Can You Prove A Negative?

In the public sphere, it is often said that you cannot prove a negative. This often arises in debates about the existence of god and other claims relating to the supernatural. For example, you might hear in a debate between a theist and an atheist the phrase, “you can’t prove that god doesn’t exist, because you can’t prove a negative.”  It is not always the theist making this statement, either. Consider this dialogue between Ayn Rand and Phil Donahue in an interview:

Ayn Rand: No.
Donahue: Now the reason you don’t is because you can’t prove that such an entity or being or energy exists.
Ayn Rand: I can’t nor can anyone else. There is no proof.
Donahue: There is no proof so therefore you’ve concluded that there isn’t one?
Ayn Rand: That’s right.
Donahue: You can’t prove there isn’t.
Ayn Rand: You are never called upon to prove a negative. That’s a law of logic.

            Nor is Ayn Rand the only atheist espousing this sort of statement. James Randi is well known to say the phrase, “you can’t prove a negative.” You can see an example of this here. These sorts of statements require an examination of the concepts of proof, the burden of proof, what a law of logic is, and looking at the differences between deductive and inductive logic. Meaningful debate about something like the existence of god cannot take place when even the skeptics have a misconception about the nature of logic.

            What does it mean for something to be proven? Proof as it pertains to deductive argument is when you have a true premise that follows from true premises in a valid argument. If an argument has an invalid form, it does not matter if the premises and conclusions are all true, because an invalid deductive argument lacks the ability to give a guarantee. Validity is a guarantee that if the premises of an argument are true, and the conclusion is true, then the conclusion is guaranteed to be true. Deductive logic does not deal with probability but with certainty. If there is a result that has a false conclusion despite having true premises, then the argument is invalid. However, it is possible to have a true conclusion despite having a false premise and still be valid. If an argument has a true conclusion despite one of the premises being false, it is unsound. If the premises of an argument are true, the conclusion is true, and the form of the argument is valid, then you have a sound argument. You can challenge an argument by testing the validity, or challenging one of the premises if the argument is valid. For instance, if an argument is valid and is believed to be sound by one person, you can challenge the truth of one of the premises. If you can succeed in challenging the truth of one of the premises of the argument, it now is not clear if it is sound or not. If you can show a premise to be wrong beyond the shadow of a doubt, or that the form of the argument is invalid, then you have performed a refutation. These concepts are easily demonstrated by using something called a truth table. A little knowledge of logic is required to understand how a truth table works, so if one has not read an introductory book to deductive logic, or has not taken a logic course, it is recommended to read the following paragraph. Otherwise, if one is familiar with how a truth table works, you can skip the following paragraph. 

If one is not familiar with how a truth table works, then here is a crash course in logical annotation. It is best to become familiar with the symbols used in logic. The symbol for material implication (⊃) is read as “if…then.” For example, p ⊃ q is to be read as if p, then q. If a negation symbol known as the tilde (~) appears, it is to be read as “not”. So, ~p would be read as not p.  The conjunction symbol (•) is to be read as and. The disjunction symbol (∨) is to be read as “or”.  The therefore symbol (∴) represents a conclusion. It is important to be familiar with the forms of the premises as well.  The basic ones that will be reviewed here are called compound statements or binary operations. A conjunction (p•q) is true only when both components are true. A disjunction (p V q) is true as long as at least one component is true. The material implication (p ⊃ q) is true in every situation except for when the first component is true, and the second one is false. In other words, p ⊃ q is only false if p is true, and q is false. In every other situation, it is a true statement. The truth table works by establishing the possible truth conditions of each variable in an argument form in a systematic manner (starting with the variable furthest to the right alternating true and then false, and then doubling the trues and falses each variable to the left, so that the first one to the left is true, true, false, false, and so on). If one wishes to learn more, a guide to further logic symbols can be found here and a page on truth tables with examples can be read here. 

