Friday, March 27, 2015

Arrow's Impossibility Theorem: Summary and Explanation

Overview
A central idea in democracy is that in voting for what the community should do, each individual gets to express their preference for one thing over another. If individual preferences don't all align then sometimes there will be a puzzle about what the community should do--assuming policy decisions are ideally supposed to represent the aggregation of individual preferences. Let me make explicit the assumption in that last sentence.  The assumption is that a community's ranking of options should be derived only from the individual rankings.

Arrow's theorem shows that, given 3 or more policy alternatives, it is sometimes impossible to sum up individual preferences to derive a community-wide ranking without violating at least 1 of 4 minimal constraints. Another way to express this is to say that sometimes you can't move from individual rankings of preferences to a community ranking of preferences without violating 1 of 4 ideal criteria for a community ranking.


Before getting all fancy, there's a simpler illustration that demonstrates that there will be cases where it's impossible to derive a community ranking of alternatives from individual rankings.


Let's suppose there's a community of 3 people. They have to vote on how to rank 3 alternative. It can be a ranking of anything: people to hold office, priorities for funding certain programs, and so on. To keep things simple we'll just call the three alternatives A, B, and C. Here are the individual rankings:


Person 1: A>B>C
Person 2: B>C>A
Person 3: C>A>B


Notice there's a problem right away.  As you might expect, people rank things differently. Democratic theory tells us that in order to figure out how the community ought to rank the alternatives (for policy purposes) we need to aggregate the individual rankings. Let's do that and see what happens.


Doh! We can't because there's a three way tie for the first priority, a three way tie for the second priority, and a three way tie for the third priority. So, now what? Do we reject democracy and do what Plato told us to do 2, 400 years ago: Put the philosophers in charge? I say "yes" but some of you might not agree. For those that disagree there's still hope...


The Condorcet Method 
Let's take each set of alternatives as pairs and compare them that way. If a majority prefer one alternative over another, that might help give us a social ranking. That is, we will apply a simple majority rule to pairs of alternatives


Look at pair (A, B). Person 1 and Person 3 prefer A to B. Even though Person 2 prefers B to A, we say majority rules in terms of generating a society-wide ranking. Perfect. Now we have part of our social ranking:


A>B


Let's now turn to the relationship between B and C. Person 1 and 2 both rank B ahead of C, majority wins so our social policy should reflect this. We add it to our social ranking:


B>C


Now we can do a little basic logic to figure out the rest. If A is preferred to B and B is preferred to C then it follows (by transitivity) that A is preferred to C. That's just elementary school logic! Amiright? The last pair in our social ranking should look like this:


A>C


Now hold on a tick. Let's look at the individual preference orderings again because that's what's supposed to generate the social ordering. Take a look and see if you notice something funky. 


I'll wait....


You should have noticed that Person 2 and Person 3 prefer C to A. Our majority rule tells us that our social ranking should then be 


C>A


Well, that's a problem. 


Ok, fine. Let's just ignore the illogicality of the social ranking and stick to our majority rule.


We now have C>A, A>B and logically it must follow that C>B. But look back to the individual preferences. They tell us that B>C. To quote the great philosopher Britney Spears, "Oops, we did it again...." (and we could continue the process ad infinitum


So, what's the point of all this? The point is that even with a very simple voting rule there will be occasions where you can't generate a social ordering of preferences that conforms a basic norm of rationality (i.e., transitivity) if that social ordering is derived from individual orderings alone...which is kinda at the heart of democracy.  Democracy can't do the very thing that it's supposed to do! Therefore, Plato was right, philosophers should be put in charge immediately. Report to your reeducation camps. Ayn Rand book-burning party scheduled at 11. 


The end.



Ok, I lied. That's not the end. But things don't get better, they get worse.  But we're going to need to get a bit fancy to show why...


Vocabulary:

A preference is a ranking of one outcome/policy/thing over another. A preference in economics is a technical term. It doesn't mean "liking". It is always comparative: E.g., Bob prefers x to y. When one thing is preferred to another it is said that x gives Bob more utility than y (i.e., it has more value); so preferences are represented in terms of utility.

A utility function is a ranking of a set of preferences in terms of utility. Suppose Bob can choose from 3 kinds of fruit: apples, pears, and bananas. Suppose Bob prefers apples to bananas and bananas to pears. His utility function will rank apples above bananas and bananas above pears. Notice that transitivity is implied. That is, x>y and y>z, then x>z.

An individual utility function is simply the utility function of a single person. That is to say, it's how a particular individual ranks all the possible options. Bob's ranking of {apples, bananas, and pears} is an example of an individual utility function. It one particular person's (i.e., Bob's) ranking of available options.

A social or welfare utility function is the utility function you get when you aggregate all the individuals in a particular community. Basically, take all the individual rankings of options and add them all up. That's the social utility function.

Minimum Criteria for Establishing a Social Welfare Function (in a Democratic Society)
Here comes the tricky part. Who needs the Quickee Mart? There are 4 (actually 5) minimal conditions that we don't want to violate is coming up with a community wide ranking of preferences.

Condition T: This one's the key.
Condition T is transitivity. It's not often directly stated but implied by the conditions of rationality. As mentioned above, if someone prefers x to y and y to z then transitivity requires that they prefer x to z.

Constrain is said to be a rational constraint on generating a social utility function. The remaining four are said to be normative constraints; i.e., they are values that constrain how to generate a social utility function.

Condition U: Let's go to the zoo.
Condition U is unrestricted domain. This means that in a democratic society, at least pre-constitutionally, no preference orderings can be precluded from the social utility function. For example, suppose a set of 3 things {apples, guns, the bible}. One person ranks them {Bible, guns, apples} and another person ranks them {guns, Bible, apple}. There are still ways to rank these options by putting apples ahead of guns and the Bible. Any ranking that puts apples ahead of guns and the Bible are clearly irrational and not socially justified, however, unrestricted domain says that you can't exclude these rankings from a social welfare function no matter how crazy they are.

Condition P: Learn it with me!

Condition P is weak Pareto principle. Without getting too fancy, the weak Pareto principle is simply that if every individual in a society prefers guns to apples (i.e., their individual utility functions rank guns above apples) then the social utility function must also rank guns above apples.

