Day 2: Naiveté, Falsificationism, and the Counterfactual

Psychiatric Inpatient
Many years ago in my early 20s I read a short story that stuck with me for its nightmarish realism. I can't remember the title or the author, although I think it was G.G. Marquez. I do know it was a Latin American author (if you know the story I'm talking about let me know so I can put a link--it's a great short story). The general plot goes something like this: 

A young woman is touring the countryside and goes into a tourist stop to get some food and use the bathroom. There's also a tour bus full of people at the stop. She happens to exit the store at the same time the tour bus is loading its passengers. One of the "tour guides" asks her where she's going. She says that she's going to her car to drive to the next town. She's on vacation. The tour guide smiles and nods and directs her to get on the bus. She's confused by this and tries to explain again. Again the tour guide gives a similar response, but this time calls over the other "tour guides".  Once again the woman tries to explain that she's not with the tour. They smile and nod.

Anyhow, to make a long story short, the people on the bus were patients from the local psychiatric hospital and the "tour guides" were the doctors. The upshot of the story is that there's nothing she could say to convince them that she didn't belong on the bus (and eventually institutionalized). When she got agitated, they interpreted this as a need for sedatives. When she explained her story, they interpreted this as delusion. 

My naive impression of the psych ward (and psych consult), in some cases, was very much like this. There was no way for the patients to answer  any questions without their answers being interpreted as evidence for some pathology.  In short, a psychiatric diagnosis was unfalsifiable. 

There were, of course, also cases were the patients did have obvious serious psychiatric problems--such as attempted suicide (and usually a history of the behavior)--but some of the patients' behaviors, to my naive eyes, seemed like totally rational responses to their difficult situations. Many were in there after a particularly traumatic and stressful event. 

Since I now have an official visitors' badge I think this entitles me to give a diagnosis. Basically, my visitors' badge plus two days of observations put me at just one level below an expert. Anyway, my evaluation was that there was very little that was abnormal about their response. The major difference between me (and many of my peers) with many of the patients is that we have access to the social and financial resources to weather a storm. Most of the patients didn't. Think about a crisis in your own life. Can you imagine having to go through that without the people that got you through it and, on top of that, having to worry about where you were going to sleep the next day? Very few individuals can bear trauma alone and under additional stressful conditions.

One gentleman's wife was dying. He was functionally illiterate and she had handled all their affairs. He had admitted himself and was very anxious. The staff kept asking questions to try to give a psychiatric diagnosis--i.e., they needed a label for his general anxiety and confusion. He keeps repeating "My wife is dying and I need to take care of her and I don't know what to do." Meanwhile the physicians are asking him to answer math questions and spell words backwards. I literally wanted to scream "Why are you asking him these questions? He's already told you 3 times what the problem is. His wife is dying and he doesn't know how to handle it. How is this pathological? This is the response we'd expect from any normal person."

Here's the thing. What I don't know (and the doctors do know) is the patient's case history. He has a history of various serious psychological problems. While this particular response is normal it could trigger more dangerous responses. And so they adjust his medication.

I don't mean to be an apologist here. Unlike food babe, I don't want to be quick to judge things that I know very little about--despite what my gut tells me (which, incidentally is never wrong--that's a scientific fact). Based on the little I observed, more than anything the guy needs social support. He needs someone to help him manage his wife's care. He isn't literate. How's he supposed to administer the medicines properly? Pay the bills? Manage everything that she had done previously for the both of them? He needs to know/feel that he isn't alone and that there are people that care about him.  Of course, this doesn't necessarily preclude medical treatment to stabilize his behaviors. But it seems to me the emphasis should be on social and emotional support.
. . . .

The psych ward is a strange place. The rooms are barren. No TVs, no radios, no paintings, no cellphones. the patients do, however, have access to books.  Think of a running track. The inside of the track is where the staff are. Most of the desks face outward so the patients can be observed while the staff work. The patients, with nothing to do, walk aimlessly around the island, again, again, and again. They're like zombies. 

This set up strikes me as odd. Why the intentional sensory deprivation? How would you act if you were in a room all day with no art, no TV, no cellphone to text or call your friends, and nowhere to go and nothing to do? I can tell you right now, whatever sanity I had going in would be long gone after a few days. 

Well, that was my first impression. It turns out, they do also get group and individual therapy. There's a stretching class. And there is a TV room--just not one in each room. Not as bad as my initial impression.

I met a woman who seemed quite normal. I can't remember how many days she'd been in the unit. Less than a week. Anyhow, the first time we talked to her, at the end she asked when she could go home. She wants to go home today. She was very polite about it but you could sense the pleading in her voice. The medical student said "I'll talk to the doctor and we'll let you know in an hour". 