The first syllogism that will be examined is known as the modus tollens. This argument form states that if p is the case, then q is the case, q is not the case, therefore, p is not the case. This is represented symbolically as: 

p ⊃ q
 ~q,
∴~p

            Notice that the conclusion of the modus tollens is a negative. The purpose of this argument is to prove a negative. Is it valid? Let’s plug the variables into the truth table and find out. 

		First Premise	Second Premise	Conclusion	Valid?
p	q	p ⊃ q	~ q	~p
T	T	T	F	F	No
T	F	F	T	F	No
F	T	T	F	T	No
F	F	T	T	T	Yes

The truth table reveals that in no instance is there a case of all true premises and a false conclusion. This shows that yes, in this case, a negative can be proven. Let’s look at an example in action. If Loch Ness contained a monster, then the monster would have been found by someone. No one has found a monster in Loch Ness. Therefore, there is not monster in Loch Ness. This is quite true. If every single inch of the lake was searched for a monster, and no monster was found, the only conclusion to be drawn is that there is not a monster there. This is also a case where absence of evidence is evidence of absence.

          Let’s also look at another type of argument called a categorical syllogism. Categorical syllogisms were formulated by Aristotle, who discovered logic. While this sort of argument is very limited in what it can do, this is because it is the oldest type of logic there is. Modern logic has made many advances since then, but these remain useful to know. A categorical syllogism is comprised of three parts known as the major premise, the minor premise, and the conclusion. Each premise is called a proposition. The proposition in the major premise will contain a predicate and a middle term. The minor premise will contain a subject and a middle term. The conclusion will contain both a subject and a predicate. The major premise always comes first, and the minor premise second. These are abbreviated by S, M, and P. If one understands how to plot a vend diagram for these sorts of proposition, then skip the following paragraph.

 

It must be tested for validity. The problem, as you may have noticed, is that it does not use the same type of terms or structure as the symbolic logic of the modus tollens. The claims of a categorical syllogism are tested by plotting venn diagrams. Since these arguments concern classes of things, it is possible to represent those classes in a venn diagram. When making a venn diagram to test a categorical syllogism, draw a venn diagram consisting of two circles on top, and one on the bottom. The top left one represents S, the top right one represents P, and the bottom circle represents M. If something is a universal claim, meaning that a claim is made about all objects in the class, then it is illustrated by coloring in the region that something is not, so that only the appropriate regions overlap. For instance, if you color in the statement All S are P, then color in any regions of P that do not overlap with S. If the proposition is No S are P, color in the sections where S overlaps with P. If the proposition is Some S are P, then put an x in the region where they overlap. If the proposition is Some S are not P, then put an x in the region of S that does not overlap with P.  If something is a particular claim, meaning it deals with only specific objects within a subset, then it is illustrated by an x placed in the circle. In the event that the x could go in either spot, the x is placed on the line between two classes, but in that event, you automatically know that the argument is invalid. Only plot the two premises, and not the conclusion. If after diagramming the two premises the conclusion is represented, then you have a valid argument. If in the process of diagraming the premises the conclusion fails to be represented, then the argument is invalid. 

Let us look at a particular form of the categorical syllogism called the modus celarent. This argument goes, “No M are P. All S are M. Therefore, No S are P.” Is this valid? Let’s look at the venn diagram. 

The conclusion that no S are P is represented after diagramming both premises. This means that the argument form is valid. Note that the conclusion is a negative statement. A practical, although silly, example of this argument would be, “No mammals are reptiles. All humans are mammals. Therefore, No humans are reptiles.” Each of the premises in this argument are true statements. The form of the argument was shown to be valid. Therefore, the conclusion must be true. The negative proposition that no humans are reptiles was just proven.  

