Gender, Sex, and Transgender Debates

Introduction

For a while, I've had the idea of writing short posts presenting right wing and conservative ideas as charitably as possible. There's a tendency on both sides of the culture wars to oversimplify and straw man (person?) the opposition. I fall prey to this just as much as the next person but in an effort to practice what I preach, here is the first is what I will try to make a regular feature of my blog. My first attempt is here. In this post, I'm going to enter the mine-field debate over sex, gender, and transgender identity.

In these pieces, my aim isn't really to argue for a particular view. I'm mainly going to try to give an overview of some of the issues and trade-offs associated with various popular positions. The hope is that I'll accomplish what I aim for in my classroom: To get people to at least feel the intuitive pull of competing positions and understand why someone might adopt them. That said, where I think a position is particularly strong or weak, I'll suggest this.

Sex, Gender, and Transgender Debates
In good philosophical fashion, let's begin by defining our terms. There is disagreement over the terms and we'll look at that later, but we need something to begin with. Here are how sex and gender are often defined.

Sex: A biological category defined by some combination of chromosomes, hormones, and genitalia. Edit: In an earlier draft I wrote that there are five sexes for humans, especially since it's not clear how to classify hermaphrodites. I'd read this several years ago in some now-forgotten articles. I've since learned from commenters that this is a contested claim. The two-sexes view holds that we can always identify the female because "she makes large gametes." For a fun twitter feed on this topic, go here. For an overview of the biological possibilities for sex determination in humans, go here. For scientific support for the two sex view, go here.

Gender: The behavioral norms typically associated with a particular sex. Norms, in this case, are often understood to be both descriptive and prescriptive. That is to say, they can describe how members of a sex do act or how they ought to act. It's important to keep the descriptive and prescriptive elements separate since most people often equivocate between the two. The most common genders are man and woman or, as adjectives, masculine or feminine.

Famously, gender is often referred to as the social meaning of sex. That is to say, when we think "female" or "male," gender represents the social roles and behaviors associated with the respective sexes.

As far as I can tell, the standard conservative position is that sex=gender. This view is usually referred to as gender essentialism. By this I mean that biology and behavioral norms do not come apart. Sort of. On the descriptive account, being female means that you will behave in certain ways and perhaps be disposed to particular gendered preferences. That is, your biology determines your gendered behavior and dispositions.

The normative account of gender usually follows: If you don't exhibit the appropriate biologically determined behaviors then you are deviating from how you should behave. This is what people mean when they say things like, "he's not a real man" or "act like a lady." These are admonitions to act according the norms appropriate to your biological sex.

Critics of gender essentialism point to a potential problem. If gendered behavior is determined by biological sex then how is it possible that some people don't behave according to the biological sex? The reply usually has to do with the effects of decadent liberal culture corrupting the youth. In other words, culture is corrupting "natural" behaviors. A problem with the reply is that it concedes the very point that their opponents often make: gender is socially constructed and the "natural" gendered behaviors don't occur in a cultural vacuum either. They occur in a cultural environment that models and reinforces particular gender norms....

This leads us to the other end of the spectrum where people argue that sex and gender can come apart. (The fact that it's possible to say "be a man" or "act like a lady" seems to tacitly support this in the descriptive sense...) We only believe that gender and sex are inextricably linked because biologically female humans are socialized to internalize the corresponding cultural gender norms just as biologically male humans are socialized to internalize their corresponding gender norms. If males and females were socialized differently, they would act differently than the gender norms typically encouraged and modeled in our society.

So, to repeat, here are the two extreme ends of the continuum: Those who say that sex determines gender and those that say that gender is entirely the product of socialization--not biology. Those who argue that sex determines gender often move from the descriptive claim to the normative; i.e., that one ought to align one's behavior with the gender norms associated with one's biological sex. Failing to do this is, to varying degrees, morally bad.

As you might guess, there's also everything in between: People argue that, in a population, some traits and dispositions are statistically correlated with one sex rather than another. Basically, some of our behaviors and dispositions are biologically determined by our sex while others are indeed the product of socialization. It's important to add that just like for every other species, most traits fall on a continuum: No one has all traits in the same amounts and so, at the population level, we should expect to find all traits in both male and female humans and in different degrees.

