Tuesday, March 4, 2014

The Monty Hall Problem and Three Common Fallacies Among "Relevant" People

There are three common fallacies used in the rhetoric of people who like to be "relevant" in this day, age, or culture: Ad baculum (appeal to force of punishment, i.e. to the "stick"), ad ignominiam (appeal to shame), and ad populum (appeal to the people/masses).

I bring this up because I've seen these three fallacies many times, and they've popped up a few times in the last month from either atheists that I know or are relatively well known.

The appeal to force of punishment (ad baculum) refers to when a person coerces another to discredit a belief on the grounds of the possibility of impending punishment for holding that belief. A popular example of this fallacy would be Henry VIII's coercion of St. Thomas More to support the king's divorce against the decision of the pope. Henry VIII threatened dissidents with the death penalty. A way this fallacy is used against believers or those who hold to "outmoded" views is to point to the consequences of holding such belief, such as loss of social status, job, or other aspects of one's reputation—or even life.

The second fallacy is very similar to the ad baculum; the ad ignominiam fallacy refers to the shame that follows for holding a belief. Thus this fallacy focuses on a specific consequence, which could be interpreted as a form of social punishment, of holding to an "outmoded" or discredited view. E.g. "It's 2014 and there are still people who believe in x, y, or z?" "What will they think of you?" "Don't be naive." "You'll be a laughing stock!" This fallacy was used in the early '90s by literally thousands of people who held Ph.D.s in mathematics against Ms. Marilyn vos Savant, who became famous for holding the world's highest recorded IQ in the Guinness World Records during the late '80s, when she attempted a response to the "Monty Hall Problem," a statistical paradox that was introduced in 1975 and was made famous when vos Savant gave her reply to the problem.

The "Monty Hall Problem" was also made a bit more famous by the TV show Numb3rs when it was featured in one of their early episodes. You can look up the problem via a Google search, and there are several possible ways of calculating the solution (see the Wikipedia article that covers the entire problem: http://en.wikipedia.org/wiki/Monty_Hall_problem), but the solution is this: if you switch doors after the host has opened a non-car door, you double your chances of winning. There even was a guest comment on this post below trying to prove to me that the chances are 50/50 no matter what (I've since deleted the post since it contributed nothing useful and in fact only gave further proof of how non-intuitive the Monty Hall problem is, no matter how pseudo-mathematical random people try to be) even though my point in this post is not to defend the validity of the solution of the Monty Hall problem but to point out the logical fallacies that were committed by Ph.D.s regardless of whether they are right or wrong regarding the answer. Here are some of the responses from the Ph.D.s (source: http://marilynvossavant.com/game-show-problem/) with my commentary following each response:
Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful.
Robert Sachs, Ph.D.
George Mason University 
The implication of this response is that vos Savant is part of the "general public's lack of mathematical skills," of which she ought to be ashamed.
You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!
Scott Smith, Ph.D.
University of Florida 
Shame! Shame! Shame! SHAME! No further comment needed.
Your answer to the question is in error. But if it is any consolation, many of my academic colleagues have also been stumped by this problem.
Barry Pasternack, Ph.D.
California Faculty Association
I.e. any consolation to the shame of being so publicly humiliated.
You’re in error, but Albert Einstein earned a dearer place in the hearts of people after he admitted his errors.
Frank Rose, Ph.D.
University of Michigan 
I.e. be like Albert Einstein, humble and apologetic, in order to neutralize the shame of being wrong.
I have been a faithful reader of your column, and I have not, until now, had any reason to doubt you. However, in this matter (for which I do have expertise), your answer is clearly at odds with the truth.
James Rauff, Ph.D.
Millikin University 
This response was actually an ad baculum since there really was no purpose in telling vos Savant that he was a faithful reader except to imply that her shameful mistake shook his "faith" in her. The implication, therefore, is: if you do not correct your error, I will "punish" you by not reading your column.
May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?
Charles Reid, Ph.D.
University of Florida
The implication of this "suggestion" is that vos Savant has the shame of making a very basic error in probability that a "standard textbook" could have helped her avoid.
I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns.
W. Robert Smith, Ph.D.
Georgia State University 
This response attempts to instill a sense of shame by stating that vos Savant could afford help, not from the Ph.D. who is writing to her, but from measly high school and college students; i.e. she is at "their" level, not the level of the Ph.D. holder who doesn't have time for people like vos Savant—except the time to send a letter to instill shame and be condescending.
You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?
E. Ray Bobo, Ph.D.
Georgetown University 
This doctor attempts to instill shame by explicitly including vos Savant within the "deplorable situation" of mathematical knowledge in the US. It is also an ad baculum: "IF" she admits her error, then all is good; if she doesn't, then she should be totally discounted. Furthermore, another attempt to instill shame comes by asking how many "irate mathematicians" are needed (to change a lightbulb?) to change vos Savant's mind as if their anger were worth anything...except to instill shame.
I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake.
Kent Ford
Dickinson State University 
This response (the responder didn't have a Ph.D. but appealed to those who had one) is both an ad baculum, an ad ignominiam, and an ad populum (see below); the appeal to force of punishment is the implication that vos Savant's gross error caused "shock" in the reader; therefore she should admit her error so that the reader may be at peace again. The appeal to shame and the appeal to the Ph.D. "masses" comes from referring to how vos Savant is still adamant despite being corrected by three (and more) professionals.
You made a mistake, but look at the positive side. If all those Ph.D.’s were wrong, the country would be in some very serious trouble.
Everett Harman, Ph.D.
U.S. Army Research Institute
Yes, the country is in some very serious trouble, not only from the fact that nearly a thousand Ph.D.s were mistaken, but that those people utilized multiple fallacies that attempted to instill shame and had no reference to solving the problem in all simplicity. Some of these comments were sexist (predictably so); others were outright personal insults (one person called vos Savant "the goat," playing off of the featuring of goats in the problem and, perhaps unconsciously, the idea of a scapegoat, a symbol fraught with connotations of shame and victimization). They were appallingly bad at mathematics AND logic.

As Dr. Peter Kreeft has remarked, "What does it take to believe in the 100 most absurd ideas ever conceived? You must have a Ph.D."

The final fallacy is the ad populum, which is the appeal to the masses or the people. It attempts to discredit a belief simply by pointing to what most people happen to believe or have believed. "Most Catholics practice/support contraception/divorce/abortion/legalized homosexual unions; therefore the Church is wrong in maintaining the opposite standpoint." "Most Catholics don't believe in the Real Presence; therefore, etc." "Who believes that today?" "That is an outmoded/ancient/medieval/irrelevant notion that we have rejected today." "You're on the losing side (of history/of the masses/of intellectual superiority/of the growing political movement etc.—this slogan is often used especially for things like gay marriage, abortion, contraception, radical feminism, and other sexually-related and/or gender-related political movements)." "It's 2014; who still believes in that today?" (Yes, this type of statement/rhetorical question is both an appeal to shame and the masses, i.e. the current masses of 2014 or whatever year it is).

Was Christ "relevant"? Well, He came in the fullness of time and offered us exactly what we need always: salvation—that is relevant. But remember what the Pharisees said: "If you let this man go, you are no friend of Caesar's!" (John 19:12).

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