We went back with the attending doctor about an hour later. We repeated that previous conversation. "How are you feeling? Are you hearing any voices? To you have any desire to harm yourself?" All these questions were answered just as you or I would answer. Again, she asked if she could go home today. The doctor replied, probably tomorrow. 

I'm thinking: this woman sees totally normal. If I'd asked to go home and someone said no, I'd get agitated (which she didn't). Of course, she probably knows that if she acts agitated it will be further reason to keep her (i.e., problem non-falsifiability).  Anyhow, later I asked the Dr. why she didn't let her go home.

Here's where case history and having more than a visitors' badge are important. When the patient was admitted she was having suicidal thoughts and hearing voices. Of course, I didn't know that part, but given the conditions of the admission, the doctor's caution isn't as odd as it appeared.

And there's more. Every day these doctors must deal with the counterfactual. If a patient comes in (most seemed to be self-admitted) and doctor releases them and something happens--a suicide or homicide--guess who has that on their conscience for the rest of their lives? If you knew that someone in your direct care was a suicide or homicide risk, would you release them the first day they said they felt better? From this point of view, keeping someone an extra day or two doesn't seem so strange.

Day 1: First Impressions

For the next month-ish I'm observing rounds in a Cleveland-area hospital. More on how I got here later, for now here are some of my first impressions

Waiting
From my notebook while I was waiting for the MICU (Medical Intensive Care Unit) rounds to start.

7:30am
People look tired but are friendly with each other. A janitor sees me sitting by myself in the atrium. "They didn't forget about you did they?"  The visitors badge combined with my nervous fidgeting must be more obvious than I'd hoped.

8:15-noon MICU Rounds
I've always had a lot of respect for doctors but observing the MICU doctors took my respect to another level. As a patient you might think your doctors barely know your name. In fact, they know every minute medical fact about you. They know your medical history. In many cases they know your family's medical history. For every chemical, endogenous or foreign, they know the concentration in your body. If you've been in the hospital for a while they know the whole history of every chemical's concentration in your body. They know what each chemical concentration might indicate about your health. They know a bunch of stuff about you that was too fancy for me to follow or remember. The amount of health-relevant data they have on you is staggering. And like I said, it's not just that they have so much data it's that they interpret and analyze all of it.

Of note: Most of the patients in MICU were alcoholics and/or smokers whose bodies eventually couldn't keep up. Most end up with partial organ failure and chronic infections all at once. If you're a hard drinker or smoker, please stop. The future isn't bright.

1:20pm-5pm Psychiatry Consult Liaison
When patients get admitted into emergency and there are obvious or probable reasons for psychiatric evaluations, these doctors are called in to do just that. For example, if someone suffered an injury that looks like a suicide attempt or someone is delusional or even extremely depressed the psychiatry evaluation team is called in.

To me this was by far the most interesting part from the point of view of medical ethics. Based on the psych evaluation, the patient can be forced to stay in the hospital. For example, if they're perceived as a suicide risk.

I filled half a notebook on my thoughts and experiences doing this round but I'm going to limit this entry to just one anecdote.

As an official observer--unless asked for my opinion--I'm expected to do just and only that: observe. But this isn't always easy.

One person I saw was extremely depressed. They (I'm intentionally using an ambiguous pronoun) were in their later years but still living independently---even working.  They had recently beaten what is for many a terminal illness. They had mustered the strength to get through the treatment. Quite a feat for anyone. They got their life back. They fought for their life back. They were exercising again and even working. Life was looking up.

During the post-treatment there was a complication and their organs failed. Now there is nothing that can be done. All that fighting for nothing. They explained to the doctor what had happened. "Do you want to live?" asked the doctor. "Not like this...not like this. I'm independent and I took care of others and now I can't even take care of myself".  The patient began to cry. I wanted to comfort the patient and hold their hand but all I could do was observe.

Evidence for Moral Claims Part 2: Coherentism

Overview of Coherentism
The issue on the table is to figure out what it takes for a moral claim to be epistemically justified. The coherentist offers the following general account of justification. A moral belief is justified to the degree that it coheres with the believers other beliefs. In short, justification is cashed out in terms of a belief's inferential relations to other beliefs. Central to the coherentist position is that there is no privileged set of beliefs that ground other beliefs. All beliefs are justified in terms of other beliefs--regardless of what kind of beliefs they are.

