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I Am the King of France [Jun. 9th, 2016|06:15 pm]
De Horror Vacui
One of the most obtuse discussions I remember from reading Russell ("On Demarcation," which I liked) and the logical positivists (where it got twisted) was the analysis of the sentence, "The King of France is bald." Or, as the positivists became more and more scholastic, "The present King of France is bald." The idea, as I remember it, is to try to understand what a sentence without referents means, which is very important if you're a logical positivist. In this case, since there is no King of France,* how can you determine if he's bald or not? And I just ran into the same problem in reading a 1994 book called Inevitable Illusions by -Massimo Piattelli-Palmarini - although not with the King of France.

This is different than saying the proposition is vacuously true, which is the fact that if you have a false antecedent, your statement is true -- but rather boring. In this case your statement is best viewed as a statement about sets (see Venn Diagrams below). The statement "The King of France is Bald" is true if every member of the set of Kings of France,** of which there can be at most one, is also a member of of the set of bald men. That is, the statement is true if the member of the sets is in the intersection of these two sets (the green region).

You can't really test this proposition unless you make one other determination, or assumption. That is unless you can say that there exists a unique King of France, you cannot ask whether or not he's bald.

And that's the problem for the positivists, because for them the meaning of the statement is the way in which you test whether or not it is true (it's operational definition). And attempts to answer what it means could often turn into a complicated mess.



You could argue that the statement should have the Kings of France wholly within the Bald Men (and I'll do something similar later), but since we're looking for a testable proposition, I think this interpretation is better for positivist purposes.

Inevitable Illusions gives me another problem that benefits from this particular analysis. I was having a problem with a particular syllogism that Piattelli-Palmarini said had a consequent. The syllogism went something like this:

(A) All MPs are thieves.
(B) No composer is an MP.
--------------------------
(C) ?

What (C) can you conclude from (A) and (B)? It could be about anything at all, but it should be logically sound. And again, there is something you can conclude.

Well this stumped me. So I went with my Venn diagrams again, finding something like this:



Some of the different categories I've drawn a little differently, but as you can see there are five regions I'm allowing for -- accepting (A) as true (which is why MPs are completely within the Thieves set here). People who are not thieves or composers, those who are thieves and not MPs and not composers, those who are thieves and composers, those who are composers and not thieves, and those who are MPs. I looked at this, and looked at this and couldn't quite figure out what the author was telling me I should say is true based on (A) and (B).

So, let's insert a population of people and see what comes up.



This shows exactly what Piattelli-Palmarini was getting at: there are people who are not thieves, there are people who are composers who are not thieves, there are people who are MPs and are thieves, and there are people who are composers and are thieves. In fact, there could be people who are not composers or MPs who are thieves, but that would muddle the issue.

What you are supposed to conclude is this: (C) Some thieves are not composers.

But, I would contest that this is no more valid an inference with (A) than without it (that (A) is unnecessary for that). The reason for this is that the existence of MPs hasn't been logically asserted.

It is completely possible that there are no MPs, a situation where the populations will look like this:



In this case there are no MPs, and there are no thieves who are not composers. Without a third line in the syllogism expressing the existence of at least one MP, the belief that there are thieves that are not composers is logically unjustified.

Think of it this way. Change "MPs" to "goblins" in (A) and (B), above. Now, (A) and (B) are missing referents and certainly doesn't imply that there is at least one composer who is not a thief. You have to truthfully assert the existence of the referent, which is a separate logical step. Logic cannot prove the existence of something from positive universals, only its non-existence.

And, of course, the first thing that I thought when I came up with this defense was to think, "is this right?" I mentally went through my old quantification rules, I thought about squares of contradiction, and so on, and it kept coming back to this is the right interpretation. And then Piattelli-Palmarini described a particularly ignorant defense of the Conjunction Effect.

The Conjunction Effect is a kind of error in our probabilistic reasoning. This is related because probability heavily depends on set theory. Let's start with an example (probably from Kahneman and Tversky):

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply converned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.

Rank these statements in order of their likelihood:

(A) Linda is a teacher in elementary school
(B) Linda is active in the feminist movement.
(C) Linda works in a bookstore and takes yoga classes
(D) Linda is a bank teller.
(E) Linda is a social worker.
(F) Linda is a bank teller who is active in the feminist movement.


Usually, people answer this in an order like B > E > A > C > F > D.

And this is objectively wrong. Because F has to be more likely than B, and here's why:



There height of the horizontal bars are proportional to the probability of having different jobs (mutually exclusive), the width of the vertical bars are proportional to the probability of having different hobbies (non-exclusive). The heights and widths are subjective (A > D and D > A are both equally valid judgments), but it's the areas are proportional to the probability. Assuming these exhaust the possibilities, the probability is the ratio of the rectangle that describes a particular situation (teacher who does yoga and pickets) divided by the size of the big rectangle with all the little boxes in it. So to figure out the relative likelihood of B and F, you look at the boxes for being a feminist (no matter what job) and the box for being a feminist bank teller. No matter how skinny the bank teller box is, it will be just as skinny as the feminist bank teller box, but longer.

By defense of the Conjunction Effect, I mean that someone says that this is not an error in judgement. And that person was Gerd Gigerenzer, psychologist. Gerd believed that since people were reasoning about an individual the theory of probability doesn't apply. In particular, he took refuge in the frequentist interpretation of probability, which says that the meaning of a probability statement is the long term average of results from repeated experiments.*** If you're reasoning about an individual, then you don't have the repetition that's required for a probability statement to make sense.

That is, we only have one Linda, so saying that Linda has a 25% change to be a grade school teacher, 5% chance to be a bank teller, a 50% chance to be a social worker, and a 20% chance to be a book clerk. She's one of them, period.

However, you could say that about any one of the composers in the previous example. Either that composer is a thief or that composer is not a thief, and knowing that there are as many composer-thieves as not isn't something you should use in deciding whether or not the particular composer is a thief.

And if you did, you'd lose a lot of stuff to composers -- they've done the test (with lawyers and doctors), and the cognitive biases do cause errors in forecasting.

Although each person is a data point, we can reason probabilistically about them by taking the things we know about them, we can move the lines on, say, the different possible jobs they could have and make a prediction about what it is they do for money. And so on. We cannot repeat the individual experiment, but we can repeat the prediction process to see how well our models work (or how badly they fail).

So, some implications require the founded assumption that a member of a set exists -- the definition of a set or an implication about its members in insufficient. But when we do have members of a set that could be part of multiple subsets (such as Linda), we can use these definitions and relationships to estimate a probabilities for the member to be in each of them.

* Or, rather, since there is "presently no King of France with executive authority recognized by the constitution of the Republic of France." Which is why the discussions started getting obtuse.

** See (*).

*** Right now, I think that even Bayesianism doesn't escape this problem: in the end, the number that is found in Bayesian decision theory still needs an objective interpretation of probability to interpret its meaning.
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