Defending the conservative position: Across all species we observe statistical behavioral differences between males and females of that species. We also know that there is a biological foundation to many behavioral dispositions. It would be weird if humans were the only species in all of creation for which sex and biology didn't play any role in statistical distributions of behaviors.

Here comes the tricky part: Humans are unique in that culture plays a huge role in determining behavior. This is why we observe different behaviors across cultures and time. So, while it's entirely reasonable to hold that many behaviors are grounded in biology, many behaviors are also a product of socialization in a particular culture. How do we distinguish behaviors that are biologically grounded from those that are socially grounded when behaviors occur in an environment where both determinants exist?

For some, the solution is to abolish all gender norms and to "let the pieces fall where they may." That is, if we tear down gender norms, people--as unique individuals--will follow a path that conforms to their intrinsic dispositions. In this way, people who might have been pushed into roles that clash with their inner disposition are free to pursue a life congruent with their unique combination of drives and dispositions. Also, those who fit well in traditional gender roles still have that available to them with the important difference that they are genuinely choosing it.

For others, gender norms offer a safe road map for harmonious family and community living. Destroying these norms provides people with no road map and eviscerates the institutions upon which family and society have historically been built.

The gender abolitionist assumes that humans can handle all that freedom and new harmonious forms of social organization can emerge (Read: The Inquisitor from Dostoyevski's The Brother Karamazov for a great take on this). The gender conservative believes that society can't flourish without certain gender roles. They also assume previous forms of social organization grounded in gender norms were indeed harmonious or at least more harmonious than any other possible form of social organization.

There's a lot more to say here but I'm trying to make this just an overview and get to the issue of transgender identity.

Transgender Identity
Ok, if I end up in a re-education camp for this, please contact my mom. She's a professor in the Department of Education at UBC so she may be able to pull some strings.

We can think of transgender identity as involving two distinct but related issues: One ontological and one ethical.

The Ontological Issue
Ontology is a fancy way of talking about the philosophy of "being." In this area of philosophy we try to figure out what makes a thing what it is rather than something else. The ontological question regarding transgender identity asks "what is essential to gender?" In fancy philosophy talk we might ask, what are the necessary and sufficient properties that a human must have such that they are one gender rather than another?

Here are the two simplified ends of the continuum. On one end, some people say that gender is fundamentally determined by how one conceives of oneself. This position is often straw personed(?) as someone merely self-declaring to be one gender rather than another. If I feel like a cat then I am a cat. Most proponents of self-declaring view hold that the self-declaring is a consequence of, amongst other things, a deep psychological self-conception as well as dispositions and behaviors that align with the gender not typically associated with their biological sex.

On the other end of the continuum gender essentialists argue that because gender is biologically tied to sex, one cannot change their gender without changing one's chromosomes. Gender has nothing to do with self-identity and everything to do with biological sex.

There are A LOT of positions in between.

Interestingly, the trans movement has created a division between some feminists. The historically dominant feminist view holds that gender is the product of socialization (often called gender critical feminism). If we accept this then self-identity in the absence of socialization cannot on its own confer gender status. This puts traditional feminism at odds with newer strains of trans-inclusive feminism. A male who is socialized as a man, on this view, cannot be a woman even if they undergo gender reassignment surgery because they have not been socialized as a woman.

Notice that this view (you can't change genders) holds the same conclusion as conservatives but for different reasons. For essentialists you can't change genders because you can't change your chromosomes. For gender critical feminists you can't change genders because you can't change how you were socialized in the past.

Notice also that, on the gender critical view, a trans person could over time potentially become their chosen gender if others treat them that way; i.e, they undergo gendered socialization. One's position here depends on how much socialization is required and at what stages in one's life it occurs.

Most gender critical feminists also disagree with the idea implicit in transgenderism that there are these two neat boxes called "gender" that we can put ourselves or others in. "Gender is a construct, we're trying to deconstruct it, and now you're trying to preserve it just like the conservatives!"