Condition I: Buzz like a fly.
Condition I is independence of irrelevant alternatives. Suppose all individuals prefer guns to bananas. So, condition P obtains. However, Bob's ranking is {guns, Bible, bananas} and Mary's is {Bible, guns, bananas}. The fact that Bob ranks guns ahead of Bible and Mary ranks Bible ahead of guns should have no bearing on the fact that when we aggregate their preferences (i.e., construct the social utility function) guns must be ahead of bananas. The alternative "bible" is irrelevant to constructing the social utility function. The only thing that matters is whether each individual ranks guns higher than bananas. If this is the case, then the social utility function must put guns ahead of bananas.

Condition D: It's so cooooold in the D....
Condition D is non-dictatorship. Consider all the possible things that individuals could have preferences over that will be relevant to social policy. Think of all the different things that get voted on. Simply put, non-dictatorship stipulates that there's no individual who gets their way for everything. In other words, there can be no individual who's individual utility function (i.e., ranking of all the possible policy options) matches the social utility function (i.e., the aggregate of all the individual utility functions). In short, there is no person that can always get their way because this would imply dictatorial power and that's unamerican! (and undemocratic). The social utility function needs to represent (by definition) the preference rankings of various individuals and so if it represents the ranking of a single individual it can't be the ranking of all individuals.

Back to Arrow's Theorem: The social utility function is defined as the aggregate of all individual utility functions. Arrow's theorem says there is no way to construct a social utility function without violating one of the 4 (5 including transitivity) conditions. Why should we care? Because if we conceive of democracy of taking the aggregate of each individual's preference ranking and enacting policy according to that aggregate ranking, we can't get the ranking without violating one of the 4 criteria. In other words, to construct a community-wide ranking of preferences, we will have to violate either intransitivity (i.e., although everyone ranks x higher than y and y higher than z, we will rank z higher than x); unrestricted domain (i.e., we will have to exclude some preference orderings as possibilities. E.g., "You aren't allowed to prefer z to y."); weak Pareto (i.e., we will have to rank y higher than x even though everyone individually ranks x higher than y); independence of irrelevant alternative (i.e., even though everyone ranks x higher than y, if some people rank z higher than y then y will be ranked higher than x); non-dictatorship (i.e., someone will always get their way: their preferences ranking will become society's preference ranking).

I'm going to do two different proofs for Arrow's Theorem. The first is an informal proof. The second is a formal proof. 

Strategy:

The first part of the proof is to prove a conditional: If an individual is almost decisive over one pair of options for f, he will be decisive over all pairs and therefore be a dictator. We apply the minimal conditions for constructing the social welfare function (f) from individual functions and show that this can't be done without violating D. The second part of the proof is to show that there is such an individual.

We're gonna need some formal definitions: Using variables to stand in for terms will help us cut down on awkward phrases...

Notes/defining variables: V=a subset of individuals in the community. xPy= x is preferred to y (in the social utility function). yPx=y is preferred to x (in the social utility function. i=individual. xPiy= some individual prefers x to y; i.e., this is a preference ranking for an individual utility function. f=social utility function.


Almost Decisive:  A set of individuals V is almost decisive for some x against some y if, whenever xPiy for every i in V and yPix for every i outside of V, x is socially preferred to y (xPy). In English this means that a subgroup (identified with the variable V) of the community is almost decisive if V's preference ranking for x over y sets the social utility function in terms of ordering x and y even if other members of the community (not V) rank y ahead of x. This is a very common scenario. Usually, we have a rule that says if there isn't agreement over a ranking then the majority wins. We can think of V as the majority in a community.


Decisive: A set of individuals V is decisive for some x against some y if, whenever xPiy for every i in V, xPy.  In English: If there's a subgroup in the community and every member of that subgroup ranks x ahead of y and the social utility function adopts this ranking (xPy) then V is decisive. Notice the difference with 'almost decisive'. With almost decisive V's preference ranking sets the social utility function if V's conflict with non-V members of the community. In Decisive, it doesn't matter whether V's ordering conflicts or not. Whatever V's ranking of x and y, that's what the social ordering of x and y will be too.

Informal Proof
In the informal proof we're going to prove the contagion principle: The contagion property explains why all four conditions can't be met simultaneously. In short, if one individual's preferences over one pair of alternatives are almost decisive (i.e., generate a ranking in the social utility function) this decisiveness spreads to all other pairs of alternatives.  

For example: suppose there are 2 people, Abe and Bob. Abe prefers guns to bananas but Bob prefers bananas to guns. If the social utility function becomes guns over bananas (maybe there's a voting rule that says in case of tie we go according to alphabetical order) then we say that Abe's preferences are almost decisive. I.e., his preferences set the social utility function. The contagion property says that once one individual's preferences become almost decisive (i.e., are the ones that set the social utility function) then that decisiveness will spread to other pairs of preferences. We can see how this result occurs by doing an informal proof.

Up until now we've talked about a subgroup (V) in the community being decisive or almost decisive. We now introduce a particular individual, J. He prefers x to y (xPjy). Suppose J's ranking of x and y is almost decisive. To symbolize this we will use this: D(x,y). To symbolize that J's ranking is decisive we'll use this: D*(x,y). Notice that if a person or group's ranking is decisive, it's also almost decisive

Why? This is almost trivial but it's because if a particular individual's ordering of preferences over 2 options set the ordering in the social utility function then it's also true that that individual's preference over 2 options set the ordering of the social utility function even if other members of the community rank those same options differently. In short, D*(x,y) implies D(x,y). Otherwise stated, if there's a rule that your preferences set the social preferences then your preferences will also set the social preferences even if other individuals don't have the same preferences as you. Seems trivial but it'll matter later.

Lemma  1
Prove that if there is some individual J who is almost decisive for some ordered pair of alternatives (x,y), then that individual's preferences will be decisive for all pairs in f.  (If one individual's ordering preferences set the entire social utility function then that person is a dictator and Condition D has been violated.)  In other words, if f satisfies U, P, and I, it can't also satisfy D. 

Proof:
Step 1: Assume that some individual J is almost decisive for some x against some alternative y. (J prefers x to y and this preference sets f even if other members of the community rank y ahead of x.) Assume also there is a third alternative z. Let's also refer to the collection of all individual utility functions other than individual J's as i. (i=the utility functions for all members of the community except J).

