Consider the sentence:
(1) Blue horses are more likely to die than red ones.
Let b be a blue horse and r be a red horse.
For convenience, let's translate this into a logical form:
∀b,r Pr(Db) > Pr(Dr)
That is, for all blue horses and red horses, the probability that a blue horse will die is greater than the probability that the red horse will die. This seems reasonable (at first glance).
Consider a one-word change:
For all blue horses and red horses, the probability that a blue horse will die is always greater than the probability that the red horse will die.
Is this redundant? Under one interpretation, no, because this precludes the possibility of new information causing a re-evaluation of the initial probability assessment. The word, always, appears to mean that, no matter what, a blue horse is always more likely to die than a red one, even if the blue one is starved and the red one is healthy. Strictly speaking, our original statement, (1), is ambiguous in this regard.
Pick a blue horse and a red horse at random. There is a greater probability that the blue one will die than the red one. This still seems reasonable to me. But let's change the verb to something involving intentionality, making our hoofed friends a bit more sadistic.
(2) Blue horses are more likely to kill you than red horses.
∀b,r Pr(Kb) > Pr(Kr)
That is, for all blue and red horses, the probability that blue horses will kill you is greater than the probability than the probability that red horses will.
Consider the following preliminary inference:
(3) #I have a blue horse and a red horse (and I have no other information about them). The blue horse is more likely to kill me than the red one.
Is this valid? Something strange is going on here, and it seems to be a deeper issue than simply updating probabilities to account for new information.
From (3), we could straightforwardly make the following entailment:
(4) Azula is a blue horse, and Rufus is a red horse. Therefore, Azula is more likely to kill me than Rufus.
Somehow, it's easier to say that Azula is more likely to die (from (1)) than it is that Azula is more likely to kill me, but it isn't quite so jarring just yet. This is, in fact, similar to a number of inductive fallacies (in particular, cherry picking), but, as we shall see, if this is cherry picking, then cherry picking is unavoidable in almost any statistics at all. I'm confident that the following examples will make my point a bit more clear.
Consider:
(5) Blue horses tend to die more than do red ones.
(6) ??I have a blue horse and a red horse (and know nothing else about them). From (5), the blue horse is more likely to die than the red one.
The inference of (6) from (5) seems passable to me, though there are some questions about it. Now, let's change the verb from die to kill.
(7) Blue horses tend to kill people more than red ones.
(I.e., Pr(blue horse kills a person) > Pr(red horse kills a person))
To me, tend to makes the intentionality extremely palpable.
Consider the following preliminary inference:
(8) # There is a blue horse and a red horse (and I know nothing else about them). From (7), the blue one is more likely to kill me than the red one.
It should also be noted that the semantics of tend to are already ambiguous.
For example: Blue horses tend to kill people simply means that blue horses are likely to kill people (whatever likely means): it could mean that each horse kills people often, or some high ratio of blue horses are likely to kill at least once.
The point for now, though, is that something seems very wrong with the inference from (7) to (8), and it seems to depend only on the verb that I've chosen.
Let's look at some other intentional vs. non-intentional inferences:
People who use computers every day tend to get carpal tunnel more than people who don't. Blevin uses a computer every day. Therefore, Blevin is more likely to get carpal tunnel than people who don't use computers.
??People who use computers every day tend to drive poorly more than people who don't. Blevin uses a computer every day. Therefore, Blevin is more likely to drive poorly than people who don't use computers every day.
??People from Podunk tend to be less bright than people from Austin. Blevin is from Podunk, and Lave is from Austin. Therefore, Blevin is probably less bright than Lave.
#70% of people from Podunk steal cars. Blevin is from Podunk, and Lave isn't. Therefore, Blevin is more likely to steal than Lave.
#Neil is black, and Daniel is Asian. Average math scores of Asian-Americans are higher than average math scores of African-Americans. Therefore, Daniel's math scores are probably higher than Neil's.
These naive inferences seem to be less bothersome (and, in some instances, less stomach-turning) when there is no intentionality. It's as though we intuitively know that probabilities are generally not accurate ways of describing individuals.
Perhaps most troubling is that we (including scientists and social scientists) tend to reason in this way, using this kind language. Probability semantics are non-trivial. We seem to intuitively know that referring to human beings as objects is inadequate when pressed to make judgments about these inferences. While, strictly speaking, when starting with a lack of information, each of these troubling inferences seems to have a valid interpretation, they are just that: interpretations. Even probability must be interpreted, and using probabilities to categorize people can be a demeaning, dehumanizing thing to do. This is done in both science and politics.
One might say that this is just a case of a lack of information. If we were to have more information, the probabilities would be more reasonable. This is exactly right, but what probabilities would we use for an individual? The probabilities most relevant, in my estimation, are the probabilities in that individual's history, at least insofar as intentionality is concerned, which permeates nearly every aspect of human existence. Human beings are incredibly complex individuals, and the probability of the group subsumes the probability of the individual in statistical reasoning. Put another way, the complex snowflake that is the individual falls into some lukewarm water and the complexity thereof is altogether missing from the sloppy probabilistic reasoning.
It's more reasonable (though still problematic) to say:
Blevin has made excellent scores on all of this tests. Therefore, his next score will probably be excellent.
It is not acceptable to say: Blevin is a member of a group known for low test scores. Therefore, Blevin's test scores will probably be low.
The most troubling fact of all is that both of these are theoretically valid statistical inferences, depending on the starting point and the paucity of information. We update the probabilities as we gather more information, but the point is, with individuals that have intentionality (i.e., human beings), we create artificial categories as a necessity for making generalizations -- and, as we've seen, the English used when describing the situation has various pragmatic consequences.
In actuality, an individual is a category unto himself or herself. There is always a lack of information, because we must always choose probabilities from a group (an arbitrary category) and apply them to the individual to create the categories by which to use our (flawed) reasoning. This doesn't mean that probability assessments about groups are useless. That would be a ridiculous assertion. It does seem to show some inherently difficulties with attempting to apply Modus Ponens to individuals based on data from a group to which such an individual happens to belong.
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