Sunday, June 24, 2012

Pseudo-Relativism

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This blog entry will be about attacking a position I’ll call “pseudo-relativism.” Although I address it in part 3 of my series on the moral argument, I thought it deserved its own entry since it seems I’ve seen it or something like it popping up often enough. What is pseudo-relativism? Suppose Adolph thinks that torturing infants just for fun is a morally right action for him and Oskar thinks that would be a morally wrong action for Adolph to do. Pseudo-relativism says that morality is relative in such a way that Oskar and Adolph are both right. Before attacking this view I’ll recap a few things about morality is and why someone might be motivated to accept pseudo-relativism.

The Motivation

To recap a few things about morality, an action is morally wrong for subject S only if S ought not to do it, and an action is morally right for subject S only if S ought to do it. Moral objectivism says that morality holds independently of human opinion. For example, a popular belief among moral objectivists is that torturing infants just for fun is morally wrong independently of whether we humans believe it to be so.

An atheist might want to reject moral objectivism on the grounds that if God does not exist, then objective morality does not exist, but finding an intellectually viable alternative to moral objectivism can be a bit tricky. One might balk at accepting “there is nothing morally wrong with knowingly torturing infants just for fun,” but to maintain that there is something morally wrong with torturing infants just for fun without accepting moral objectivism, one would have to accept some form of ethical relativism. Ethical relativism (or moral relativism) agrees with moral objectivism that morality exists but unlike moral objectivism says that moral truths are relative to human opinion. For example, a belief called cultural relativism (which goes by various other names, such as conventional ethical relativism and conventionalism) says that morality is relative to cultural opinion in such a way that the culture believing that a certain action is morally right/wrong/good/bad/etc. makes that action morally right/wrong/good/bad/etc. for that culture. If for instance a culture believed violent anti-Semitism to be morally right, then it becomes morally right for that culture to do it. Someone else might think the culture is doing something morally wrong, but such a person would be mistaken because the culture thinks it is morally right and that is enough to make it morally right for that culture.

Another version of moral relativism called ethical subjectivism (which also goes by various names, e.g. subjective ethical relativism) says that the individual believing an action to be morally right/wrong/good/bad/etc. makes that action morally right/wrong/good/bad/etc. for that person. So if Adolph thinks that killing Jews is morally obligatory, then it is morally right for Adolph to do it. If Oskar believes killing Jews is morally wrong, then it is wrong for Oskar to do it. Under ethical subjectivism, Oskar can believe it is wrong for Adolph to kill Jews, but that belief would be mistaken because Adolph thinks otherwise, and so on ethical subjectivism Adolph has a moral duty to kill Jews.

It’s understandable then why someone might want to reject both ethical subjectivism and cultural relativism; who wants to say that e.g. it is morally right for Adolph to kill Jews? That wouldn’t make for a very plausible system of ethics. Examples of unfavorable implications could be multiplied. If someone believed torturing infants just for fun is morally right and, after killing everyone who disagreed with him, he tortured infants for the sheer fun of it, then according to ethical subjectivism there is nothing morally wrong with the man torturing infants just for fun. If a community of men believed torturing infants just for fun was morally right and, after killing everyone who disagreed with them, they tortured infants for the sheer fun of it, cultural relativism says that there’s nothing morally wrong with such torture.

Help From Some (Painless) Symbolic Logic

Symbolic logic helps to get a view of the structure of the sort of arguments being used against ethical relativism. Some quick symbolic logic stuff:

EnglishSymbolic Logic
 
If p is true, then q is truep → q
If p were true, then q would be truep □→ q
Not-p (p is false)¬p


That there is a difference between “If p is true, then q is true” and “If p were true, then q would true” can be revealed by observing that one can believe “If Oswald didn’t shoot Kennedy, someone else did” without believing “If Oswald didn’t shoot Kennedy, someone else would have.” A useful rule of logic:

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

Therefore, not-p
p → q
¬q

∴ ¬p


The above arguments against the alternatives to moral objectivism take the following form, where P is some challenged proposition (e.g. “ethical subjectivism is true”), T denotes a thought experiment (e.g. a man who believes it is morally right to torture infants just for fun, does such torture, and has killed everyone who disagreed with him), and R denotes the questionable result of the thought experiment (e.g. the man torturing infants just for fun isn’t doing anything morally wrong):
  1. P → (T □→ R)
  2. ¬(T □→ R)

  1. ¬P 1, 2, modus tollens
The conclusion (line 3) follows logically and necessarily from the premises (lines 1 and 2) by the rules of logic, and thus for the conclusion to be false a premise must be false. Putting the argument in symbolic logic also helps see why certain objections don’t work. One could object saying that the state of affairs described in T (e.g. the man torturing infants just for fun believing it to be morally right after killing everyone who disagreed with him) has never happened and will never happen. That may be true, but such an objection doesn’t attack either premise of the argument, and if both premises are true the conclusion follows whether one likes it or not. One could also criticize the argument by noting that the state of affairs described in T is an extreme and contrived example. Maybe that’s true, but again that doesn’t attack any premise of the argument.

Consequently, it’s difficult to convincingly attack “If a man believed it was morally right to torture infants just for fun…” type arguments. One wants to avoid promoting an alternative with unacceptable consequences. One also wants to avoid the apparently unacceptable consequences of denying moral objectivism. For example, a quick case for moral objectivism could go like this:
Consider this hypothetical scenario. The only humans who ever existed are a community of men and infants, both of which are grown in a lab. To replace the men who die, the community creates fully grown men in the lab. Throughout all time, every human has agreed that both torturing infants just fun and killing them are morally right. The community of men grow infants in the lab and then torture them to death just for fun. As these men sadistically torture infants for the sheer fun of it, would this action be morally wrong in spite of their opinion to the contrary? Would that action be morally wrong independently of their opinion?
If the answer to both questions is “Yes” that would seem to suggest that there is at least one objective moral truth. To see why, first we can ask ourselves this question: would moral objectivism be true in that scenario? If the scenario were actualized, every existing human’s opinion would be that the act of torturing infants just for fun is morally right, and so for that action to be morally wrong regardless of their opinion, it would have to be morally wrong regardless of any human opinion that exists (bear in mind we are talking about human opinions that exist in this scenario). But if in that scenario the action’s moral wrongness would hold independently of any existing human opinion, the only way to prevent the action’s moral wrongness from holding independently of all human opinion in that scenario is if the action’s moral wrongness were somehow dependent on nonexistent human opinion, which doesn’t seem very plausible. Since in that scenario the action’s moral wrongness would hold independently of human opinion, moral objectivism (a moral property existing independently of human opinion) would be true in that scenario. Yet if in this scenario the moral wrongness of torturing infants just for fun would exist independently of human opinion, it seems that the moral wrongness of such torture would hold independently of human opinion in the real world also.

In spite of the above argument, one might still be tempted to say that if this community actually existed alongside dissenting opinion like ours, the action would be morally wrong relative to our opinion, but it would be morally right relative to the hypothetical community of baby torturers. Hence pseudo-relativism, where Oskar can correctly say it is morally wrong for Adolph to torture infants just for fun but the “it is morally wrong for Adolph to torture infants just for fun” is a relative truth rather than an objective one. Adolph thinks it is morally right for him to torture infants just for fun and Oskar believes it is morally wrong for Adolph to torture infants just for fun, and the beliefs of both people are true relative to their opinion.

The Problem with Pseudo-Relativism

But this form of ethical relativism doesn’t work. Contrast pseudo-relativism with ethical subjectivism. If Adolph thinks that torturing infants just for fun is morally right, then it is morally right for Adolph to do it. If Oskar believes torturing infants just for fun is morally wrong, then it is wrong for Oskar to do it. Whether ethical subjectivism is correct, it at least produces a non-contradictory answer to the question “Should Adolph torture infants just for fun?” In this way ethical subjectivism is a coherent form of ethical relativism. But what is not coherent is the belief that right and wrong do exist, where Adolph thinks its morally right for him torture infants just for fun and Oskar thinks it is morally wrong for Adolph to torture infants just for fun and have them both be right. Pseudo-relativism cannot give a coherent answer to question “Should Adolph torture infants just for fun?” It might be able to reiterate the beliefs of people about whether Adolph should torture infants just for fun but it cannot answer the question. Pseudo-relativism is not a coherent ethical system.