Here's another interesting twist in the debate. Some trans-inclusive views can sort of align with gender essentialists. Our psychology is grounded in our brain biology. We know that in a population, traits are distributed along a bell-curve--regardless of biological sex. This means that some humans with male chromosomes will have a "female" psychology. Gender identity becomes tricky here. What's more important to what we most fundamentally are? Our chromosomes or our psychology? Both are grounded in biology.

On the one hand, you are you because of the psychology particular to you. For example, if you are a shy person it doesn't make sense for someone to call you an outgoing person. You both feel and behave like a shy person. Here, biology points in two directions: The (biologically grounded) brain structures underlying a person's psychology might be what our society associates with femininity while their XY chromosomes point in the other direction. If we weigh psychology and underlying brain structures more heavily, then gender is determined this way. The other position weighs chromosomes more heavily in determining gender identity.

Both replies assume that one or the other is more fundamental to gender identity. Notice that both positions also sort of agree that there is something essential about gender: masculinity and femininity are identifiable clusters of properties grounded in biology. The disagreement is over which is fundamental.

The deep psychological view of gender presents its critics with the following challenge. How do we explain the fact that despite socialization and despite chromosomal sex some people deeply and sincerely identify as the gender not typically associated with their sex? If gender is primarily the product of socialization, then how do we explain gender dysphoria in those who were never socialized for that gender? If gender is primarily chromosomal, how do we explain the existence of a psychology (grounded in brain structures) that can resist a life-time of conditioning in the other gender direction? On the essentialist view, chromosomes, by definition, code for brain structures that underlie the psychology of typical gender identity for that sex. But there exist people for whom this doesn't appear to be true.

The central task for trans-inclusive feminists, with respect to the ontological question, is to show a disanalogy between race and gender. Almost no one thinks that self-identity can determine one's race. So, trans-inclusive feminists need to argue that gender and race differ in some important respect where gender can be determined by self-identity but race can't.

These arguments exist but disagreement over their soundness still abounds--even in the neo-Marxist post-modernist universities (i.e., all of them, according to Jordan Peterson). Regardless of one's position on the issue, I think it's unfair to vilify conservatives and people on the right over the ontological issue when there isn't even consensus on the liberal left.

The Ethical Issue
That said, the left generally agrees on the ethical question: Should I refer to someone according to their preferred gender pronoun? Regardless of whether someone actually believes a trans person is really the gender they believe themselves to be, most people on the left hold that basic norms of dignity and mutual respect imply we call people by their preferred pronoun.

A loose analogue might be someone who self-identifies as a Christian but acts contrary to Jesus's teachings and has never read the Bible. If they want me to identify them a Christian, norms of basic dignity and mutual respect suggest that I do so if that's their preference. I gain nothing by insisting that they are not TRUE Christians. Of course, being a Christian isn't a biological category but it's the norms of dignity and mutual respect that ought govern behavior towards one another regardless of what my ontology tells me. That said, if I want to write a respectful philosophical paper on the necessary and sufficient conditions for being a Christian, I should be able to do this without Harris-Mint.

Finally, the norms of dignity and mutual respect hold people should not be discriminated against based on their self-identity--even if we disagree with how they self-identify. This is the benefit and responsibility of living in a free society. We cannot escape interacting with people with whom we disagree but we can choose to treat them the way we would want to be treated. As an itinerant Jew from Israel once said:

All things whatsoever ye would that men should do to you:
do ye even so to them: for this is the law and the prophets.

and

Thou shalt love thy neighbour as thyself

Amen.

One final point regards individual liberty. Americans luuuuuuuuvz them some freedom talk. Consider what a genuine commitment to freedom entails. Respecting individual freedom to do only the things one likes and agrees with is no commitment to freedom. It's thinly disguised prejudice. A genuine commitment to individual liberty and its real test implies imparting dignity and mutual respect to those who make choices and live in ways we strongly disagree with.

USA! USA! USA!

FREEDOM!!!!

Pew! Pew! Pew! Pew!

