# Possibility and Free Will

What is possibility? What do humans mean when they say "This is possible" or "That's impossible!"?

There are many definitions of "possibility", so in polite conversation it's useful to use them precisely. Let's start with "physical possibility" as one form. This is basically saying "The known Laws of Physics either permit or do not forbid this outcome from occurring." As an example, suppose you're pondering the possibility of picking up a rock and throwing it such that it lands on the moon. This is physically impossible because your body and your arm do not contain the required energy to get the rock there. As another example, suppose you are considering the possibility of sustained nuclear fusion in the lab providing a huge supply of cheap energy. This is physically possible because we have physical proof of this process already happening--that is, the sun does it already.

Another form of possibility is "Logical possibility". This is basically saying "Using two-valued True/False logical propositions and their rules, we can logically deduce whether some proposition questioning the logical possibility of something is true (that is, it's possible) or false (that is, impossible)." As an example, suppose you assume for the sake of argument that a human arm can generate a million petajoules of energy on a whim, then you assume the fact that it takes some amount of joules to throw a rock to the moon. Then you can logically deduce that because you have a sufficient amount of energy to throw a rock to the moon, you can do so. It is logically possible under this set of assumptions. As another example, suppose for the sake of argument that penguins can fly. Then it's logically possible for a penguin in a zoo to take off and fly back to its home in wherever it came from. As another, assume for the sake of argument that crows can't fly, and so it's logically impossible for it to escape a glass container you put it in.

My three examples are all on their face absurd. This is intentional. Logical possibility and logical impossibility are only as valid as the assumptions used to arrive at those conclusions. If you assume nothing supernatural can exist, then it's logically impossible for most definitions of God to exist. But this has no bearing on the empirical question of whether God exists, supernatural things exist, the empirical question of whether penguins or crows can fly, the empirical question of whether a human arm actually can generate petajoules of energy within seconds. Therefore physical possibility is stronger than logical possibility. This doesn't mean logical possibility is useless: you can pick less controversial assumptions that most people agree are obvious and use the many tools of logic at your disposal from there to find out interesting facts. You can even use the very useful probability theory tools. If you as a cop know and assume that people who break store windows late at night are likely robbers, you can conclude that you should arrest first and ask questions later if you see two thugs break a store window.

You can even intermingle logical and physical possibility by assuming physics as we know it is completely correct and working out logical implications that you can then experiment. If your logical implication says something is impossible, that's great news! Now you can go test it and see if your assumption that physics is correct is empirically sound.

There's another form of possibility I want to talk about, which is the most general kind. Sometimes people qualify it with "That's theoretically possible or theoretically impossible", but that assumes a theory and often logic so it's not the precise form of possibility I'm after. I don't really have a handy qualifier for it apart from "general", so I'll have to explain it. This most general form of possibility is closer to "I can imagine it to be the case."

"General possibility" is basically saying "this abstract world-configuration is a way out of all the available ways some state-space can be configured." More concretely, it's kind of saying "This universe is made of atoms. This particular arrangement of atoms, with one here, and one there, is a way to arrange the atoms. It is therefore possible." Thus it is generally possible that the Eiffel Tower could hover over the Golden Gate Bridge tonight at 6pm and stay there indefinitely, because I can imagine the universe having the same arrangement of atoms that is the Eiffel Tower but in a different place--namely above the bridge.

Thus it is generally possible for me to throw a rock to the moon in this sense: at one particular slice of time, the arrangement of the atoms in the universe has me holding the rock. In another slice of time (not necessarily in the past or future, just an arbitrary readout of the state of the universe's atoms), the rock could be 100m in the air and my arm in the follow-through motion of a throw. At another slice, the rock could be on the moon and I, still on earth, would have just completed a throw. If you took these three pictures together you could interpret them as "I threw a rock to the moon".

So this general notion of possibility isn't that useful, either from a logical or empirical standpoint. Logical possibility at least has useful tools, and physical possibility actually tells you about the world we live in. At best this general possibility is like the existential quantifier in math: general possibility says that something exists in the set of ways the universe could have its atoms arranged, and general impossibility says that something cannot exist in the set of ways the universe could have its atoms arranged.

Unfortunately this general sense of "possibility" seems to be the most commonly used in everyday discourse when people say "That's possible!" or "That's impossible!"

Okay, maybe not quite as general, but close. And so people are easily deceived (lotteries), scared (password cracking), or confused (quantum physics), or argumentative (free will). Let's look at each of those in turn.

Here's a lottery bet for you. I have a fair coin, I will flip it twice. If heads comes up twice (25% of the time), I pay you $1000. Otherwise, you pay me$500. (75% of the time.) Do you want to take this bet? You are free to take it as many times as you like.

Las Vegas' popularity indicates that there are indeed people who will take this bet. They are fools and Sure Losers, by both empirical observation and logical deduction.

But they can grandstand and say "Ahh, but it's possible for heads to come up twice 200 times in a row and thus it's possible for me to win $200,000!" Logic says this: your expected earnings is mathematically calculated by multiplying the probability of your wins times the value of your winnings, then subtracting the probability of your losses times the value of your losings. In this case,$1000*0.25 - $500*0.75 = -$125. This is a losing strategy, you are more likely to have losses than gains, so you shouldn't take this bet even once let alone multiple times.