According to U (unrestricted domain) we can order x, y, z in any way possible. Individuals are allowed to order them however they want even if preferring z to x seems loco. So, let's suppose the following preference orderings:

xPjy,  yPjz  and
yPix,  yPiz

Notice that transitivity imposes a complete ordering on J's preferences. If he prefers x to y and y to z then he must also prefer x to z. In the case of i, however, things are different. Transitivity doesn't impose any ordering between x and z. All we know is that y is preferred to both x and z but we don't know how x and z are ranked against each other.

We've stipulated that J's ranking of x and y is almost decisive--meaning that even if other members of the community rank them differently, f is set by J's ranking. Since J ranks x ahead of y, f will also rank x ahead of y. So, xPy for f.

Next we notice that both J and i rank y ahead of z. This activates condition P (weak Pareto) which stipulates that if every individual in the community ranks y ahead of z then  f also ranks y ahead of z. (Recall that i=all individuals except J). So, now f has two rankings xPy and yPz.

Our social utility function ranks x ahead of y and y ahead of z. Condition T (transitivity) now kicks in. If x>y and y>z transitivity demands that x>z.

Recap so far: We started out by saying that J's preference for x over y is almost decisive and so part of f is xPy. Applying condition U told us that f must take into account all possible orderings of x, y, and z. Applying condition P added yPz to f.  So now f contains xPy and yPz. Transitivity completes the ordering for f by setting xPz.

Why should we care? Notice that for all individuals except J the relationship between x and z is still open. Some might prefer x to z while others might prefer z to x. However, because we applied conditions U, P, and T, the social utility function (f) reflects J's complete ordering of the three variables. This shows that if one individual is almost decisive in setting the social ordering (f) for a single pair, this decisiveness spreads to all other rankings in f (i.e., the contagion principle is instantiated).  Well, not quite. We still have to finish the proof...

We said that applying conditions T, U, P, I would lead to a violation of Condition D. That is, you can't simultaneously uphold condition T, U, P, and I without undercutting D. We still need to apply condition I to finish the proof.  So, let's do eet!

Recall that condition I (independence of irrelevant alternatives) stipulates that if everyone ranks some option x above some option y then it doesn't matter to the social utility function (f) where they rank a third option so long as x is always ahead of y. So, maybe you have 3 people with the following ordered sets A {x, y, z}, B {z, x, y}, C {x, z, y}. In each of these individual utility functions x is ahead of y and so when we construct the social utility function, the ranking of z should have no bearing on the ordinal ranking of x and y; i.e., f=xPy.

Also, (and this is the important part for the proof) f can only be informed by individual rankings. Notice that for i we stipulated yPix and yPiz. The relationship between x and z was left undetermined. Maybe people in i are indifferent to x and z. The point is that the ordering of x and z for the social utility function can only come from individual orderings. By definition, it is merely a representation of those orderings and so if individuals don't have an ordering of two options (as in i) then those individuals aren't relevant to the social ordering.  There's nothing for the social ordering to represent!

What follows from this is that the social ordering of x to z is purely a consequence of J's orderings. That is, since i doesn't have an ordering of x and z it can't be represented in f and so the actual ordering in f is a consequence of only one individual's ordering (i.e., J's). In short, by applying T, U, P, and I we end up with a violation of D. The contagion principle effectively makes J a dictator. The social utility function and J's individual utility function are one and the same. J gets everything he wants.

Since J sets the social ranking of x in respect to z then he is decisive (not just 'almost decisive'). In short, if J is almost decisive for x and y (that is, there is some voting rule that lets J's preference for one set of options be represented in f) then J's preferences will be decisive for some other set of options (x and z).

Step 2:
Let's again assume that there is some voting rule that makes it so J is almost decisive for x and y (xPjy).

Let's further assume some new preference orderings

zPjx,  xPjy   and
zPix,  yPix      (recall that i=everyone except J)

Since J is almost decisive for the ordering of x and y and xPjy the social utility function is ordered accordingly, xPy.

Condition P tells us that f must also order zPx because both i and J order z ahead of x. So far, our f looks like this: zPx and xPy. Transitivity now kicks in (if z>x and x>y then z>y) so the social ordering of z and y will be zPy.

We now apply Condition I. The only thing relevant to a social ordering are individual ordering. Since i doesn't have a ranking of z and y it can't factor into how f ranks them. It follows that the ordering of z ahead of y in f is a consequence of J's preferences alone. And so J is a dictator in terms of setting the social utility function.

J started out with almost decisive power to set a pair ordering in f. In this case it was xPy. To order the unsettled options in f we applied conditions T, U, P, and I.  The result was that where there wasn't total unanimity, the ordering (zPy) was set based on J's ordering alone. No other individual had an influence on f. And so condition D (non-dictatorship) was violated. It also follows that if one individual is almost decisive for setting one pair ordering in f then they will be decisive for the entire ordinal ranking of f.

Part 2
If you're like me when I first saw the first part of the proof, I was like, "Ok, sure maybe if someone were almost decisive over one pair they'd end up being decisive over all pairs, but how likely is it that such a person exists?" Well, as it happens, this is the second part of the proof. You can prove that such an individual must exist!  

Before doing that, however, I'm going to present the formal version of the 1st part of the proof. Feel free to skip it and go straight to the last part of the proof; that there is indeed a dictator. 


Formal Proof:
Symbolization
1. x>y = x is preferred to y (xPy). 
2. x<y = y is preferred to x (yPx).
3. J= a particular individual.
4. i= everyone in the community except J.
5. xPy=x is preferred to y in the social utility function (f); xPjy= J prefers x to y; xPiy=everyone except J preferes x to y.


Step 1
J: x>y>z
i: x<y>z
Therefore
f:  If xPy (because J is almost decisive for x,y), yPz (because condition P) then xPz (because condition T and I).
Therefore if D(x,y) (i.e., if someone is almost decisive for x, y) then D*(x,z) (then they are decisive for x,z) then D(x,z) (because D* implies D).


Step 2
J: z>x>y
i: z>x<y
Therefore, 
f: zPx (from condition P), xPy (J is almost decisive for x,y) implies zPy (Condition T and I).
So, if D(x,y) then D*(z,y) then D(z,y).


Step 3
J: y>x>z
i: y>x<z
Therefore,
f: yPx, xPz implies yPz
So, if D(x,z) then D*(y,z) then D(y,z).


