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.

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