To drive the point home further, imagine that Oskar is a pseudo-relativist who is incapable of lying. Adolph comes to believe that pseudo-relativism is true but isn’t quite sure he understands it, so Adolph asks Oskar, “I believe I am morally obligated to torture infants just for fun, but should I do it?” What can Oskar as an honest pseudo-relativist say? Oskar can’t say “Yes” because he believes Adolph shouldn’t do it, but Oskar can’t say “No” because Adolph believes Adolph should do it. On pseudo-relativism, there doesn’t appear to be any coherent answer to Adolph’s question and thus no coherent fact of the matter as to what Adolph should do here. When it comes to moral obligations, pseudo-relativism is an incoherent mess. In contrast, both cultural relativism and ethical subjectivism handle the “moral obligation is relative to a given framework” notion in a coherent manner.

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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Sunday, June 17, 2012

Spooky Action at a Distance (p. 2)

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Spooky Action at a Distance
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As it turns out, something called Bell’s theorem shows that local hidden variable theory makes different predictions than quantum mechanics. Suppose that two entangled electrons fly very far apart from each other after. Two randomly chosen different axes from among F, L, and R are picked to measure the two electrons (e.g. it might be that the spin of F is detected for electron #1 and the spin of R is detected for electron #2). What are the odds that the measured spin orientations of the two electrons will be different from each other, i.e. what is the likelihood that one is spin up and other is spin down? With “Pr()” symbolizing “the probability that,” let Pr(measured spins are different)” represent “The probability that the measured spins of electrons #1 and #2 are different.” Quantum mechanics and local hidden variable theory give different predictions as to what that probability would be. How does local hidden variable theory make its prediction? To start out with, let’s look at a specific example: electron #1 with F+L+R– and electron #2 with F–L–R+. We can determine the likelihood of finding different spins along two randomly chosen different axes by checking all possible axis pairs (e.g. checking electron #1 on the F axis and electron #2 on the L axis) as follows:

electron #1: F+L+R–FFLLRR
electron #2: F–L–R+LRFRFL
spin difference: 2/6d d   
Pr(measured spins are different) = 13


For example, if in the case above we measured electron #1 on the F axis and electron #2 on the L axis, we would have found that the two spins are different (hence the “d” above), since electron #1 is F+ and electron #2 is L–. If however we measured electron #1 on the F axis and electron #2 on the R axis, we would have found both to be “spin up” (electron #1 is F+ and electron #2 is R+). We can see from the above table that if we randomly picked two different axes, there’s a ⅓ chance of finding different spins. Let’s check all possible spin orientations of the electron pairs:

electron #1: F+L+R+FFLLRR
electron #2: F–L–R–LRFRFL
spin difference: 6/6dddddd
Pr(measured spins are different) = 1
 
electron #1: F+L+R–FFLLRR
electron #2: F–L–R+LRFRFL
spin difference: 2/6d d   
Pr(measured spins are different) = 13
 
electron #1: F+L–R+FFLLRR
electron #2: F–L+R–LRFRFL
spin difference: 2/6 d  d 
Pr(measured spins are different) = 13
 
electron #1: F+L–R–FFLLRR
electron #2: F–L+R+LRFRFL
spin difference: 2/6   d d
Pr(measured spins are different) = 13
electron #1: F–L+R+FFLLRR
electron #2: F+L–R–LRFRFL
spin difference: 2/6   d d
Pr(measured spins are different) = 13
 
electron #1: F–L+R–FFLLRR
electron #2: F+L–R+LRFRFL
spin difference: 2/6 d  d 
Pr(measured spins are different) = 13
 
electron #1: F–L–R+FFLLRR
electron #2: F+L+R–LRFRFL
spin difference: 2/6d d   
Pr(measured spins are different) = 13
 
electron #1: F–L–R–FFLLRR
electron #2: F+L+R+LRFRFL
spin difference: 6/6dddddd
Pr(measured spins are different) = 1


So no matter which electron pair we get for our experiment, we get this inequality:
Pr(measured spins are different) ≥ 13.
For reasons I’ll explain a little later, the above sort of inequality is called Bell’s inequality. Our inequality is different from the predictions of quantum mechanics, which calculates the probability this way:
Pr(measured spins are different) = cos2(0.5 × angle between #1 and #2)
The angle between electrons #1 and #2 will always be either 120° or 240°. Using the above equation both turn out to have a probability of ¼. So who’s right? Quantum mechanics or local hidden variable theory? The answer of course is quantum mechanics.