Final Remarks
There is no way to cover this entire debate in a single blog post. This topic is massive. The intent here is simply to give people an overview of some of the major positions and what they entail. If you have something you'd like to add, feel free to write me something in the comments.

Doing What Feels Right and Sartre's Existentialism

Introduction
Existentialism can be summarized in one phrase, "existence precedes essence." But what does this mean? Pre-existentialism, philosophical systems presumed Man had an essence. To understand what that means consider a chair. Chairs don't just randomly pop into existence...no sir!  Before they are created someone or something has to have a concept of what a chair is (i.e., something for sitting), then the chair is created in conformity with the concept. The essence of a chair is that for which it is designed--which is sitting, if you didn't know.

Notice two things: (a) having an essence, in the sense I've described, implies a creator and (b) that the creation's essence is determined before it comes into existence. And so, artifacts--things like chairs, computers, cars, iphones, etc--all have creators and have an essence/nature/purpose before they come into existence.

Let's return to "existence precedes essence." For existentialists, human beings are unlike artifacts in that we exist before we have an essence/nature. We are not designed and so there is no predetermined essence that defines who or what we are. Who/what we are comes after we exist. We are thrust into the world and create ourselves through the actions we choose. This is what "existence precedes essence" means: first we exist, then we will acquire a nature (though our acts).  If I do harmful acts and act selfishly, then this is what I am. If I create and share, then this becomes my nature. There is no predetermined nature beyond what I actually choose to do. Again, contrast this with artifacts: first they have a nature/essence/purpose then they are brought into the world.

For existentialists, the human condition is (a) understanding that we are free to choose our own essence (through our actions) and (b) figuring out how we ought to create ourselves given we have no intrinsic nature. Another way of thinking about this is to say that we are responsible for who we are.

Radical Freedom and Responsibility
Although we have no common nature among us we all share the same conditions: We are "condemned" to be free thus we have to make choices about how to live and 'be'. There is nothing objective in the world to grab to guide us in our choices. Every choice is exactly that--our choice. And because every action is a consequence of our own choice, we bear absolute responsibility for it. Pretending you don't have a choice is what Sartre calls "bad faith". Bad faith is lying to yourself about the reality of your radical freedom. Even when you act in bad faith you cannot escape responsibility for your choices which define who you are. It was your choice to lie to yourself.

The existential condition is that there are no objective values to guide our decisions and no one but our selves to decide which values we adopt. Now some might say that we can turn to religion or authorites to guide us. There are a few problems with this. First, you are shirking your responsibility as a free being for decisions. By deferring to an outside source for your decisions your are denying your responsibility to choose for yourself.

One might say that choosing to follow this or that text or leader is a choice--and it is. But this doesn't allow you to escape the subjectivity of the human condition:  how you choose to interpret the various texts and advice will also be a matter of your own choice. There's no external source you can lean on to tell you how to interpret. You cannot escape the subjectivity of your existence and so you still must bear the responsibility of choosing one interpretation over another.

Furthermore, your essence is nothing more that the sum of the things that you choose to do. And if trying to deny your radical freedom and offload responsibility are part of your actions, then you are a coward. Anguish, for the existentialist, in part comes from knowing that he bears full responsibility for what he does and who he is and that this responsibility is inescapable.

Moral Choice
If there are no objective values in the world, it seems like anything goes. Again, just like with any other choice, the ethics that you choose define your essence. If you choose and act on a selfish ethic, then that's what you are and you are responsible for everything that comes from it.

But, in a sense, our actions aren't completely unconstrained. Here's where we exit subjectivity. Every choice that I make not only defines who I am but also defines the essence of Man as a whole because I am a part of that whole. This demands that I consider the consequences of my choices on what will be the nature of Man.

For every man, everything happens as if all mankind had its eyes fixed on him and were guiding itself by what he does. And every man ought to say to himself "Am I really the kind of man who has the right to act in such a way that humanity might guide itself by my actions? (Existentialism)

I am responsible, through my choices, for how Man is defined for my time in history because I am a part of Man.