Coherentism is perhaps best understood in contrast to foundationalism. A foundationalist holds that certain kinds of beliefs are--well--foundational while others are either inferred from or not as privileged as the foundational set. For example, in terms of justificatory status, Descartes foundationalism privileges a priori beliefs over empirically derived beliefs. That is to say, if an empirically derived belief conflicts with an a priori (or derived from a priori) belief, the latter belief is more justified because a prior beliefs have more justificatory status (i.e., they're more foundational). In contrast, the coherentist would say that the best justified belief is the one which best coheres with the totality of your beliefs--regardless of kind.

I want to emphasize that the coherence theorist (of the type advocated by Sayre-McCord) isn't a theory of truth rather it is a theory of justification.  Coherence doesn't make beliefs true, coherence simply justifies a belief for an agent.

Emotional Experiences, Justification, and Coherentism
In my previous post I argued that it's plausible that our emotional reactions to certain situations play an evidential role for moral beliefs. In short, in some cases we use emotions to support moral claims or emotional experiences can change our moral positions.  If we think that emotions can sometimes count as evidence for moral claims then, I will argue, coherentism isn't able to capture this type of evidence in its account of how moral claims are justified. In other words, coherence is sufficient for justification but not necessary. The reason for coherentism's difficulty in accounting for emotions as evidence is that (obviously) emotions are very different sorts of things as beliefs. And, on a simple version of coherentism, only beliefs can justify other beliefs.

Let me briefly make explicit the problem. Beliefs--moral or otherwise--are propositional in nature thus can, in principle, have a truth value. Emotions, on the other hand, are not propositional and so cannot have a truth value.  So, (obviously) emotions are not the same sort of thing as beliefs. There doesn't seem to be any obvious way on the coherentist model that something non-propositional could count as evidence for a proposition because the only thing that can justify a belief is its relationships to other beliefs.

Coherentism and Perceptual Experience
Well, not so fast. In some cases--namely perceptual experience--it doesn't seem far-fetched to say that something non-propositional could justify a proposition. For example, the experience of seeing red seems to support the proposition "I see red".  Or maybe the experience of stubbing your toe plausibly supports the proposition "I'm in pain".

So, even though perceptions aren't--strictly speaking--propositional, the tight connection between direct observation and propositional beliefs should make us willing to allow such experiences as able to justify certain beliefs. (More on this later.)

However, even if we allow perceptual experience to justify beliefs on the coherentist model two problems arise. The first problem is general to all types of experience as evidence and the second regards only extending coherentism to allow emotional experience to play a justificatory role. Let me address the first problem...uh...first.

The Isolation Problem
Suppose all my life I've believed that brick walls are soft.  Maybe my parents are the world's greatest pranksters or I was home-schooled or something...Anyway, I've never actually seen a brick wall but I've read a lot about them. As an toddler I was lulled to sleep with stories about their pillowy softness. At dinner my father would regale me with tales of the Great Wall of China which is the longest man-made pillowy structure. In short I have many many beliefs about the pillowy softness of brick walls.

Anyhow, one day I go off into the world and encounter a brick wall, which is perfect timing because I'm really sleepy and could use something comfortable to rest against. I can't wait to feel that pillowy softness. I triumphantly yell "Geronimo!" and sprint head first into the brick wall. To my shock and awe, the wall does not feel pillowy soft. In fact, it feels as hard as a cloud (my parents also taught me that clouds are very very hard).  On the coherentist model, despite my experience of hardness, I shouldn't reject my belief that brick walls are pillowy soft. Why? Because I only have one belief "walls are hard" yet I have many many more beliefs that brick walls are pillowy soft and that support "walls are pillowy soft". If justification is a matter of coherence with other beliefs then the latter belief coheres better with the totality of my beliefs than the former.

In short, a problem for coherentism is that there will be at least some cases (perhaps not as far-fetched as my example) where we think a single experience-derived belief warrants over-riding many beliefs. However, on the coherentist model, we shouldn't do this because the single belief won't cohere as well with the totality of beliefs as will the contrary proposition. The single (recalcitrant) belief will not be justified.

It may be that a sophisticated account of coherentism can handle this objection but it's not clear that it can. I'll leave the matter as it is for now and move to my next point.

Emotional vs Perceptual Experiences
Earlier, I suggested that we should allow coherentism to admit perceptual experiences as being capable of justifying beliefs because of the short inference from the experience to the belief. In fact, Sayre-McCord says as much:

It's true, coherentism doesn't allow experience as relevant to justification unless and until the experience comes into the person's cognitive economy. Yet, especially in its recognition of cognitively spontaneous beliefs, coherentism leaves room for experiences to enter that cognitive economy unbidden, either thanks to the experiences themselves having cognitive content (in which case it is the content of the experience that serves as evidence or by their being the content of an appropriate cognitive attitude (in which case it is the fact that such an experience occurred that serves as evidence). (Ibid, p. 122)

Ultimately, the coherentist is able to admit experience as evidentiary but its justificatory status will still be cashed out in terms of its relations to other beliefs. Let's accept this. Now, what about emotions? They're a kind of experience and so if we've admitted experience it looks like my objection has been met.