While our main target was local hidden variables theory, Bell’s inequality can be used to argue for nonlocality in general. One might think the localist could just reject hidden variables theory and locality is saved, right? Well, it’s not quite that simple. Consider the cases where both electrons have their spins measured along the same axis. If observing one electron’s spin doesn’t causally influence the other, how is it that the spins are always opposite of each other regardless of which of the three axis pairs we pick? How does the other electron always “know” which spin is opposite of its partner? One could say that the entangled electrons had the three opposite spin outcomes “built-in” for each axis in advance, but that of course leads to Bell’s inequality which conflicts with quantum mechanics. Because the spins are always opposite of each other whenever the two entangled electrons are measured on the same axis, Bell’s inequality provides a powerful tool for arguing against locality. I’ll construct a more rigorous argument for nonlocality a little later.

As a further illustration of why just rejecting deterministic hidden variables isn’t enough, consider the Copenhagen interpretation which denies hidden variable theory. The Copenhagen interpretation says that when the entangled electrons fly apart the electrons have no definite spin on any axis before any of the electrons have their spins measured, and so when both electrons are to be measured on the same axis, if electron #1 is measured first and found to be spin up (where the measurement randomly determines the spin result) this measurement influences the other electron’s measurement to be spin down when measured on the same axis. So the Copenhagen interpretation rejects hidden variables but embraces nonlocality. Indeed, it is difficult to see how to plausibly avoid some sort of nonlocality from taking place here.

What is Bell’s theorem?

Bell’s original theorem proved that a certain inequality follows from local hidden variable theory and showed how that inequality is inconsistent with quantum mechanics. “Bell’s inequality” now refers to a family of inequalities conceptually similar to the one used in Bell’s original theorem (all such inequalities assume locality and some form of hidden variable theory). These inequalities are also known as “inequalities of Bell’s type.” The term “Bell’s theorem” refers to a family of methods showing that (a type of) Bell’s inequality follows form (a form of) local hidden variable theory and that this inequality is inconsistent with quantum mechanics. Incidentally, it’s not just electrons and spin that exhibit this sort of entangled behavior. Similar behavior occurs with photons and polarization and one could use that to illustrate Bell’s theorem. In my version of Bell’s theorem, the inequality I derived from local hidden variable theory is this:
Pr(measured spins are different) ≥ 13
Some forms of Bell’s theorem (like the one I used) show that given quantum mechanics, locality is incompatible with counterfactual definiteness. What is counterfactual definiteness? Counterfactual definiteness comes in varying degrees, but in its strongest version, it says that for any sufficiently specified conditions C, there are invariably true statements in the form of “If in (sufficiently specified) conditions C a measurement M was made, then R would have been the result.” Applied more narrowly to Bell’s theorem, counterfactual definiteness says that for any angle A and electron E, there are always true statements of the form “If we were to measure electron E at angle A, result R would have obtained.” An example of an interpretation that embraces this counterfactual definiteness is hidden variable theory largely thanks to its fully deterministic outlook, hence local hidden variable theory implying Bell’s inequality. An example of an interpretation that rejects it is the Copenhagen interpretation (it holds that observations randomly determine outcomes in some cases, such that “If we were to measure electron E at angle A, result R would have obtained” are not true, because different results are possible under those same circumstances). Bell’s theorem shows that counterfactual definiteness and locality cannot both be true if quantum mechanics is true thanks to Bell’s inequality. Though quantum mechanics doesn’t require counterfactual definiteness of the sort described, there is a (much weaker) sense of counterfactual definiteness that quantum mechanics does demand: whenever the entangled electrons have their first spin measurements along the same axis, the spins are always opposite of each other.