But this is vague. How do I decide what to do in specific cases? Unfortunately, general moral principles can't tell us how to decide particular cases. Sartre give the following case:

[The young man's] father was on bad terms with his mother, and, moreover, was inclined to be a collaborationist; his older brother had been killed in the German offensive of 1940, and the young man, with somewhat immature but generous feelings, wanted to avenge him. His mother lived alone with him, very much upset by the half-treason of her husband and the death of her older son; the boy was her only consolation. The boy was faced with the choice of leaving for England and joining the Free French Forces--that is, leaving his mother behind or remaining with his mother and helping her to carry on.
He was fully aware that the woman lived only for him and that his going off--and perhaps his death--would plunge her into despair. He was also aware that every act that he did for his mother's sake was a sure thing, in the sense that it was helping her to carry on, whereas every effort he made toward going off and fighting was an uncertain move which might run aground and prove completely useless; for example, on his way to England he might, while passing through Spain, be detained indefinitely in a Spanish camp; he might reach England or Algiers and be stuck in an office at a desk job. As a result, he was faced with two very different kinds of action: one, concrete, immediate, but concerning only one individual; the other concerned an incomparably vaster group, a national collectivity, but for that very reason was dubious, and might be interrupted en route. And, at the same time, he was wavering between two kinds of ethics.
On the one hand, an ethics of sympathy, of personal devotion; on the other, a broader ethics, but one whose efficacy was more dubious. He had to choose between the two. Who could help him choose? Christian doctrine? No. Christian doctrine says, "Be charitable, love your neighbor, take the more rugged path, etc., etc." But which is the more rugged path? Whom should he love as a brother? The fighting man or his mother? Which does the greater good, the vague act of fighting in a group, or the concrete one of helping a particular human being to go on living? Who can decide a priori? Nobody. No book of ethics can tell him. The Kantian ethics says, "Never treat any person as a means, but as an end." Very well, if I stay with my mother, I'll treat her as an end and not as a means; but by virtue of this very fact, I'm running the risk of treating the people around me who are fighting, as means; and, conversely, if I go to join those who are fighting, I'll be treating them as an end, and, by doing that, I run the risk of treating my mother as a means. (Existentialism)

And so a general ethical principle can't help us decide specific cases. And besides, even if we do choose a general ethical principle, there is no guide to tell us which to choose as either a general guide of action or for particular cases. We also have to make that choice and we bear responsibility for it and accept that it now defines us in so far as we act on it.

Sartre and Emotions
General ethical principles can't tell me what to do in particular cases. Maybe I ought to do what feels right to me. If the young man feels that his love for his mother is great enough to sacrifice his other desires, then he should do that. But if the feeling of love for his mother isn't enough to make him give up everything else, then he ought to leave.

But how will he know that he loves his mother enough to give up everything else unless he actually does it? To know that his feeling leads him to the right choice he has to live that choice. He has to see how it plays out. He might try it and find out that the desire to avenge his brother overwhelms him and he's unhappy staying behind and so the feeling misguided him. But he can't know this before he lives it. The feeling can't tell him in advance what to do.

Consider another case that many can relate to. How do you know if marrying someone will be the right choice? Consulting your feelings before you're married can't tell you. You'll only know if it's the right choice if you actually do it. If it turns out well, it was the right choice. If it turns out poorly, it wasn't. The feeling can't tell you what's going to happen and so can't tell you in advance what the right thing to do is. Besides, if feelings were a good guide to marriage, we'd expect the divorce rate to be substantially lower.

It Isn't All Doom and Gloom
So, we are hurled into the world, condemned to be free with no fixed points to guide us in how we ought to live, yet we are somehow responsible for everything we do and are. This is the existential forlornness and anguish.  Forlorn because we cannot turn to anyone to make decisions for us and anguish because of the tremendous responsibility that comes with creating both our own essence and that of Man.

But chin up, butter cup! The good news is that existentialism is a philosophy of action. You might not have any guides but you get to create yourself--with every choice. Bonus, if you didn't like where things were going, you can reinvent yourself at any moment, if you so choose.