Step 4
J: y>z>x
i: y<z>x
Therefore,
f: yPz, zPx implies yPx
So, if D(y,z) then D*(y,x) then D(y,x).


Step 5
J: z>y>x
i: z>y<x
Therefore,
f: zPy, yPx implies zPx
So, if D(y,x) then D*(z,x) then D(z,x).


Step 6
J: x>z>y
i: x<z>y
Therefore,
f: xPz, zPy implies xPy
So, if D(x,z) then D*(x,y) then D(x,y).


Part 2: Finding the Dictator
Part 1 of the proof was to prove the hypothetical: If an individual is almost decisive over one pair of options he will be decisive over all pairs (via contagion). In Part 2 we prove that there is such an individual. To do this we'll need to show that there really is an individual that is almost decisive over one pair.


We know that for any set of ordered pairs in f there is a decisive set of individuals. We also know that if a group is decisive they are also almost decisive over that pair. In short, the fact that pairs are ordered in f implies that some voting rule made some group of individuals decisive for that pair. We scan all the ordered pairs in f and pick the one that has the smallest decisive set of individuals. Call these individuals V and let's assume they're decisive for (x,y).


If V contains just one individual, we've found the dictator. If V contains more than one let's divide it into two groups with the following preference orderings (recall everyone in V ranks x>y):
V1 (only one person): x>y>z
V2 (everyone who is decisive for xPy except the person in V1): z>x>y
and
O is everyone else in the community: y>z>x


By definition, f = xPy because V is almost decisive for (x,y). Where do z fit into f? Let's first look at the relationship between y and z. Can f = zPy? We've already defined V as the smallest decisive set and since V2 is a subset of V it can't be decisive. Since f can't be zPy, f must be yPz. Now, f = xPy and yPz. Transitivity implies further that f = xPz. In short, f = x>y>z. If that ordering looks familiar it's because it belongs to V1. Notice also that neither V2 nor O rank x ahead of z. Only the individual V1 does. So, despite the fact that everyone one except V1 ranks z ahead of x, f = xPz. We've found our dictator.



From the first part of the proof we proved that if an individual is almost decisive over one pair of options by contagion he will be decisive (and almost decisive) over all other pairs in f. We've proven that there is such an individual and so the proof is complete: You cannot achieve a social ordering of alternatives by simply aggregating all individual rankings without violating one of the 4 ideal conditions! (5 if you include transitivity.   






Friday, January 30, 2015

Review and Summary of "Ethics and Observation" by Harmon

Introduction
The major debate in ethics is whether there are objective moral facts. There are a variety of defenses and objections to either position. Those who say there are objective moral facts are called "realists" while those who deny realism are called anti-realists or nihilists (there are actually more positions such as constructivists and non-cognitivists but let's not worry about that).

A popular strategy realists use is to say that moral reasoning is analogous to either scientific reasoning or mathematical reasoning. In super condensed form, the former strategy plays on the idea that scientific reasoning is a recursive spiraling toward the truth. There's a continuous interplay between hypothesis/theory and observation, the one impacting the other. Hypotheses and theories are tested, confirmed or rejected based on the best available observations. Similarly, hypothesis and theories impact how we interpret our observations. 

Moral reasoning is no different. We begin "in the middle" with both a moral theory and observations. Particular observations (judgments) influence our moral theories and theories influence how we interpret our observations. In both domains, there's a back and forth between theory and observation and the trajectory or both enterprises aims at truth.

A similar but slightly different argument applies to the analogy between mathematical and moral reasoning. We just as we can't directly observe mathematical facts, we can't directly observe moral facts. We reason our way to them. Also, both moral and mathematical facts don't have a physical existence yet we are sensitive to them: they inform our actions and our view of the world.

Ok, so those are very simplified versions of the arguments. In "Ethics and Observation," Gilbert Harmon challenges both analogies, although he focuses mostly on the first. Harmon's conclusion (at least not in this article) isn't that there's not such thing as moral truth, rather that the analogy between scientific observation and moral observation doesn't hold.

Let's check out his argoomints...

Part 1: The Basic Issue
Can moral principles be tested and confirmed the same way scientific principles can? On the surface it seems they can.  For example, I want to know if the principle "whenever it's possible, you should save 5 lives rather than 1."

How do we test this? Well, we can do a thought experiment and see what our verdict would be. Harmon gives this example:
Suppose you're a doctor in the emergency section.  6 patients are brought in. All six are in danger of dying but one is much worse off than the others. You can just barely save that person if you devote all of your resources to him and let the others die. Or, you can save the other 5 if you are willing to ignore the most seriously injured person.
What do? It seems like this case confirms the moral principle in question.  However, this being philosophy, there will be counter-examples a-plenty.
You have 5 patients in the hospital who are dying, each in need of separate organ. ONe needs a kidney, another a lung, a third a heart, and so on. You can save all five if you take a single healthy person and remove his heart, lungs, kidneys, etc... to distribute to these five patients. Just such a healthy person walks into the hospital for routine tests. His test results confirm he'd be a match as an organ donor for all 5 dying patients. If you do nothing he'll survive and 5 will die. If you apply our principle of action, that you ought to always save 5 instead of one whenever you can, then, well, you'll save 5 and only lose one life...
What do? It seems like we've tested the moral principle and the test disconfirms it. The moral principle is false (or needs to be modified).

So, is this kind of testing the same as is done in the scientific realm? It doesn't look like it. Scienticians test their hypotheses and theories against the real world not just in their "imagination".  It doesn't seem like we can perceive the rightness or wrongness of an act the way you might perceive the color, shape, and mass of an object.

How do scienticians know bluebirds are blue? Because they can look out into the world and see the blueness of the bird in question. But how do we know if an act is wrong? It doesn't seem like you can point to the wrongness or rightness the way you can point to an object's physical properties. And so it seems as though there's a way in which scientific and moral observation are different.

How do we make moral judgments anyway? Harmon gives the following case. You're walking down the street and you happen upon two teenagers (damn teenagers!) pouring gasoline on a cat then ignite it.  It doesn't seem like you perform any reasoning process. It's not like you go. Hmmm, let me see, there's a moral principle that
P1. causing unnecessary suffering of innocent creatures is wrong, and 
P2. these youths are engaged in an instance of causing unnecessary suffering to an innocent creature,
C. therefore, what they are doing is wrong.
What actually happens is you move directly to the moral judgment. You just see it's wrong. The process is a direct observation that the act is wrong. No  reasoning required.