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Saturday, June 16, 2012

Spooky Action at a Distance

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Spooky Action at a Distance
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What is Spooky Action at a Distance?

The phrase “spooky action at a distance” comes from Einstein describing the idea, inspired by quantum mechanics, that in some cases measuring one particle can instantly (as in taking literally zero seconds) influence another particle regardless of how far away they are from each other, even if the two particles are light-years apart! The causal influence, if real, would be such that the cause is simultaneous with its effect. Stranger yet, one can’t exploit how this works to send an information signal faster than light. But how could the causal influence work this way and how is it we can have grounds for thinking this spooky action at a distance is real? In this article I’ll explain that, but to do that I’ll briefly introduce some simple quantum mechanics.

Introducing Some (Simple) Quantum Mechanics

Electrons have something called “spin,” a property that is not to be taken literally but in some ways electrons act as if they have spin. Along any given axis on which the spin of an electron is measured, an electron is either “spin up” or “spin down.” A Stern-Gerlach device is something that measures electron spin. This device can be rotated to measure an electron’s spin at any angle from 0 to 360 degrees, which allows for some interesting experiments. For example, suppose we have an electron passing through Stern-Gerlach devices which we can label #1 and #2. Suppose device #1 is titled at a 90 degree angle from device #2. If an electron measures “spin up” at Stern-Gerlach device #1, there is a 50% chance that this same electron will be measured as “spin up” by device #2 also. If we let Pr() symbolize “the probability that” we can let Pr(the measured spins are the same) represent “The probability that measured the spins are the same.” Given some angle A, the equation for how this works is given by the following:
Pr(the spins are the same) = cos2(0.5 × A)
That means we take cos(0.5 × A) and square it. In our case, the angle was 90°, and cos²(0.5 × 90°) = cos²(45°) = 0.5, and thus there’s a 50% chance that the measured spins will be the same between device #1 and device #2.

It’s possible to have a pair of electrons in a strongly correlated state so that when they fly apart from each other these two electrons have the interesting property such that if both have their first spin measurement be along the same axis, the spins will always be opposite of each other; these two electrons whose spins are correlated are “entangled.” If entangled electrons #1 and #2 have their spins measured on the same axis, then if #1 is measured as spin up then #2 will be measured as spin down. How is it that these spins are always opposite whenever the two electrons are measured on the same axis? One could say that the entangled electrons are in such a state that their spin measurements are in a sense “built-in” in advance. To help illustrate this I’ll symbolize three angles the Stern–Gerlach device can be used: let F represent an angle of 0°, L represent 120° (slanting upwards somewhat to the left), and R represent 240° (slanting upwards somewhat to the right). Let + represent spin up and – represent spin down. So an electron having a F+L–R– property means that if the electron were measured on the F axis the result would be spin up (+) and if the electron were measured on the R axis the result would be spin down (–).

Earlier I mentioned the probability of finding an electron’s spin being the same with a certain equation. Is this real randomness where identical physical conditions can produce different outcomes, or are there “hidden variables” that only make it look like it’s random? This is a matter of some controversy in philosophy of quantum mechanics that I won’t resolve here, but I can at least describe two (among many) different interpretations of quantum mechanics. The “hidden variable” theory says that the randomness is only apparent and that each electron has some property that deterministically determines the spin result before it enters the Stern-Gerlach device. For example, if an electron has a F+ property then it has the property that determines a “spin up” result if measured on the F axis. One version of hidden variable theory says that locality is true—which in this case means that a measurement on electron #1 has no physical effect on electron #2 (the denial of locality is called nonlocality). The conjunction of locality and hidden variable theory is called local hidden variable theory. So how does local hidden variable theory account for the electron spins always being opposite when the entangled electrons are measured on the same axis? By having the potential opposite spin results “built-in” in advance. So if electron #1 is F+L+R+, then electron #2 would be F–L–R– to guarantee that if the electrons were measured on the same axis the spin measurement results would be opposite. In contrast, something called the Copenhagen interpretation says (among other things) that the randomness is real and that when the two entangled electrons fly apart they don’t have a definite spin until measured, and the measurement randomly determines the spin of the electron. The Copenhagen interpretation also denies locality, unlike local hidden variable theory.