There is no reality except in action. Man is nothing else but his plan; he exists only to the extent that he fulfills himself; he is therefore nothing else than the ensemble of his acts, nothing else than his plan. (Existentialism)

Before you go skipping off into the sunset, existentialism is a demanding philosophy if you take it seriously. In creating yourself and defining Man, with every act you are saying "this is what I think all mankind ought to be--follow me!" When taken seriously, that is a massive responsibility to bear (hence, all the angst).

It also means that there is nothing that exists that is not expressed in action. There are no great authors with unwritten great books, no charitable people with kind deeds undone, no great loves who have not loved.  "Reality alone counts and [...] dreams, expectations, and hopes warrant no more than to define a man as a disappointed dream, miscarried hopes, as vain expectations" (Existentialism).  In other words, coulda woulda shoulda is worthless. At the end of the day, what matters is what you did with your life and whether it was an example for others to follow.

Day 3: Psychological Jade and The Arrogance of Ignorance

Introduction
Before reading this, I suggest reading my post from yesterday since most of what I talk about here relates to it (and I end up retracting most of what I said).

'The arrogance of ignorance' is one of my favorite phrases. I'm not sure of its origins but I heard it first from Dr. Steven Novella. I think the phrase is the best way to capture a cluster of common cognitive errors. In no particular order: a) Assuming that because you are knowledgable in one domain that you know a lot about another (or are able to correctly evaluate another). b) Moving from small data sets/anecdotal experiences to broad conclusions. 'The arrogance of ignorance' is a close cousin to the Dunning-Kruger effect: You have so little knowledge of a particular domain that you are unable to assess how little you actually know and grossly overestimate how much you do know, in turn leading you to wildly wrong conclusions.

Anyhow, yesterday I was guilty of all of the above crimes. You'd think a guest pass to a hospital and a few hours of observation would give me enough authority and knowledge to correctly evaluate an entire subfield of medicine. Strangely, it didn't.

Today I went back to inpatient psych, and boy am I glad I did (from a pedagogical point of view). Let me try to both convey my experience and undo some of the misconceptions I had.

Sample bias
Most of the patients I met yesterday had been in the hospital for over a week. I was meeting them after they'd undergone treatment and had been stabilized. Of course they seemed normal to me! As I learned (and hopefully you will too once you read this post), what I did was the equivalent of walking into a surgery unit and looking only at the patients about to be discharged, then asking"why did they need surgery? They look fine to me!"  

So, what are patients like on admission and early in treatment?

Obviously there are a variety of disorders but all of them are severe. Here's the thing, unless you work in a hospital or have someone in your family with a severe mental illness you've probably never actually seen severe mental illness. To most of us, this is an invisible population because most of their lives are lived in care homes or in institutions and, unfortunately, in the streets.

The patients range from very well-spoken with linear thought to having only elementary vocabulary with disjointed unintelligible thought, and any combination of the above. Regardless of where they fall on the spectrum, most of them suffer from severe delusions. Some examples: (1) Being part of an intergalactic group of assassins being pursued by the (intergalactic) mafia, (2) believing that a family member is dead who isn't and all the evidence they have should lead them to conclude the opposite, (3) being pursued by terrorists and (actually) destroying windows and cars to avoid/prevent the terrorist plot, (4) having voices in their head telling them to kill others or kill themselves. There were more but this should suffice.

Interestingly, the ones that had grand conspiracy delusions, e.g., (1) and (2), were extremely pleasant to talk to. If you were to have a conversation with them and the content of the delusion never came up, you wouldn't suspect a thing. It was as though they simultaneously inhabit two realities. When you ask them, they know where they are and why they're there. They'll say "I just want to get better" but at the same time they'll discuss their delusions as though they're just as real as the chair they're sitting on.

Unlike what I hypothesized yesterday, these people don't "just need a little more social and material support." That's the equivalent of saying someone with cancer can be healed with a back rub.

