumstances, no matter the universe, no matter the laws of nature, and no matter anything else. They are impossible and that’s the end of story.

The problem, usually, comes when a claim is about something that might, just might, be physically possible, but never observed in any shape or form. Then all sort of funny ideas pop out of people’s minds.

Let’s have a look at the these two categories of impossibilities.

In a well-defined logical system, it is trivial to produce the following proof:

If the system harbors a contraction, then it is trivial to prove any falsehood by using the statements in the system, both as contradictions and falsum ⊥.

The proof itself is trivial to produce in propositional logic, but nontrivial to discuss in this post; so I’m going to skip it.

The proof entails that any set of premises that by virtue of its members can produce a single contradiction, will necessarily produce infinite number of other contradictions. That’s to say, if we assume p and q, and if assuming such, eventually results in ¬p and or ¬q, then it’s trivial to prove (a ∧ ¬a), (p ∧ ¬p), (x ∧ ¬x), (y ∧ ¬y), (z ∧ ¬z), … , and any other contradiction as well as false statements, i.e. ⊥, would be trivial to be proven in this system.

This is a very big deal in philosophy. Since it’s rather technical, most people, even many philosophers, either don’t know about it or don’t care about it if they do. The implication of this negligence is that given a set of contradictory concepts, anyone can conclude absolutely anything.

Logicians try to mitigate contradiction bombs either by removing contradictions from the systems they study, or by resorting to exotic forms of logic that, in principle, disallow contradictions either by limiting the scope of the effects of contradictions to local domain of discourse, disallowing negation, disallowing conjunction or disjunction, denying the law of excluded middle, converting the statements’ truth values into numbers, computing any functions necessary, converting the resulted number back into statements (e.g. fuzzification and defuzzification in fuzzy logic), and so on.

These exotic logics do work. However, they all have very narrow applications. Paraconsistent logic, fuzzy logic, and the rest are all proposed. Some of them, such as fuzzy logic, is being successfully used in technology (e.g. digital camera, ABS brake system, control systems, AI, and so on). However, none of them can avoid the fact that they are all practical solutions to day to day life problems, and cannot properly avoid the challenges they face. One challenge is that, for technical reasons, they all seem to fundamentally rely on natural deductive logic, and they do their tricks only by limiting the scope of natural deductive logic. As such, they all are in limited use. Necessarily so, since they paint themselves in a corner and make themselves special cases of natural deductive logic. In short, they are subsets of natural deductive logic, which in turn is directly affected by axiomatic logic.

Now, let’s investigate the distinction between logical impossibilities and physical impossibilities.

When people claim something, they use sentences to propose the claim. The sentences will have implications. The implications can be contradictory or not. Also, implications can be true or false. The truth values of the implications can be related to the matters of fact and physical reality. The contradiction aspect relates to the logical foundation of the claim.

Here is an example:

There exists an X, such that X belongs to set D. D is a set of properties such that D{a, b, c, …}.

Now, the questions are as follows:

Q: Are a, b, c, … contradictory?
A: if yes, then the case is closed. If no then,

Q: Can we prove a contradiction from what a, b, c, … imply?
A: If yes, then the case is closed. If no, then,

Q: Does the set D entail absolute physical impossibilities?
A: If yes, then the case is closed. If no then,

Q: Does the set D entail a contingent physical impossibility?
A: If yes, then we might choose to take the claim into consideration because such an impossibility might be overcome later. If no then,

The claim is a physical possibility and might be proven to be true by evidence .

The last two options are the only ones we can discuss with any hope of getting anywhere.

Here is a concrete example:

There exists an X, such that X is a swan, and X is black.

Does it contain a logical contradiction? No.
Does it entail a logical contradiction? No.
Does it contain or entail an absolute physical impossibility? No
Does it contain or entail a contingent physical impossibility? No.

Then, it is physically possible. Then the claim could be true if evidence is provided. Note that we cannot say the claim could be true if and only if evidence is provided (I’ll skip explaining the reason why not).

So, to answer the main question, i.e. is it possible to prove a negative, such as proving A does not exist, we only need to look at the set of A’s properties to see if it contains a contradiction. Then we need to investigate the possibility of the set’s implications to see if they entail contradictions. If any logical contradiction is found in the set or in the implications of the set, then A cannot be what the claim says it is. Such an A with the given set of properties cannot exist. The same would be true about the physical impossibilities.

I hope that you have noticed that the kinds of implausibilities are those that are not human-dependent. They are not universe-dependent either. For instance, it is not physically possible to fit a larger solid object inside another smaller solid object of the same matter without adding tons of ifs and buts to the proposal. A mouse cannot swallow a star. It doesn’t matter what universe we are in and what its laws are. Give a set of laws, and one can immediately discover infinitely many events that are physically impossible to happen in that particular universe. So, no matter the universe and its laws, there will always be physical events that are absolutely impossible to happen in a given universe.

This only leaves us with possibilities bounded by the certain immutable limits. As such, we can easily cross out many claims, including many universal existential claims, especially those related to certain deities.

Further readings

Levy, S. (1971). Logical Impossibility. Philosophy and Phenomenological Research,32(2), 166-187. doi:10.2307/2105946

Rinaldi, F. (1967). Logical Possibility. Philosophy and Phenomenological Research,28(1), 81-99. doi:10.2307/2105325

[M07] Types of possibility. (n.d.). Retrieved from

Wikipedia contributors. (2017, December 2). Proof of impossibility. In Wikipedia, The Free Encyclopedia. Retrieved 18:59, July 10, 2018, from

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