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Saturday, June 2, 2012

Introductory Logic, Part 2

Home  >  Philosophy  >  Logic

This is part 2 of my series on logic and critical thinking.
  1. Introductory Logic, Part 1—Introducing both logic in general (such as the difference between a deductive and inductive argument) and propositional logic in particular
  2. Introductory Logic, Part 2—More propositional logic
  3. Introductory Logic, Part 3—A defense of the material conditional

Bait

To bait both atheists and theists into reading this article on logic (though I hope you already have an interest in learning logic), consider this argument from evil:
  1. If God exists, then gratuitous evil does not exist.
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
A theist cannot deny both premises on pain of irrationality (it is logically impossible that both premises are false) and I’ll show how one can prove it via symbolic logic in this article. Similarly, consider this moral argument:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
An atheist cannot deny both premises on pain of irrationality (it is logically impossible that both premises are false) and I’ll show how one can prove it via symbolic logic in this article.

Review

In my previous entry I talked about the connectives and various rules of logic. To recap the connectives:

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.”
↔ (biconditional)p ↔ qp, if and only if qThis means the same thing as “p → q and q → p.”
¬ (negation)¬pNot-pThe negation of P is ¬P, and ¬P means “not-P” or “P is false.”


To recap the rules of inference:

modus ponens
 
In English In Symbolic Logic
If p then q
p

Therefore, q
p → q
p

∴ q
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
simplification
 
In English In Symbolic Logic
p and q

Therefore, p
p ∧ q

∴ p
p and q

Therefore, q
p ∧ q

∴ q


conjunction
 
p
q

∴ p ∧ q
constructive dilemma
 
(p → q) ∧ (r → s)
p ∨ r

∴ q ∨ s


hypothetical syllogism
 
p → q
q → r

∴ p → r
addition
 
p

∴ p ∨ q
absorption
 
p → q

∴ p → (p ∧ q)


Some equivalences:

equivalencename of equivalence
 
p ⇔ ¬¬pdouble negation
 
p → q ⇔ ¬q → ¬ptransposition (also called contraposition)
 
p → q ⇔ ¬p ∨ qmaterial implication
 
p ↔ q ⇔ (p → q) ∧ (q → p) biconditional equivalence
 
¬(p ∧ q) ⇔ ¬p ∨ ¬qDe Morgan’s laws
¬(p ∨ q) ⇔ ¬p ∧ ¬q
 
p ⇔ p ∧ p idempotence
p ⇔ p ∨ p


And some more equivalences:

equivalencename of equivalence
 
p ∧ q ⇔ q ∧ pcommutation
p ∨ q ⇔ q ∨ p
 
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ rassociation
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
 
p → (q → r) ⇔ (p ∧ q) → rexportation
 
p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r)distribution
p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r)


Next I’ll introduce some cool new rules of inference: conditional proof and indirect proof (also known as proof by contradiction, among other names). I’ll also show how to use symbolic logic to prove a statement without using any premises.

Conditional Proofs

Recall that the conditional is symbolized as p → q where p is called the antecedent and q is called the consequent. The conditional proof aims to prove that a conditional is true, with the antecedent of the conditional being the conditional proof assumption which is often used to help show that if the antecedent is true then the consequent is true also. The structure of a conditional proof takes the following form below:

conditional proof
 
a) p conditional proof assumption
b)
 ...
 q
c) p → q a-b, conditional proof


For example, suppose we want to prove A → (B ∧ C) from premises 1 and 2 below:
  1. A → B
  2. A → C

  1. A conditional proof assumption
    1. B 1, 3, modus ponens
    2. C 2, 3, modus ponens
    3. B ∧ C 4, 5, conjunction
  1. A → (B ∧ C) 3-6, conditional proof
Notice that the validity of a conditional proof does not rely on the conditional proof assumption actually being true; rather it relies on the fact that if it is true then it properly leads to the consequent. Nothing in the proof above, for example, relies an A actually being true, which brings us to this caveat: once the conditional proof has been ended, no lines inside it can be used again. Lines 3-6 are inaccessible for lines 7 onward. For example, note the mistakes below:
  1. A → B
  2. A → C