I think my reaction yesterday is probably analogous to what happens with many deniers of modern medicine. They've been in the hospital to visit a friend or they read some article online--maybe even spoken to a disgruntled doctor.  But they've only seen 1 billionth of the data set, and only from the patient side of the bed. Things look very different from the doctor's side of the bed and as you get a larger data set...

Tip of the Iceberg
Another factor that led me to my (wayward) conclusions yesterday was I didn't ask enough questions about case histories. Once you read the case histories, your perspective will change very quickly. Every patient in there has a lifetime history of psychosis that is well documented. Almost all have been suffering the same symptoms since adolescence. Some have their condition for unknown or unknown biological reasons (usually genetic, as it runs in their families), others (there were 2 there) had suffered major brain injuries at some point earlier in their lives and haven't been the same since, others have their condition as a consequence of a life-time of substance abuse. For many it was a combination.

Someone who thinks that a little positive thinking or mere talk-therapy is going to solve these people's problems is extremely naive--like I was yesterday. Someone who thinks along these lines is mistaking people who have one-off breakdowns or depression with this other population. Like I said, unless you work in a hospital or have a family member (or work in law enforcement, probably) it's unlikely you've ever met anyone from this unfortunate population. Until you do, you can't fathom just how serious it is.

Philosophy of Science Lesson: Depression and Jade (Bear with me, You'll See How this Relates in a Moment)
What is jade? Up until the 19th century it was believed to be a kind of mineral. However, a French mineralogist (Alexis Damour) discovered that it was in fact two distinct minerals: Jadeite and nephrite, each with distinct chemical and structural properties. Nephrite is a microcrystalline interlocking fibrous matrix of the calcium, magnesium-iron rich amphibole mineral series tremolite (calcium-magnesium)-ferroactinolite (calcium-magnesium-iron). Jadeite is a sodium- and aluminium-rich pyroxene. The gem form of the mineral is a microcrystalline interlocking crystal matrix.

This isn't Rocks for Jocks 101, so why should we care? I'm getting there. Notice that both structurally and chemically, nephrite and jadeite are different. What does this mean? Well, we know that different chemical compositions will react differently and different microstructures will also also behave differently. Jadeite and nephrite have different fundament properties. From the point of view of science, if we think that science divides and studies the world in terms of its fundamental rather than superficial properties there's no such thing as jade. There's no one structure or chemical structure that is jade.

Here's another way to illustrate what I'm getting at. Why isn't there a science of green things? Why aren't there green-ologists? The reason is that green is a superficial property. Knowing that something is green gives us no predictive power in terms of how other green things will behave. It also provides no explanatory power for why it behaves the way it does. 

For example, green algae has very different fundamentals chemical properties from a green glass. Suppose I put a HCL on the green algae. Based on the chemical reaction, would I be able to predict what will happen if I put HCL on green glass? Does the algae's greenness explain why it reacts the way it does to HCL? Of course not. In science, I want to lump things into categories that are going to allow me to make generalizations and predictions about other things in that category.  

Learning about green algae doesn't help me learn anything important about green glass except what I already knew--they're both green. We don't lump the two into a scientific category because science is only concerned with "lumping" things in terms of shared fundamental properties rather than superficial properties. We ought to "split" superficial categories that contain objects that have different fundamental properties.

Ok, so what does all this have to do with psychiatry and psychiatric diagnoses?  Consider that a common diagnosis we hear about is depression. Most cases we (by we, I mean non-medical professionals) have encountered are probably infrequent non-pathological affairs, perhaps set off by a traumatic event. We know that, with support, most people eventually work through the depression and end up fine. The problem is, 'depression' is psychological jade. 

Different types of depression can manifest the same superficial symptoms but the underlying causal structures are different. (No, alt-meders, this isn't the same "root cause" you're thinking of but it's the one you ignore).  So, the mistake is to think, "ah, depression...we just need to treat it with x, that's what we did with the last case". But this is to treat depression like jade--i.e., as a homogenous category based only on superficial resemblance.

For example, I learned (the very surprising fact) that for many types of deep depression the most effective treatment is ECT (electroconvulsive therapy)--yes, you read that right! I had to ask the doctor twice because I couldn't believe my ears. Apparently, it's well studied. Of course, the current procedure is quite different from how it was in the early days but still...who'da thunk? 

