The Cat That Killed De Morgan

According to google (number of hits), it's supposed to be "the cat who killed", does anyone know for sure?

• Part two of Quantum Mechanical Logic. (Part one here).

In logic (classical/ordinary logic) we have something known as De Morgan's laws:

Let P be the statement 'It's raining outside', or any logical statement

Let Q be the statement 'The sun is shining', or any other logical statement

Then saying 'not(P and Q)' is the same as saying '(not P) or (not Q)', in words:

'It can't be both raining outside and sunny' is the same as

'It isn't raining outside, or It isn't sunny'

(yes, none of the sentences are necessarily true, but they are equivalent)

(in logic, 'A or B' means 'either A, or B, or both')

Does this work in Quantum Mechanics? No. Remember our perfectly grey cat. If you ask the cat

P: Are you black

Q: Are you grey

The statement 'not(P and Q)' [not both black and grey] is true. In the Quantum Mechanical sense, you cannot observe that it's grey and black at the same time (that is not a legal outcome). Hence (grey and black) is false, and 'not(grey and black)' is true.

The statement '(not P) or (not Q)' [not black or not grey] is true in 50% of the experiments. The part (not grey) is never true, as it starts out with being perfectly grey, and you cannot then observe it to be not grey. The statement 'not black', however, is true 50% of the time as it is a 50% chance that when we measure it to be 'not black' the cat will spontaneously become white. In total '(not P) or (not Q)' is true 50% of the time, and false the other 50% of the time.

So in Quantum Mechanical logic De Morgan's laws are not valid, the two statements are not equivalent. But in logic we require this law, so what to do? Well, even though I have called it Quantum Mechanical logic, it isn't really logic, but something else that has a strong similarity to ordinary logic; and it has some differences as we just saw.

To be fair to any mathematically inclined readers I want to add a comment. The "logical" 'or'-statement in Q.M. is usually taken to be a join of subspaces (the least subspace containing both the subspaces of Q and P), instead of a 'or' between two experimental outcomes, so that in our example the most natural thing to say is that (not black) or (not grey) is the linear span of the two, namely the whole 2-dimensional black/white subspace. Observing whether it is in this subspace would give a 'yes' with a 100% probability. It was, after all, grey. This seems to make my point moot, but alas, even with this more refined notion of "logical 'or'" you can find contradictions to De Morgan's laws (where meet and complement of subspaces is not the same as complement and join), see for example this page.