Monday, June 18, 2012

Spooky Action at a Distance (p. 3)

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Spooky Action at a Distance
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A More Rigorous Argument

For a more rigorous argument for nonlocality I’ll define some terms. Let spin property be whatever it is that is responsible for the electron spin measurement to be spin up or spin down (the spin property might be the electron having the spin prior to being measured, or it might be something in the electron that interacts with the spin measuring device that creates and randomly determines the spin, or it might be a combination of properties of both the electron and the measuring device etc.). Call a determinate spin property a spin property that definitively determines the electron’s spin measurement result (the electron having the spin prior to observation, or something in the electron interacting with the measuring device to deterministically cause the spin measurement result etc.). Let the phrase “an observation affects spin” denote the observation deterministically or indeterministically interacting with the electron in such a way that it is not the case that a determinate spin property was present prior to the spin measurement for the axis measured; an example of an observation affecting the spin measurement would be the observation randomly determining the spin. With those terms defined, we can begin a deductive argument.
  1. It is not the case that observation never affects spin.
  2. If observation at least sometimes affects spin when the axes are different, but never when they are the same, then nonlocality is true.
  3. If observation at least sometimes affects spin when the axes are the same, then nonlocality is true.
  4. Therefore, nonlocality is true.
In a little bit I’ll prove that the conclusion (line 4) logically follows from the premises (lines 1-3) but first I’ll try to justify the premises.
  1. It is not the case that observation never affects spin. By an observation “not affecting” spin I mean that the determinate spin property was already present prior to measurement as opposed to e.g. the measurement randomly determining what the spin result would be. Does this idea work? Consider the case where the entangled electrons are measured along the same axis. If observation never affects the spin, how is it that the entangled electrons always have opposite spins regardless of which of the three axes are chosen? If observation never determines the spin orientation and in all cases the definite spin properties are present prior to our observing them, the only plausible answer seems to be that all three axes had definite spin properties set prior to the measurement so that no matter which axis is chosen the measured spins will be opposite of each other, but this yields Bell’s inequality and conflicts with quantum mechanics. So the first premise appears true and is more plausible than its denial.
  2. If observation at least sometimes affects spin when the axes are different, but never when they are the same, then nonlocality is true. Trying to maintain that locality exists when observation at least sometimes affects spin when the axes are different but never when they are the same has a number of problems. If measurement never affects spin when the entangled electrons are measured along the same axis, how is it that the spins are always opposite of each other regardless of which of the three axes we choose? We could say that all three axes had definite spin properties set prior to the spin measurement, but then we get into the same problem I mentioned above. Even apart from that, there’s another problem with locality here. Granting that observation at least sometimes affects spin when measuring on different axes but never on the same axis, then when measuring electron #1, how does electron #1 “know” whether electron #2 is being measured along the same axis? How could the “don’t let the observation affect spin when #1 and #2 are being measured on the same axis” prohibition get enforced? One could say that measuring electron #1 causally prevents an observation from determining the spin of electron #2, but then we’re right back at nonlocality again. The second premise thus appears to be true, and is more plausible than its denial.
  3. If observation at least sometimes affects spin when the axes are the same, then nonlocality is true. To illustrate, suppose entangled electrons #1 and #2 have their first spin measured along the same axis and electron #1 is measured first. If observing electron #1 makes it spin up, how does electron #2 “know” to have the opposite spin? To explain the correlation in accordance with the laws of physics, making one electron spin up in this case would also have to (directly or indirectly) make the other electron measurement result spin down. Otherwise the correlation becomes inexplicable. (What if both electrons are measured at the same time? Perhaps then we have two simultaneous causes yielding a given measurement result.)
With justification for all three premises, the next task is to show that the conclusion follows. I plan on creating an Introductory Logic, Part 3 entry, but while it may seem that this article is a complete digression on my series on symbolic logic, I’ll actually use symbolic logic here to prove that the conclusion follows from the premises. Those of you who have read Introductory Logic, Part 1 and Introductory Logic, Part 2 may wish to skip the crash course in some symbolic logic below.

Crash Course in Some Symbolic Logic

While I strongly recommend reading Introductory Logic, Part 1 and Introductory Logic, Part 2, a crash course in symbolic logic seemed like a good idea to have in this article anyway. First, explaining the connectives ∨ ∧ ¬, and → below:

ConnectiveSymbolic
Logic
English
Meaning
Notes
∧ (conjunction)p ∧ qp and qThe p and q parts are called conjuncts.
∨ (disjunction)p ∨ qp or qThe p and q parts are called disjuncts.
→ (conditional)p → qIf p, then qThe p part is called the antecedent and the q part is called the consequent. Sometimes p → q is read as “p implies q.”
¬ (negation)¬pNot-pThe negation of P is ¬P, and ¬P means “not-P” or “P is false.”