"Jadists" about depression might think all cases of depression can be treated with ECT. This would be a mistake. There are different kinds of depression with different etiologies (underlying causal structures). It turns out that depression in manic depressives doesn't respond to ECT. Depression has its own jadite and nephrite (and more). The "root cause" of depression in manic depression is fundamentally different than it is in other kinds of depression.

Beside this overview of a famous philosophical argument, why am I talking about this? Because if you're a human being you're probably going to commit the same cognitive error that I made when it comes to psychological diagnoses. You hear that a patient (or someone you know) is depressed or has some other general psychological problem and you think about how you or someone you know dealt with it. You think, well, all they have to do is x (whatever worked for you or your friend). It's so simple! 

But you're treating the diagnosis like psychological jade. The diagnosis might have the same symptoms but it doesn't mean it has the same underlying fundamental structure and thus, there is no reason to suppose it will respond to the same intervention. It's a different kind. 

Worse yet, someone could dogmatically claim the all disease and/or psychological diagnoses share the the same "root cause". Such dogmatism precludes any chance of recovery since the same ineffective treatment will only be applied more and more vigorously. What's more, this way of thinking is the opposite of scientific thinking. Saying everything has the same "root cause" is just like being a green-ologist. You're confusing superficial similarity for fundamental similarity. To use the lingo of metaphysics, you're lumping when you should be splitting. 

We see green-ology all over the place in alt-med. For chiropractors, the "root cause" of all disease is some sort of spinal misalignment, for Ayurvedic medicine the "root cause" is chakra alignment (or some shit), for reflexologists the "root cause" is something to do with your feet (WTF? How are these people even a thing?), etc... (While I'm pointing the finger I should make clear that I have my own "root cause" default. I have a tendency to lump various problems as being caused a general lack of meaningful social relationships, belonging to a community, and sense of purpose.) And then there's alt-med's favorite: stress. The "root cause" of all disease--physical and mental is stress. More green-ology. 

To be charitable we can say that stress can trigger or make people more susceptible to disease but this is to confuse notions of causation. Let me illustrate. Suppose someone is in the hospital with a broken leg because they got hit by a car. What "caused" the broken leg? Being hit by a car, right? Now, just because the car was the trigger for the broken leg no one in their right mind would believe that removing the car will heal the leg. 

I can just imagine the doctors at an all-alt-med hospital: "We've cured your leg by getting rid of the 'root cause'--the car has been destroyed! You can walk now!"

So, while it's true that stress can trigger certain reactions, it doesn't follow that the solution to the problem is merely to remove the trigger. Yes, doing so may decrease the likelihood of the same event from occurring again, just like not getting hit by a car will prevent you from breaking your leg again (that way); however, this insight is often of trivial value. No one with more than two brain cells to rub together thinks chronic stress is good for them. What's the next great insight? A poor diet isn't good for your health? Revolutionary! Please collect your Nobel Prize.

The Lesson
The causes of many diseases, physical and mental, have to do with their fundamental underlying structural properties. This is why people respond differently to different treatments. Superficial similarities can cause us to lump when we really should be splitting. Overzealous lumping leads to failed treatment and frustrated patients. Overzealous lumpers are green-ologists. Don't be a green-ologist. 

Anyhow, this is just one more cautionary tale for me to heed. Hopefully, it gives you pause too the next time you diagnose someone (including yourself) and assume that superficial similarity implies fundamental similarity...

Also, hopefully this little digression shows the value of philosophy to science. You can't do one without the other.

In a Nutshell
The conclusions I drew in my last post were wrong. But I'm leaving that post up as a cautionary tale to both myself and to anyone reading this. My hope is that it reminds us how easily we can get things wrong when we only have a little bit of information, particularly about areas where we are not experts. People think it's "being a sheeple" to defer to experts. It's not. It's smart and good epistemic practice. Only arrogance fueled by ignorance would lead a person to think that they know more than an expert in that expert's domain.

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.