Both and Neither

Here’s a strange logical phenomenon I thought of. I admit it is pretty random, but I thought I would like to share it.

Suppose you have these two sentences:

A. Statement B is false.

B. Statement A is false.

So far, this is ordinary, and is not a paradox, as one statement can express the truth while the other can express a falsehood. You can imagine this situation as two people, both saying the other is lying. If one person is telling the truth while the other is lying, both would say that the other is lying. Now I introduce the following pair of statements:

C. Both A and B are true.

D. Neither A nor B is true.

Because we know that one and only one of A or B is true, both C and D must be false. Continuing in like fashion, we have:

E. Both C and D are true.

F. Neither C nor D is true.

In this case, E is false while F is true. If we continue this one step more, we find we are in a repeating cycle:

G. Both E and F are true.

H. Neither E nor F is true.

Only one of E and F is false, so it follows that both G and H are false.

Statements Number of True’s
A, B 1
C, D 0
E, F 1
G, H 0

In the previous example, we started with two somewhat unusual sentences. I’ll change the initial sentences, for example, to

A. Statement B is true.

B. Statement A is true.

Now if two people both say the other is telling the truth, then they’re either both telling the truth or both lying. Thus it’s possible that either A and B are true, or neither A nor B is true.

C. Both A and B are true.

D. Neither A nor B is true.

The trouble is that we don’t know whether C or D is true, but we know one of them must be true.

E. Both C and D are true.

F. Neither C nor D is true.

Because only one of C and D is true, both E and F must be false.

G. Both E and F are true.

H. Neither E nor F is true.

It follows that H is true.

Statements Number of True’s
A, B 0 or 2
C, D 1
E, F 0
G, H 1

We notice the alternating between 0 and 1 again, with a possible 2. The even-odd pattern is still kept intact, though the second example begins with an even number while the first began with an odd. What happens, however, when we are not 100% sure that whether the first step will have an odd or even number of true statements?

A. Schrödinger’s cat is alive.

B. Schrödinger’s cat is dead.

Actually, Schrödinger’s cat is 50% dead and 50% alive.

Meta Writing and Logic

A couple chapters into GödelEscherBach by Douglas Hofstadter, I’ve become intrigued by the idea of meta writing, which is writing about writing. We end up going to this topic from logical paradoxes. Let’s begin. Normally, when we write a statement, it is either true or false.

One and one make pie.


The sky is an orange.

In these cases, the statement refers to something else. But what if the statement refers to itself? Here we have the infamous paradoxical statement:

This statement is false.

Think about this. If it’s true, it must be false, and if it’s false, it must be true! Thus, it can be neither true nor false.

What about referring to other sentences? This is allowed, generally.

(1) The next statement is true.

(2) All horses are white.

(3) The previous statement is false.

Now we have three sentences, and this set of sentences makes sense. Each one is either true or false: (3)—true; (1) and (2)—false. As long as statements are either true or false, logicians are content. Normally we might care whether a statement is true, but that is beyond the point.

Okay, so where does meta writing come in? Simple. In the previous example, (2) was a regular old statement, while (1) and (3) were about it. And they weren’t about the content in (2)—they were about statement (2) in itself. If they only described the content, they can still be contradictory, but they wouldn’t be meta statements. Consider the following:

(a) All horses are white.

(b) Not all horses are white.

These two statements present no logical paradox. One is right, and the other wrong.

Escher Hands
Drawing Hands, Escher, 1948.

But, what if, in the original example, we removed statement (2), leaving only (1) and (3)? This leads to logical disaster:

(1) The next statement is true.

(3) The previous statement is false.

Now, we cannot simply dismiss one as true and the other false. If (1) is true, then (3) must be true, so (1) must be false. And if (1) is false, then (3) is false, so (1) must be true. Obviously (1) cannot be both true and false. Same goes for (3): if (3) is true, then (1) is false, so (3) must be false, etc.

So far the book has covered one way to eliminate this paradox. I’ll try to explain the gist of it. Take the set of all normal statements, i.e. non-meta statements. We’ll call this set zero. Now take all statements about statements in set zero, and put them into set one. Set one will thus be the set of meta statements. Then take all statements about statements in set one, and put them into set two. These are meta meta statements, or statements about meta statements. Set three will consist of meta meta meta statements, and so on.

The rule is basically that all meta statements must refer to a statement in a lower set than itself. If not, then the statement has no meaning.

Consider the disaster example above. We don’t know what set (1) belongs to—say it belongs to set N, where N is a positive whole number. Because (1) refers to (3), (3) must be one level lower than N, so it is of the set N – 1. But (3) refers to (1), so (1) must be one level below (3). That means (1) must be of the set N – 2. But we began with the assumption that (1) is of set N, and N cannot equal N – 2. Thus, the two statements (1) and (3) together have no meaning.

Another way of looking at it is this: (1) refers to (3), so it must be of a higher set than (3). But (3) also refers to (1), so it must be of a higher set than (1). They cannot both be of higher sets than the other, so the combination of the two sentences makes no sense.

Now it’s possible to make sense out of the statement, “This statement is false.” It is referring to itself, so it must be a meta statement of itself. That is not possible. A statement cannot belong to a higher set than the set to which it belongs. Thus the statement makes no sense.

Try this one:

(P) Q is false.

(Q) R is false.

(R) P is false.

This is again paradoxical. The explanation will be left to the reader as an exercise.

Okay, well, to explain it anyways: if P is true, then Q is false, so R is true, so P is false… Again, we reach a statement being both true and false, which doesn’t work. And, since P refers to Q, it must be of a higher set than Q, and Q must be of a higher set than R, which in turn must be of a higher set than P—this does not mathematically work.

Last puzzle:

(P) Q is true.

(Q) R is true.

(R) P is true.

AT FIRST, this set appears to make sense. After all, if P is true, then Q is true, and R is true, and P is true—check, it works. But, we know from our earlier observation that if statements refer to each other in a loop like this, it’s bound to be meaningless; it did not matter whether statements were true. In other words, this puzzle should have the same explanation as when each true was a false. When we look at it, however, it seems different. It seems to work.

So what’s wrong with it? Assume P is false (after all, we do not know whether P is true or false). This implies that Q is false. And if Q is false, then R is false. And if R is false, then P is false. At first glance, this works. All three statements could be false. I suppose this does make logical sense. I’m not sure what to make out of this.