Here's the issue: Are you perceiving something objective or is your reaction simply a product of your particular psychology? That is, if you'd been around when people tortured cats for fun, you might not have made the same judgment from your observation. Likely, your judgment is a reflection of facts about you, not about the act. More on this in a bit...

Part 2: Theory-Laden Observation
Let's get clear on what is meant by observation. In philosophy of science, it's widely recognized that there are no "raw" observations. All observations are "theory-laden." This means that, implicitly or not, we interpret all of our experiences though the lens of a theory of the world. The most basic one is that there are physical things that cause my experiences. This can be a bit difficult to wrap your head around for people who don't live in the wacky world of philosophy but check it out: you don't perceive objects directly...

You have mental representations in "the theatre of your mind" which you interpret as being representative of external physical objects which are the ultimate cause of your perceptual experience. You could be in the matrix with the exact same experiences being piped into your consciousness and nothing would be different. The worlds would be indistinguishable. The leap from the experience as though there are physical objects in my visual field to there are physical objects in my visual field is an unconscious one. But it is nevertheless a theory. You interpret your experiences of the world with the theory that it is physical objects that are causing what you see in your head.

Anyhow, the point is that even at the most fundament level we interpret observations through the lens of a theory. This happens in everyday life and it also happens in science. When a scientician sees a vapor trail in a cloud chamber, he thinks "there's a proton". He has a theory about the fundamental units of matter and the ways they interact with other matter, and that colors his interpretation of the observation. The observation itself doesn't tell him there's a proton. He adds that as a way of interpreting the raw observation. I.e., that he perceive a "proton" is the product of his theory.

Another example comes from biology. Famously (although he has since rejected it), Dawkins interpreted all evolution as being grounded in the selfishness of genes. In short, all selection call be explained by appealing to genes. If you want to know why one trait is more prevalent than another it must be because the genotype responsible for that trait confers greater fitness. His theory of selection (i.e., that genes are the fundamental unit of selection) colors how he interprets evolution.

On the other hand, a competing theory says that, at least for social animals, you have to also take into account group selection; i.e., the unit of select can also be the group. The reason is that altruistic behavior is inexplicable at the genetic level. How can you explain why a genotype that codes for disadvantageous behaviors/traits could outcompete selfish behaviors/traits? The gene-view can't explain it. Disadvantageous traits by definition are disadvantageous and so should be outcompeted by selfish traits. And so this theory views the same evolutionary observations from the level of group selection.

In short, your theory about the unit of section for evolution will color how you interpret the raw data. Same data/raw observations, different interpretation. Your observations are theory-laden.

Similarly, in the moral domain, we interpret our raw observations though the lens of a theory about the world.  You have a theory of the moral domain that causes you to judge the teenagers torturing the cat as wrong. If you had a different (moral) theory of the world that excluded animals from the moral domain, you'd interpret your observation differently. Your moral observations are theory-laden.

In that observations are theory-laden, moral and non-moral judgments are alike. The theory we hold of the world colors how we interpret the raw observations in both domains. If there's a difference between moral and scientific observation, it must be something else...

Part 3: Observational Evidence
Here's the difference: In science you need to make assumptions about certain physical facts in order to asplain an observation. In ethics you don't have to assume there are moral facts; alz you need to do it know something about a person's psychology; you can explain someone's observation that an act is wrong based merely on facts about the observer.

The way I like to think about this is to imagine (some of you won't have to imagine cuz you already have this theory) that there are no objective moral facts. There are only moral opinions that arise out of upbringing and psychological disposition. Would the world appear to be any different? People would still opine on what is right or wrong. People would agree on some things, disagree on others. Everything would seem exactly the same...and we could explain people's observations too. We could say, this person believes x is right because his mommy told him so or this person thinks y is wrong because she has an aversion to it.  There'd be no unexplainable observations even if there were no moral facts.

Things are different in the case of science.  To make a scientific observation you have to assume that there are objective facts about the world.  I.e., there is matter and energy, and so on. When I observe that there's a cloud of vapor in a cloud chamber, the only way I can explain it is if I assume there are objective facts about the world: I.e., there's matter. I have theory that there are protons; I observe "stuff" happening that conforms with my theory. Thus, my theory is provisionally confirmed.. And unlike moral observations, you can't explain my observations by merely appealing to facts about my particular psychology and upbringing. You can't explain my observation without also assuming there are protons...there has to something there for me to observe!

To summarize thus far: you can asplain the moral observation ("that's wrong!") by merely appealing to psychological facts about the observer. Moral facts don't need to be "real" to explain why someone might make the judgment. On the other hand, you can't asplain why someone would observe what they take to be a proton unless there actually was something real that existed. They could be wrong about it being a proton, maybe it was a neutron but they had to observe something to explain why they report observing a proton. There must be some fact about the world in order to explain their observation. Merely appealing to psychological facts about the observer won't explain why they had the scientific observation/judgement they did.

Observations as Evidence for Theories
If I have a scientific theory that predicts a particle with such and such properties, observable under such a such conditions, observation of such a phenomena counts as confirming evidence for my theory.  But is this the case with moral observations and theories?

Does my observation/judgment that burning the cat is wrong lend credence to my theory that says that it's objectively wrong to burn cats? Let's return to imagining there are no objective moral facts. My judgment doesn't confirm my theory. I'm just extending my theory (i.e., beliefs) to the observation. This will happen regardless of whether there is an objective moral fact about the matter or not.

Again we can contrast this with the scientific observation. If my theory predicts protons but there are no protons (or nothing with the properties my theory predicts), this will impact my theory.  Whether there are or are not objective facts about the world matters for scientific observation and their effect on theory.

Let's slow down and make a distinction here between types of observation. There's a sense in which my observation that "it's wrong to torture cats" confirms my moral theory.  For example, I have a theory with the principle that it's wrong to cause unnecessary harm and suffering to innocent sentient beings. I see the teenagers lighting the cat and I immediately observe that the act is wrong. It is an instance of causing unnecessary harm and suffering and I observed it to be wrong. That confirms my moral principle. I said that such acts are wrong, I witnessed an instance of such an act, and I also observed that it was wrong. Moral principle is confirmed!