  1. A conditional proof assumption
    1. B 1, 3, modus ponens
    2. C 2, 3, modus ponens
    3. B ∧ C 4, 5, conjunction
  1. A → (B ∧ C) 3-6, conditional proof
  2. (A → B) ∧ A 1, 3, conjunctionMistake!
  3. (A → B) ∧ C 1, 5, conjunctionMistake!
It’s possible for a symbolic logic proof to have many conditional proofs, even conditional proofs inside other conditional proofs, provided they obey the accessibility restriction mentioned earlier (e.g. once an “inner” conditional proof is ended, the lines of that inner conditional proof can’t be accessed by the “outer” conditional proof). Sometimes one handy rule when doing conditional proofs is reiteration.

reiteration
 
p

∴ p


At first blush reiteration might not seem like a very handy rule, but its use comes from being able to put one proposition from the “outside” to the “inside,” as illustrated below
  1. B

  1. A conditional proof assumption
    1. B 1 reiteration
  1. A → B 2-3, conditional proof
Of course, the rule about accessibility applies with reiteration. Note for example the mistakes below:
  1. A → B
  2. A → C

  1. A conditional proof assumption
    1. B 1, 3, modus ponens
    2. C 2, 3, modus ponens
    3. B ∧ C 4, 5, conjunction
  1. A → (B ∧ C) 3-6, conditional proof
  2. A 3, reiterationMistake!
  3. C 5, reiterationMistake!

Indirect Proof

Another proof method 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
Suppose we wanted to prove B from premises 1-3 below:
  1. ¬B → ¬D
  2. ¬B → E
  3. [¬B → (¬D ∧ E)] → D

  1. ¬B indirect proof assumption
    1. ¬B conditional proof assumption
      1. ¬D 1, 5, modus ponens
      2. E 2, 5, modus ponens
      3. ¬D ∧ E 6, 7, conjunction
    1. ¬B → (¬D ∧ E) 5-8, conditional proof
    2. D 3, 9 modus ponens
    3. ¬D 1, 4, modus ponens
    4. D ∧ ¬D 10, 11 conjunction
  1. B 4-12, indirect proof
As the above example illustrates, you can have conditional proofs inside indirect proofs; you can also have indirect proofs inside conditional proofs. The same sort of accessibility restrictions go for indirect proofs as for conditional proofs. To illustrate, note the following mistakes:
  1. ¬B → ¬D
  2. ¬B → E
  3. [¬B → (¬D ∧ E)] → D

  1. ¬B indirect proof assumption
    1. ¬B conditional proof assumption
      1. ¬D 1, 5, modus ponens
      2. E 2, 5, modus ponens
      3. ¬D ∧ E 6, 7, conjunction
    1. ¬B → (¬D ∧ E) 5-8, conditional proof
    2. D 3, 9 modus ponens
    3. ¬D 1, 5, modus ponensMistake! 5 is inaccessible!
    4. E 7, reiterationMistake!
    5. D ∧ ¬D 10, 11 conjunction
  1. B 4-13, indirect proof
  2. D ∨ Z 10, additionMistake!
The next section has some interesting things to say about modus ponens and modus tollens.

Demonstrating Inconsistent Premises

One proves that the premises are inconsistent with each other by deriving a logical contradiction. This has an interesting application for modus ponens and modus tollens. First, consider the representative modus ponens argument below:
  1. P → Q
  2. P

  1. Q 1, 2, modus ponens
It turns out that it’s logically impossible for both premises of a modus ponens argument to be false. How can we prove it? Simply negate both premises and derive a logical contradiction.
  1. ¬(P → Q)
  2. ¬P