Some rules of inference I’ll use in my symbolic logic proof:

modus tollens
 
In English In Symbolic Logic
If p then q
Not-q

Therefore, not-p
p → q
¬q

∴ ¬p

Disjunctive Syllogism
 
In English In Symbolic Logic
p or q
Not-p

Therefore, q
p ∨ q
¬p

∴ q
p or q
Not-q

Therefore, p
p ∨ q
¬q

∴ p
conjunction
 
In English In Symbolic Logic
p
q

Therefore, p and q
p
q

∴ p ∧ q


A quick example of how they can work:
  1. A
  2. B

  1. B ∧ A 1, 2, conjunction
Notice that the order of the premises doesn’t matter when using rules of inference. Rules of inference aren’t the only rules of logic though. There are also logical equivalences which have the handy property of being replaceable anywhere within a line.

equivalencename of equivalence
 
p ⇔ ¬¬pdouble negation
 
¬(p ∧ q) ⇔ ¬p ∨ ¬qDe Morgan’s laws
¬(p ∨ q) ⇔ ¬p ∧ ¬q


One method of proof known by various names as indirect proof and proof by contradiction begins by assuming the opposite of what you want to prove and then obtaining a logical contradiction, i.e. a contradiction of the p ∧ ¬p or ¬p ∧ p sort. The structure of an indirect proof:

indirect proof
 
a) p indirect proof assumption
b)
 ...
 q ∧ ¬q (or ¬q ∧ q)
c) ¬p a-b, indirect proof
a) ¬p indirect proof assumption
b)
 ...
 q ∧ ¬q (or ¬q ∧ q)
c) p a-b, indirect proof


For example, suppose we wanted to prove ¬(H ∧ R) from premise 1 below:
  1. (H ∧ R) → ¬(H ∧ R)

  1. H ∧ R indirect proof assumption
    1. ¬(H ∧ R) 1, 2, modus ponens
    2. (H ∧ R) ∧ ¬(H ∧ R) 2, 3, conjunction
  1. ¬(H ∧ R) 2-4, indirect proof
And that’s it; those are the only rules of logic I’ll use in my symbolic logic proof.

The Symbolic Logic Proof

Next I’ll use symbolic logic to prove that the conclusion follows logically from the premises. The first step is to define some letters to represent propositions.
  • S = Observation at least sometimes affects spin when the entangled electrons are measured on the Same axis.
  • D = Observation at least sometimes affects spin when the entangled electrons are measured on Different axes.
  • N = Nonlocality is true.
So the claim that “observation never affects spin” can be represented as ¬S ∧ ¬D, and the claim “It is not the case that observation never affects spin” can be represented as ¬(¬S ∧¬D). With that in mind, we can represent premises 1-3 of the deductive argument...
  1. It is not the case that observations never affect spin.
  2. If observation at least sometimes affects spin when the axes are different, but never when they are the same, then nonlocality is true.
  3. If observation at least sometimes affects spin when the axes are the same, then nonlocality is true.
...like this:
  1. ¬(¬S ∧ ¬D)
  2. (D ∧ ¬S) → N
  3. S → N
We can then use the rules of logic to show that N follows from premises 1-3 like so:
  1. ¬(¬S ∧ ¬D)
  2. (D ∧ ¬S) → N
  3. S → N

  1. ¬N indirect proof assumption
    1. ¬¬S ∨ ¬¬D 1, De Morgan’s laws
    2. S ∨ D 5, double negation
    3. ¬S 3, 4, modus tollens
    4. D 6, 7, disjunctive syllogism
    5. D ∧ ¬S 7, 8, conjunction
    6. ¬(D ∧ ¬S) 2, 4, modus tollens
    7. (D ∧ ¬S) ∧ ¬(D ∧ ¬S) 9, 10, conjunction
  1. N 4-11, indirect proof
Even if nonlocality is real though, with the way it works we can’t use nonlocality to send an information signal faster than light. For example, suppose it is true, as I think it is, that when the entangled electrons are measured on the same axis, the first measurement of an electron causes one electron to be spin up and the other to be spin down. This would be useless in sending a communication signal. There’s no way to make an electron spin up or spin down the first time it is measured after the entangled electrons fly apart. Barring precognitive powers, there’s also no way to predict in advance what the spin of the first measured electron will be. And there’s no way to tell just from observing the electron whether that electron was measured before the other. So even if spooky action at a distance is real, we can’t exploit it for the purposes of instantaneous communication.

Spooky and Relevant

So why this article? First, spooky action at a distance is just plain cool. Second, it lends some rational support to the idea of simultaneous causation, i.e. where a cause is temporally simultaneous with its effect, since one plausible explanation is that observing one electron simultaneously causes the other to have the opposite spin when measured on the same axis. Even if the correct explanation isn’t simultaneous causation, it is (I think) more plausibly true than false that nonlocality of some sort is real, in which case we would at least have something like simultaneous causation. This is helpful for some theists who believe that when God created the universe the cause was simultaneous with its effect (I’ll get to this more late when I discuss the kalam cosmological argument). An atheist might object to this by saying it’s impossible for a cause to be simultaneous with its effect, but the evidence of spooky action at a distance suggests it is plausible and that we should at least be open to the possibility if we don’t have any good argument for its alleged impossibility.

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