However, there's another sense of observation in which my observation doesn't confirm my theory because it doesn't explain why I think the act is wrong.  My observation conforms with my moral theory, that such acts are wrong, but it isn't evidence in favor of my principle. We can imagine another person a few centuries ago who thought it's great entertainment to light cats on fire. They can witness the exact same act, judge that it's not wrong and say "ah! that confirms my theory that it's not wrong to cause harm and suffering to animals! My judgment confirms it!"

Not so in the case of science. If a theory predicts an entity with certain properties and those properties are observed, observing the properties explains why someone has the observation they do. And, bonus, the observation is evidence in favor of their theory/hypothesis. If no entity were observed with the predicted properties, the observation couldn't be made.  In short, the scientific (physicalist) theory explains why you observe the entity and properties that you do because the theory is about how the world is. Even if you believed the theory to be false, you'd still have the same observation:

Think Galileo. When the Church officials looked through his telescope, they didn't believe his theory yet what they observed confirmed Galileo's theory, rather than theirs. Their theoretical beliefs couldn't prevent them from seeing the evidence that disconfirmed their own theory and confirmed Galileo's. (Although, in the end they decided to reject the evidence rather than their theory...)

Possible Objection: Scienticians can (and have for all of scientific history) observe the exact same phenomena yet disagree on the theoretical interpretation of the observation. For example, both gene-based and group selection-based evolutionists observe the exact same phenomena. They don't disagree about what's happening. Evolution through differential reproduction and natural section is happening. They disagree about what theory best explains differential reproduction and natural selection. And so, it looks like, in some respects, scientific observation is similar to moral observation.

Two people can observe the exact same thing and disagree as to what theory the observation supports.  Just as two people can observe kids torturing a cat and disagree as to whether the case supports their particular moral theories on the treatment of animals, two scientists can both observe the exact same instances of differential reproduction and natural selection and disagree over the unit of selection to explain the observation.  That is, they can disagree at the level of theory about how to interpret the observation.









Tuesday, January 13, 2015

Relativism, Nihilism, and Realism: You Think You're a Relativist but You're Not.

Introduction
If I had a nickel for every time I hear something like “Well, if it's right for him, who am I to judge?” I'd still be poor. Not because I don't hear it a lot but because I'd donate all those nickels to charity. Anyway, relativism--the position that values are relative to an individual or culture-is by far the most dominant view among the general population. In fact, back in my pre-philosophical days I too was a relativist. If you were born to a relatively secular or liberal religious household chances are you *think* you're a relativist too.


Before getting into the problems with this view, I'll quickly go over some of the (perceived?) virtues of relativism. (a) It promotes a culture of tolerance and avoids/tempers dogmatism and (b) by way of (a) it can reduce conflicts. However, there are two main problems with relativism: (a) no one really believes it when push-comes-to-shove and (b) it's logically incoherent: instead you must pick either realism (there are objective values) or nihilism (there are no objective values).

EDIT: Some commenters have correctly pointed out that realism and nihilism aren't the only positions available if you reject relativism. In the interest of simplicity for my non-professional audience I intentionally lumped non-cognitivist positions in with nihilism. So, for those of you familiar with the distinction, please read "nihilism" as including non-cognitivist positions for the purposes of this post. For non-philosophers, if you'd like to learn about the difference I've given a brief summary here.


Part 1: There Are No True Relativists
OK, let's grant the virtues of relativism for the moment. Unfortunately for relativists (i.e., most Westerns) you aren't really relativists. Relativism leads to the absurd conclusion that there's no normative difference between a sadist's values and a humanitarian's values. Hey, if tormenting people for fun is perceived as good and right by the sadist, then it's good. Who are we to judge his values? The relativist might reply, “No, what I mean is you can do what you want so long as you don't harm other people.” That's a fine response but you've just given up relativism. You've just conceded that there is at least one objective moral truth.

Another response might be that people can do what they want so long as it makes them happy (however you define it). Once again you've conceded the argument because you've committed yourself to the objective value of happiness. I.e., when actions conflict with happiness, we ought to favor happiness; happiness is more important than all other things. You are actually a realist/objectivist.


Again, relativism leads to the unsavory position that the Gestapo and Medicins Sans Frontieres are organizations of equal moral worth. If the Gestapo thinks it's good and right to “throw the Jew down the well,” then, hey, who are we to judge? Punching someone in the face is no less praiseworthy as giving someone a helping hand. I doubt very much that anyone truly thinks that, beyond personal beliefs and preferences, there are no important differences between the values in the above examples. If you think there are important differences you're a realist because you just made a judgment about one set of values having more value than another. If you don't then you're probably a nihilist. But you aren't a relativist. More on that later.


Why the Virtues of Relativism Aren't Virtues
Tolerance, aversion of dogmatism, and reduction of conflicts are not genuine virtues in the face of obvious evil. If we discovered that our neighbors had child slaves most of us do not think it would be virtuous to tolerate the practice. In fact, we should dogmatically oppose it and we should confront those who practice it. In short, most of us think that confronting evil and injustice is virtuous while tolerating it is not. And so, the virtues of relativism are not virtues after all. They are contingent on the objective goodness of the practice in question.


If we think a practice is good or perhaps value neutral then tolerance for diversity appears good. But it's not relativism that grounds the virtue of tolerance, non-confrontation, etc... It's our recognition of a practice that brings about some good. And so, the virtuousness of our response is grounded in the fact that we are tolerant of things that are good (or value-neutral) and intolerant of things that are bad. In short, it is the goodness or badness of the act that grounds the virtue of our response to it.


Part 2: The Logical Invalidity of Relativism and Why You Must Be Either a Realist or a Nihilist
Argument 1: The Multiplicity of Things
Let's get the ball rolling with a little Plato...


Now if a man believes in the existence of beautiful things, but not of Beauty itself, and cannot follow a guide who would lead him to a knowledge of it, is he not living in a dream. Consider: does not dreaming, whether one is awake or asleep, consist in mistaking a semblance for the reality it resembles?
I should certainly call that dreaming.
Contrast with him the man who holds that there is such thing as Beauty itself and can discern that essence as well as the things that partake of its character without ever confusing the one with the other—is he a dreamer or living in a waking state?
He is very much awake.
So may we say that he knows, while the other has only a belief in appearances; and might we call their states of mind knowledge and belief? Republic V. 476

Ok, there's a bunch of stuff going on here that I'll possibly address in a later post but for now I just want to focus on one idea. How is it that someone can say of many particular things, actions, or events that they are good/bad/just/unjust/beautiful/ugly, etc... if he isn't referring to one objective standard? That is, by indicating that several particulars are good then they must share the property of goodness just as all red things must share the property of redness. 