Nevertheless, I want to argue that the category "experience" is too vague and is unable to satisfactorily capture both emotional and perceptual experiences.  Lumping both into the same category relies an an implicit analogy between the two types of experiences. However, it's not obvious that these two types of experiences share any properties relevant to our purposes. For example, the inference from the experience of seeing a red pen to the belief "I see a red pen" or "the pen is red" is short and obvious (philosophy of perception aside). It's not going to be so obvious in the case of emotions.

Perception is, at its core, representational, yet it's not clear what it is that emotions are representing. Here's where I think the analogy falls apart.  Because of this disanalogy, it's not obvious to me how an emotional experience gets us to a propositional belief.  Think back to the three ways I suggested emotions function as evidence in our moral reasoning. In the first way we ask our counter-part how they would feel if they were in such and such circumstances. In the second, a feeling (e.g., love) overwhelms a consistent set of beliefs (e.g., that gay marriage is wrong)In the third type of case, visceral images or experiences evoke strong emotions such that we come to endorse a position inconsistent with all our other beliefs. Suppose we are in a debate about the moral permissibility of drones.  For example, a proponent of drones might be shown interviews and footage of what families in a small Yemeni village endures on a daily basis as a consequence of drones and reverse his endorsement. To be clear, these interviews and images don't come in the form of argument, rather they are a montage that elicit certain powerful emotions.

In none of these cases is it clear how the emotional experience translates to a propositional belief. (I need to expand on this). And even if we can give some sort of account of how it might we're left with the isolation problem from above. There will be cases where all my previous beliefs cohere best with my previous position on the issue at hand yet the emotional experience over-rides them. The coherentist model tells us that I should reject my new position.

I suppose the coherentist could (coherently) reply that, "yup, you aren't justified in endorsing your new position because it doesn't cohere with the totality of your beliefs." At this point I'm not sure where to go. It looks like the issue turns on who can pound their fists hardest on the table.  I want to say that the person's post-emotional experience view is justified by the experience while the coherentist will pound back, with equal vigor, no it isn't!

It seems like we've ended up back at the normative/descriptive divide. As a matter of anthropology, yes, humans do use emotional experience as evidence for beliefs. Whether we should is another matter. And so we return to my conditional claim: If we think that emotions ought to count as evidence for moral claims, coherentism can't accommodate them as such.


Concerns:
1. I'm equivocating on what it is for a belief to be justified and what it is that justifies a belief.

2. What's the phenomenology of adopting a new belief in response to an emotion? How would you characterize it. In many cases it doesn't seem as though any of your other beliefs have changed, yet you adopt a new position.  In some cases, is the emotional reaction simply highlighting or giving greater weight to some beliefs rather than others?

3. Coherentist can say "yup, emotions are important to moral reasoning in that they can get us to reflect more deeply on the beliefs we ascribe to yet the emotions themselves don't justify beliefs." In which case I need to throw this whole paper away and start from scratch.  But still I want to resist this and say (while pounding my fist on the table) that at least in some cases the emotional experience is providing evidence for a new (?) moral belief.

What Counts as Evidence for a Moral Belief?

For a couple of years now I've been perplexed by the following problem: What counts as evidence for or justifies a moral belief?  Even in asking the question I'm a bit confused because are two possible interpretations of the question (possibly more). We might say that what justifies a moral belief is that it is well-supported by an argument. In other words, it's supported by some other beliefs that are presented in a systematic way. But this isn't a very satisfying answer because inevitably we are going to wonder what supports the justifying beliefs. There's a danger of regress which I'll address later.

What I really mean to ask is, what sorts of things count as evidence for a moral claim? Arguments using other beliefs are one sort of 'thing'. But is that it? Are beliefs the only things that can justify beliefs? I'm compelled by the view that we should admit other sorts of things as evidence, namely, emotions.

At the face of it, it sounds crazy. Imagine you're in court and you believe someone wronged you and the judge asks you to please justify your claim.  "What evidence do you have for your claim that Mr. X wronged you?" he asks. "Well," you reply "it just feels like he did." I doubt your case will do to well.

But let's pause for a moment and think about how, from a very early age, moral education proceeds. One way we do things (both as children and as adults) is that we offer arguments and introduce facts in order to support moral conclusions. For example, when a child punches another child we might explain to him that you shouldn't punch other people because punching hurts them. And hurting people is baaaaaad!

