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.

The Color Of A Cat

I'm late for my two week-appointment with my blog, so here's something special.

- How logic in Quantum Mechanics differs from the 'real world'.

Today I want to tell you one of the big secrets of Quantum Mechanics, using a parallel with a cat in it. By the time you have read this page (a couple of times) ordinary Quantum Mechanics will hopefully be clear, if not, don't hesitate to ask. Schrødinger's cat is another well known parallel with a cat, but it is about something else (in Q.M.).

We all use probability in our daily life, it's a handy tool. There is 1/6 chance of a die landing on a 6, there is 1/2 chance of a slice of bread landing with the peanut-butter-side down on the floor. Here, common sense dictates several wrong claims, like getting a 6 two times on a row (on a die) makes a third 6 less probable. Forgetting those fallacies, we all think that having enough information removes the probability. If I know the exact speed(s), air currents and form of the die and the table, I could (in theory) predict exactly on which side it would land.

So the classical world ('real world', 'everyday world') probabilities are really hidden variables. Stuff we don't know. Probability is in the map and not in theterritory.

How does this differ from Q.M.? Let us give a parallel.

Say you have a perfectly gray cat. Perfect in the sense that it is exactly halfway between white and black on your gray-scale. If you ask 'is the cat gray?' what happens? The answer is 'yes', and, of course, the cat doesn't care. If you ask 'is the cat black' what happens? You get the answer 'sort of' or 'halfway black', and, again, the cat doesn't care.

Let us assume this cat is an electron, and color is some property of that electron. The cat is still perfectly gray. If you ask 'is the cat gray?' what happens? Well, the answer is 'yes' and the cat doesn't care. It's the same, so no surprises yet! If you ask 'is the cat black?', two things can happen:

1. Answer: 'yes, black' and the cat instantly changes color to black.
2. Answer: 'no, not black' and the cat becomes non-black, which, in this case (starting with a grey cat) would actually give you a white cat.

Poor cat. But which of the answers do you get? If you had 1000 such cats and asked them all, you would get answer 1) about 500 times, and answer 2) about 500 times, so we say that the probability of getting 1) is 1/2 and same for 2).

To digress, what Schrødinger's cat is about (if I understand it correctly), is whether this is actual probability. Are there any hidden variables determining which of the cats come out black, or is there an inherent True Probability in Nature? 'God does not throw dice' -Einstein. If anyone cares, I believe Einstein to be wrong about this, and that these experimental outcomes are determined by probability. I also believe the Schrødinger's cat experiment to be a bad argument, as the cat would measure whether it was alive or dead. You don't have to be a person to do an 'experiment', and not a cat either; any two molecules on a collision course will do an experiment to see whether they collide or not.

What is special about Q.M. logic? Grey can be a 'superpositon' of white and black. How do we model this? There is a certain thing in mathematics called a Hilbert space, where colors are unit vectors (or subspaces), and a vector [1,1] can be viewed as a superposition of [1,0] and [0,1].

Why? Well, experiments show that... But why? This borders on philosophy. From a scientific point of view, this is our best model – it works (there's a friggin' flag on the Moon and a rover on Mars).