  1. ¬Q conditional proof assumption
    1. ¬P 2, reiteration
  1. ¬Q → ¬P 3-4, conditional proof
  2. P → Q 5, transposition
  3. ¬(P → Q) ∧ (P → Q) 1, 6, conjunction
It is likewise impossible for both premises of a modus tollens argument to be false. A representative modus tollens argument:
  1. P → Q
  2. ¬Q

  1. ¬P 1, 2, modus tollens
As before, we negate both premises and derive a logical contradiction:
  1. ¬(P → Q)
  2. ¬¬Q

  1. P conditional proof assumption
    1. Q 2, double negation
  1. P → Q 3-4, conditional proof
  2. ¬(P → Q) ∧ (P → Q) 1, 5, conjunction
So the only way to deny both premises of a modus ponens or modus tollens argument is to reject a rule of logic, and that is (presumably) much too high a price to pay. Next I’ll talk about how to prove things in symbolic logic without using premises.

Symbolic Logic Proofs Without Premises

So how do you prove things in propositional logic without premises? Easy: you can use conditional proofs and indirect proofs. For example, suppose we want to prove (A ∧ ¬A) → (K ∧ I):
  1. A ∧ ¬A conditional proof assumption
    1. ¬A 1, simplification
    2. A 1, simplification
    3. A ∨ (K ∧ I) 3 addition
    4. K ∧ I 2, 4, disjunctive syllogism
  1. (A ∧ ¬A) → (K ∧ I) 1-5, conditional proof
As you might suspect, anything follows from a contradiction. Using the sort of strategy above, we could have had A ∧ ¬A imply anything we wanted to thanks to rules of logic like simplification, addition, and disjunctive syllogism.

In propositional logic, proofs without premises are called theorems. As you might guess, not every statement is a theorem, but the famous law of logic the law of noncontradiction is. A famous rule of logic called the law of noncontradiction says that for any proposition p, it is impossible for p and not-p to be true at the same time and in the same context. In symbolic logic, the law of noncontradiction is expressed as ¬(p ∧¬p). Here’s how one can prove it:
  1. P ∧¬P indirect proof assumption
    1. P ∧¬P 1, reiteration
  1. ¬(P ∧ ¬P) 1-2, indirect proof
OK, so maybe that was too easy. Another rule of logic called the law of excluded middle says that for any proposition p, either p or not-p is true. Expressed in symbolic logic, the law of excluded middle is p ∨ ¬p. One way to prove the law is the theorem below:
  1. ¬(P ∨¬P) indirect proof assumption
    1. ¬P ∧¬¬P 1, De Morgan’s laws
  1. P ∨ ¬P 1-2, indirect proof
Still not much of a challenge, but oh well. At least we have the proof.

No, I Won’t Bait and Switch

First, for the atheist:
  1. If God exists, then gratuitous evil does not exist.
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
Let G be “God exists” and E be “Gratuitous evil exists.” Then using symbolic logic the premises are:
  1. G → ¬E
  2. E
If we want to show that both premises being false is impossible, we negate both premises and derive a logical contradiction.
  1. ¬(G → ¬E)
  2. ¬E

  1. G conditional proof assumption
    1. ¬E2, reiteration
  1. G → ¬E 3-4, conditional proof
  2. ¬(G → ¬E) ∧ (G → ¬E) 1, 5, conjunction
So at least one of the premises must be true. Of course, the sword of logic cuts both ways. The moral argument:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
Using G to symbolize “God exists” and O as “Objective morality exists” the premises are these:
  1. ¬G → ¬M
  2. M
As before, if we want to show that both premises being false is impossible, we negate both premises and derive a logical contradiction.
  1. ¬(¬G → ¬M)
  2. ¬M

  1. ¬G conditional proof assumption
    1. ¬M ∧ ¬G2, 3, conjunction
    2. ¬M 4, simplification
  1. ¬G → ¬M 3-5, conditional proof
  2. ¬(¬G → ¬M) ∧ (¬G → ¬M) 1, 5, conjunction
The rational atheist would have to agree that at least one of the premises is true.

Symbolic Logic Summary

To summarize the rules of logic learned in this entry:

conditional proof
 
a) p conditional proof assumption
b)
 ...
 q
c) p → q a-b, conditional proof

reiteration
 
p

∴ p
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