On this model we support our claim "one ought not to punch others" with other beliefs. If you read much of the philosophical literature on moral reasoning you might think this is the only tool available for moral education.

A little reflection reveals that it isn't. I submit that in large part support for moral claims comes from the emotions. That is, emotions led support to moral claims and therefore are a kind of evidence. Let's revisit the child puncher to see why. There's a much simpler--and I would argue--more effective way to get him to see why it's true that he shouldn't punch other people. We ask him "how would you feel if someone did that to you?"  No argument needed.  And I don't think it's unreasonable to suggest that he sees the strength of the moral claim more quickly and forcefully than if only the first method had been used.

Appeals to emotions as evidence aren't restricted to the moral education of children. As adults we appeal to emotions in conjunction with arguments as well as when arguments fail to convince others of our moral claim. In fact, we often use the same technique we employ with children. When arguments fail us, to the recalcitrant party we ask "how would you feel if I did that to you?" When the other party makes a genuine effort to introspect on what it would feel like to have x done to them they might come to endorse the claim that they shouldn't x. The change of heart comes about without appeal to argument or beliefs as evidence.

Here's another way we might think that emotions are being used as evidence for a moral position. Most people are aware of the fact that animals are often mistreated on factory farms. They also believe that it would be wrong to support any institution that mistreats animals. Nevertheless, as is often the case, many people don't come to oppose factory farms until they are actually shown videos and images of what occurs inside some of the farms. 

What's going on here? It's not as though they acquired any new beliefs. The already knew that animals are often mistreated in factory farms and they already believed that it's wrong to support institutions that mistreat animals.  My suggestion is that emotion has played an important evidential role in their position change.  So, more generally, it looks like emotions play an evidential role in moral reasoning. It is, admittedly, another step to say that the emotional response to the horrific treatment of animals justifies their changed belief. I will address this concern shortly. 

I want to suggest one more way in which I think emotions count as evidence for moral claims. Consider a case of a conservative religious opponent to gay marriage (is there any other kind?). Let's call him Dick. Dick dearly loves his daughter. At some point in her adult life, Dick's daughter confesses to her father that she is gay. Fairly quickly Dick reverses his position on gay marriage. What happened here? Did Dick all of a sudden come in contact with a new, never-before-heard, compelling argument for marriage equality? Did he acquire some new beliefs that support the claim that marriage equality is just? Doubtful. It's also doubtful that he just learned that he has the belief "I love my daughter and I want her to be happy."  No, that's not likely it.

Likely, it is his love for his daughter and his desire that she be happy that's doing the work. In fact, it's not implausible that in his entire set of beliefs the only belief that changed was his belief regarding the permissibility of gay marriage.

What I've outlined are a few ways which I think capture how emotions are employed in our everyday moral reasoning. As I alluded above, it is a separate question as to whether emotions ought to be used as evidence; that is to say, whether emotions able to justify moral claims. I don't want to argue for that positive claim here, but I do think it would be odd to say that in our moral reasoning we ought never to take into account our emotions. 

For this reason I simply want to defend a conditional claim:  if our emotions can sometimes count as evidence for moral claims then several contemporary models of moral reasoning can't adequately account all the ways in which we come to endorse moral claims.

In closing, I want to flag some potential problems with my view. First, I want to quickly return to considering emotions as justificatory. From the fact that we use emotions in moral reasoning it doesn't follow necessarily that we ought to. We might think that emotions can lead us to bad moral conclusions, not just good ones. To say that we ought to use emotions as evidence I'd have to show that our emotions get it right more often than they get it wrong.  Or at least pick out the types of cases where they tend to be more reliable than not. I'd also probably have to go through each emotion because some (possibly the reactive emotions) might be less reliable than other emotions. There's no reason to suppose that each emotion is as likely to lead us to a 'good' conclusion.

And while I'm at it, there's a further problem. My account presupposes a moral view. For example, in the gay marriage case we (most people reading my blog) think that Dick's love for his daughter got him to the right answer only because we happen to endorse marriage equality. Someone who didn't endorse that view would say that Dick's love for his daughter blinded him to the truth.  On the other hand, any theory of evidence will have to contend with this same problem. Whether a belief leads one to endorse the 'right' position will depend in large part on what you (dear reader) think the right position is. 

But on the third hand, there's no need to suppose that a belief has to be true in order to be justified. We can be anti-realist about moral claims and still think that for moral claims some things count as better evidence than others and some claims are better justified than others.

Meh...ethics is complicated.

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.

Transitivity 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 transitivity (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.