For example, when someone says x was good, y was good and z was good, all these things (are perceived to) share the quality of goodness. And so goodness must be one objective thing, not many things. There is one concept of goodness and particular acts or events can partake, to varying degrees, in goodness. (Note: some philosophers contest this view suggesting that there are only particulars but let's roll with it for now. It's not the only argument against relativism.)

The simple response is to say, yes but people might disagree over what goodness/rightness/justice is. That is, they might have different beliefs about what those normative concepts contain.

First of all, the fact that people disagree over something isn't evidence that there's no objective truth about it. For example, just because people might disagree over whether Sweet Baby Jesus created the universe or whether it was Indra (those are the only possible choices) doesn't mean that there's no right answer to the question. (Indra did it)

Disagreement only is evidence for three possibilities: (a) one person is right while the other is wrong; (b) both people are wrong; (c) there is no right or wrong answer. You can't just jump to (c) from the fact that there is disagreement over something. You must also consider that (a) and (b) might be true.


Argument 1: “True”--You Keep Using That Word But I Don't Think It Means What You Think It Means
The fact is, relativists often express their position in the following mantra “If Bob thinks X is good then it's true that X is good for Bob.” In short, each individual (or culture) is the arbiter of value. Let's see why this argument doesn't work...

I covered the first reason to reject the relativist argument in the first section. Relativism means we can make no judgments between the most evil and the most benevolent actions. Let's assume for the moment that it's true that we can't make such objective judgments. There's no way to distinguish between what we perceive as benevolent and evil actions. Let's try a different argument.

Another reason to reject the relativist mantra is that just because someone believes something is good for them doesn't make it so. If you can't think of at least a few times you believed something was good for you but at a later date realized it was a mistake, either you are still an infant or you're lying. Scientifical fact from test tubes and beakers: People can be mistaken about what is or isn't good for them and so just because a person believes something is good doesn't make it so.

Here the relativist can reply, OK so maybe people can be wrong about what's good, bad, right, or wrong for them but it doesn't follow that there is some objective good, bad, right, wrong for everyone. The person that got it wrong just got it wrong relative to them.

Let's use logicalization to address this counter-reply:
Again, at the heart of relativism is the idea that believing something makes it so. Thus, believing that abortion is wrong makes it true that abortion is wrong (for that person or culture). But this is to misunderstand the notion of truth. When we say that something is true we mean that it's true regardless of your beliefs. For example, if someone believes that there were weapons of mass destruction in Iraq their belief has no bearing on the truth of the proposition. The proposition “there were weapons of mass destruction in Iraq” is either true or false regardless of what anyone believes because the truth value of propositions depends on states of affairs in the world not on our beliefs about states of affairs. It can't be both true and false. It must be one or the other.

Consider:
Person A says “I believe abortion is wrong (in circumstance C), therefore it's wrong.” They say the proposition “abortion is wrong” is true.
Person B says “I believe abortion isn't wrong (in circumstance C), therefore it isn't wrong.” They say the proposition “abortion is wrong” is false.

The most basic rule of logic is the law of non-contradiction: that something cannot both be true and false at the same time. Relativism leads us to violate the most fundamental rule of logic. It makes it so a proposition can both true and false at the same time. Under relativism “abortion is wrong” is both true and false.

Accepting relativism means we have to reject the most basic rule of logic and this is not a good outcome. It means you can't even communicate because everything is both true and false. Squares both do and do not have 4 corners; Circles both are and are not squares; I both exist and don't exist at the same time; I both will and will not meet you for lunch. My car both is and is not a car. Those jeans both make your butt look fat and not fat. Nothing makes any sense. All meaning is drained from language. This is the logical cost of accepting relativism.

The relativist can reply. No! No! No! You don't understand! What I mean is “abortion is wrong" is true for me! (Me! Me! Me! Everything revolves around me! Even truth and falsity). But this is to completely abandon what we mean by true and false. Nobody takes seriously the person who says “the earth is flat is true for me!”. To such a person we simply say that they are confusing belief for factual knowledge. They believe that the earth is flat but this doesn't make it true. And they certainly don't know that the earth is flat. There is a fact about the universe that the earth either is flat or it isn't. It can't be both—regardless of how hard you believe and regardless of what you read in "The Secret"...

To summarize: An assertion cannot both be true and false (law of non-contradiction). Relativism violates the law of non-contradiction because some people will say of a moral statement that it is true while others will say it's false. There are two alternative: (a) both are wrong and there is no moral truth at all, just preferences (i.e., some form of nihilism is true); (b) one is right and the other is wrong (i.e., some form of realism is true).

Realism or Nihilism: Pick One but You Can't Be a Relativist
We want language to have meaning so we must abandon relativism. Our two choices are realism or nihilism. Realism (in its many flavors) is the position that we can make objective judgments about some values being better than others. Contrary to what simplistic caricatures would have us believe, realism doesn't necessarily commit you to hard and fast rules. 

Realism can be very coarse grained. It might say that for certain situations there is no one right answer there are several but there are also wrong answers. If you think that there are some situations where there is fact about what's good or right or that there can be wrong answers, you are a realist. Maybe you think it's wrong to kill a bunch of innocent civilians for drawing cartoons of a person who lived in the 7th century. If you think it's wrong, and not just a belief that it's wrong but are willing to say it's wrong for anyone to do, then you're a realist.

If you don't think there are any objective moral facts or values then you can never say one set of values is better than another. As a nihilist you can't say that making one decision over another matters (from an objective point of view) because there are no values and so there's no reason to prefer one outcome over another. People might have personal preferences but we can never objectively evaluate their preferences. The person who chooses to spend their life counting blades of grass has a life no less meaningful than the person who cures cancer.


What you can't be is a relativist. Either there are objective moral facts and/or values or there aren't. It's one or the other. It can't be both. People can have beliefs/preferences about what's good/right/bad/wrong but this doesn't tell us anything about the objective state of affairs. If you're a realist, people can have mistaken beliefs about what's good/right/bad/wrong. If you're a nihilist, people only have preferences and preferences can't be “true” or “false” any more than my preference for strawberry ice cream is true or false.