                Those are just two instances of a negative being proven in different types of logical syllogism. A point needs to be made that in inductive logic, there is no discussion of proof. Nor is there talk of validity and invalidity. Instead, there is talk of strong and weak induction. Why is this? Induction does not deal with proof or disproof, but with substantiating and denying claims. It is also used to find things out about the world. It is one of the key types of logic used in the scientific method. However, a claim can be inductively falsified. Suppose that all through your life, you see tables as rectangular shaped. Then, one day you are invited to a friend’s house for dinner. They have a round table, instead of a rectangular one, and you have never seen a round table before and did not know that round tables existed. You likely would have the idea “all tables are rectangular” in your head, but now that would be falsified. This is induction by simple enumeration, and is perhaps the easiest type of induction to perform, as well as easiest to falsify. The conclusion that all tables are rectangular is an inductive generalization. Note how this differs from anything in deduction. Take into account the mathematical field of statistics. Statistics collects data and tries to find correlations between several variables. If possible, statistics tries to look for causal relationships, but there is a classic mantra in statistics: correlation does not equal causation. For instance, aggressive crimes increase with the increased sale of ice cream. Is there a direct link there? No, it wouldn’t make sense for ice cream sales to affect murder rates. However, a third cause could be responsible for the two: summer heat. The purpose of this is to demonstrate the uncertainty of induction. It wouldn’t make sense to speak of proof or disproof. 

It is likely that when people repeat the mantra “you can’t prove a negative”, they are thinking along the lines of inductive logic. For instance, take the claim that there are aliens out there, and you are discussing this with someone. You say that there is no evidence of intelligent life on other planets, and your friend responds, “You can’t prove that there isn’t life on another planet.” You respond that “you are never called upon to prove a negative. You can’t prove a negative.” Your friend’s mistake would be shifting the burden of proof onto you. Your mistake would be saying that you are never called upon to prove a negative and can’t do that. The proper response would be, “no, you can’t just shift the burden of proof onto me. We are debating the proposition that there is intelligent life elsewhere in the universe. You are making that claim, and it is your job to substantiate that claim with evidence.” If you are skeptical of the proposition that intelligent life exists elsewhere in the universe, that doesn’t mean you believe there isn’t intelligent life on other planets, only that you lack any reason to be certain about it. The same goes for debating the proposition that a god exists. For the existence of a deity, things get trickier. If the definition of a deity offered by the believer is a specific one and contains contradictions, then the skeptic has to point out the contradictions in the definition of that deity to show that the deity does not exist. If the debate is just over the proposition that there is “someone up there”, then burden of proof is on the theist, and the skeptic wins if the theist fails to establish any reasonable evidence. 

The same applies for a sketchy premise in a deductive argument. If a premise in an argument fails to be demonstrated as true, and you challenge it, it would be wrong for the argument's giver to shift the burden of proof to you. Take the previous example of the Loch Ness monster, but in a different valid argument form called modus ponens, which takes the form, "If P, then Q. P, therefore, Q."

   This time, let the argument be, "If there were a monster in Loch Ness, then someone would have found it. A photograph was taken of it, therefore, there is a monster in Loch Ness." Granting the premises to be true, the conclusion must also be true. And there have been claims that the Loch Ness monster has been photographed. However, as a good skeptic, you would challenge the second premise’s truth. As it turns out, these photographs are faked, which undermines the soundness of the argument.

Hopefully this clears up the confusion amongst people about the concept of proof and the burden of proof. Again, it is possible to prove a negative as shown by the two examples from deductive logic. Those are not the only two ways to prove a negative, as there are many more. This was not meant to be an exhaustive list of every way to prove a negative. It has also been shown that inductive claims can be falsified (this is a driving force behind scientific progress, by the way!). Hopefully, those who read this will walk away with a better understanding of how to conduct a debate an analyze an argument.

Recommended reading: Introduction to Logic by Irving M. Copi & Carl Cohen -- any edition should suffice. There's also an online logic textbook here, and the youtube channel of Dr. Jason Campbell has a few lecture series on logic. If you are unable to take a logic course at a college, these resources are recommended to learn more about logic.