Tuesday, January 6, 2015

Annual Fitness Post: The Most Efficient Fitness Workouts

Introduction
Hey guys, it's that time of year when you go out and buy an annual gym/yoga/taichi/zumba/crossfit membership that you'll only use for a month or two. *This* time you're gonna make it work! Well, I hope you do anyway.

As my throngs of adoring fans know, every year I do an annual fitness post.
http://missiontotransition.blogspot.ca/2014/01/annual-fitness-post-injury-prevention.html
http://missiontotransition.blogspot.ca/2013/01/annual-fitness-advice-post-using-social.html
http://missiontotransition.blogspot.ca/2012/01/no-nonsense-fitness-guide.html
http://missiontotransition.blogspot.ca/2010/10/nutrition-rant-simple-vs-complex-carbs.html
http://missiontotransition.blogspot.ca/2010/06/i-was-planning-on-attacking-some-recent.html
http://missiontotransition.blogspot.ca/2010/07/health-and-nutrition-part-2.html
http://missiontotransition.blogspot.ca/2010/07/health-and-nutrition-part-3-hopefully.html


This year's post is about the most efficient workouts in the woooooooooooooorld. Basically, my patented high-tech workouts will teach everything you need to know to lose fat, gain muscle, and cure cancer. Ok, maybe I'll just teach you how to get the best workout in the shortest amount of time.

You can use these workouts a couple of ways. They can be your primary workouts or you can use them the way I use them. If I'm short on time or need a change from my usual routine, these workouts kick my ass in under 25min. For most people it'll be less.

The common theme of all these workouts is high intensity. I'm too busy to do the literature search but basically, the best evidence on fitness training is that short high intensity training gives as much if not more benefit as up to 2 hours of low intensity training. Trust me. I'm a philosopher.

Why waste time working out when you can waste time doing other things? Now, hold on to your panties cuz these workouts are intense. Some of them will finish you in 10 or 15 minutes (as long as you're pushing yourself and not whining like a baby).

World's Most Efficient Workout #1: Running Stairs
Back in my high school wrestling days, after a full practice Coach Buono would make us run stairs for 15min. It is one of the most hellish workouts you can do.  Let me be clear. I'm not talking about no sissy stair climber. I'm talkin' real stairs. There's a very big difference. You'll find out...

How to Run Stairs
Find some stairs. Run to the top. Run back down. Repeat.

Ok. So lets talk specifics so you get the best workout possible in the shortest amount of time.

First of all I like to find at least four stories of stairs to run (more is gooder). Find a building or outdoor parking garage or maybe the building you live in.

Second, mentally commit to the number of times you'll climb the stairs. You aren't allowed to quit until you've finished. At the low end it should add up to maybe around 30 flights. I like to do around 60. Back in the day I lived in an 18 story apt building and I'd run up 10 times. Don't do that. You'll die. The first time I did it, I couldn't walk for days.

Third. This is the important part. You must ascend 2 steps at a time. Running! No walking! Ok, so maybe near the end when you're starting to die you can walk the flights at the end of a rep but you must walk up 2 at a time. Otherwise, you won't be engaging a lot of the muscles in your legs and you won't be working out with enough intensity. MORE INTENSITY!!!1111!1!!!

What I like to do is I'll go up the stairs to the rhythm of the song I'm listening to. This forces me to keep a certain pace and not give up like a baby. Also, when you start to get tired, you can walk to the rhythm between flights but once you get to the next flight get back to 2 at a time and to the beat! Stop crying, big baby!

Fourth: Descending. For most people I don't recommend running down unless you're used to it. If you do, make sure you keep one hand on the handrail. I've fallen before. It's no fun.

I usually do my 60 flights in around 25 min and I'm done after that.  Lungs are on fire, legs like jelly (u jelly?). So, if you're looking for a quick workout that will give you results; this is a good one.

Reminder. RUN up. Yeah, your legs and lungs will hurt but you wanted to get in shape didn't you? Oh, did you think it was going to be easy? Did you think you could just sashay up those stairs and you'd get all the results? Listen cry baby: All those fitness sophists telling you that you can get in good shape by doing comfortable low intensity exercise are lying to you. You're going to have to push yourself past your comfort zone if you want results. Now stop crying and go run some stairs!

World's Most Efficient Workout #2: Sambo Conditioning 
What is the sambo conditioning circuit? 10 Jumping Squats, 10 Push ups, 10 Sit ups. Repeat for minimum 10 minutes. I dare you to do 15 minutes. Count how many times you can get through the circuit so next time you have benchmark and a target to beat.

How to do the squats: Go down into a regular squat but with your arms at your sides until your finger tips touch the ground. Jump up. That's one.

Lets be real. You're not going to be able to jump the whole 10 minutes. So, as you begin to fade, do as many jumping squats as you can at the beginning of the set until you fail then finish the set with regular squats. For example, you're already starting to cry cuz you can't do 10 full jumping squats. In your next set of squats, jump for the first 3 or 4 then finish the set with regular squats but make sure your finger tips touch for each one.

How to do the push ups.  You know how to do push ups. Chest to the ground then full extension of the arms. That's one rep. Don't let your midsection sag. Better to stick your ass up a bit then let it sag. You can vary the hand placement too: narrow, neutral, wide.

Sit ups: Well, they don't have to be sit ups. Pick another ab exercise if you like: crunches, leg raises, V-ups. Whatevz. Just make sure you get a full range of motion and a good contraction at the top of the movement (that's what she said).

If you can do 10 cycles in 10 minutes you are Jesus. I think my best was 8.

World's Most Efficient Workout #3: Randy Couture Weightlifting Circuit for Wrestling
Maybe you can't find a staircase or maybe you just want to do weights but not your usual work out. This weight cycle will kick. your. ass.

Do this circuit 6 times:
If you live. Please tell me about it.


World's Most Efficient Workout #4: Versa-Climber
I just discovered this one a month ago. If your gym has one, I challenge you to go just 5 min at 90 rpm (that's what I did the first time and I died the death).

The way I'd recommend using this one is as interval training. Go 2 min at 90 rpm then rest for 1 min. Repeat 5 times. Go home, you're done.