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).

Fr. Antonio Royo Marin's Commentary on the Hail Mary, pt. 7: "Jesus"

Jesus

This word was added by the Church to the end of the first part of the Hail Mary because it is effectively the Name of the blessed fruit of Mary: Jesus.

The Gospel of St. Luke tells us that "after eight days were accomplished, that the child should be circumcised, his name was called Jesus, which was called by the angel, before he was conceived in the womb" (Lk 2:21).

Etymologically, the name of Jesus means "the salvation of YHWH," which the angel hinted at when revealing to St. Joseph the mystery of his virgin spouse: "And she shall bring forth a son: and thou shalt call his name Jesus. For he shall save his people from their sins" (Mt 1:21).

St. Thomas Aquinas argues for the fittingness of the name of Jesus given to the Savior of the world since names ought to correspond to the internal essences of things, reminding us of their natures. Since we know that He is the Savior of the world, no name is more fitting than Jesus (cf. Summa Th., III, q. 37, a. 2; http://www.newadvent.org/summa/4037.htm#article2).

Fr. Luis de León [1527–1591] ends his immortal work The Names of Christ mostly explaining all the other meanings applicable to which the name of Jesus encompasses and summarizes. St. Bernard has commented on this name with his usual devotion and tenderness in his Sermons on the Canticle of Canticles, and the Church uses his other sermons in the liturgical Office for the feast of the Most Holy Name of Jesus, but these inimitable pages truly ought to be read in their entirety.

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Source: Fr. Antonio Royo Marín, La Virgen María, trans. by R. Grablin (Madrid, Spain: BAC, 1996), 455.

Fr. Herve de la Tour Summarizes St. Thomas Aquinas's Philosophy of Teaching

The better is the way of independent investigation, which he calls "discovery." It is remarkably illustrated in the exploits of gifted children who teach themselves to read or, like the three-year-old Mozart, to play a musical instrument before having had any instruction. We all employ this method in less spectacular fashion when we acquire some store of knowledge or some skill through our own experience and effort. This procedure not only manifests greater intellectual power in the learner, St. Thomas thought, but is also more perfect. For we learn in this case through an immediate contact with the realities in question, whereas, when we are taught, the teacher’s "signs" (generally verbal ones ―illustrations, explanations, etc.) intervene and, at best, point us toward those realities. It is a rare talent, nonetheless, that can wholly dispense with a teacher’s help and to do so is, in any case, time consuming. So that the chief value of this second way of learning, that is to say, learning-through-teaching, is one of economy. Most men would have neither the leisure nor the courage to learn all they need to know if teachers did not ease and accelerate the process for them. St. Thomas Aquinas and Education by John W. Donohue, S.J. (Random House, 1968).

St. Thomas’s recommendation makes clear the primacy of that personal way of discovery. The teacher, he says, should pattern his method after the one naturally used, when the student learns by himself. This is based on the general principle that whenever an effect can be produced either by natural process or by artificial method, the method should as much as possible be the same as that of nature. The great medieval scholar stresses the difference which exists between a work which can only be produced by artificial means (for instance the construction of a machine) and a work which can also be produced by nature (for instance the growth of a plant). Teaching belongs to the second category. A teacher will strive to assist and not replace the natural energies of his student. He will not be like an engineer but more like a farmer who helps the growth of the tree through watering, pruning, weeding. The principal cause of growth is the natural vigor of the tree, the farmer is only assisting it.

We have been victims since Descartes of an exaggerated emphasis on "methods." The student has been viewed with a mechanistic perspective. Teaching is synonymous with trying to cram information into the student as if he was a passive machine instead of a living mind. St. Thomas is quite opposed to this caricature of true teaching. He explains that the procedures of teaching will be most effective when patterned after those of independent search. A good teacher will make the difference between learning by discovery and learning by instruction as narrow as possible. In other words he will assist his students in their learning so that their experience will seem to them a discovery. Some great teachers really have this gift. Their classes are so good that you‘re constantly finding out new things and getting excited about them. The challenge and the enthusiasm of learning are kept alive by the evident love that these men have for their subject. The teachers cannot but communicate to their students who come out of the classroom with a desire to learn more.

Maybe we could use another illustration from St. Thomas to understand better his conception of the teacher’s role:

The example is that of a physician ministering to a man brought down by an infection. In many such cases, if the patient went unattended, his body would mobilize its restorative forces and eventually heal itself. The art of the teacher is like that of this internist, since all the teacher can hope to do is to strengthen the student’s resources and facilitate their exercise. The import of the analogy is clear enough whether it is medically accurate or not. Whenever true learning occurs, its principal cause is the learner himself. In St. Thomas’s terminology the teacher is called a secondary and instrumental cause ―helpful but not indispensable. He cannot transfer his own knowledge to the student but only help him achieve similar learning for himself. St. Thomas Aquinas and Education by John W. Donohue, S.J. (Random House, 1968).

We see that we are far removed from the modern conception of teaching where the teacher exclusively relies on artificial means (workbooks and other devices) and slowly loses sight of his true function: to stimulate the living mind of the student, to guide the development of his intellect. The emphasis on "measuring" education with grades obtained through tests is partly a result of this false philosophy. Knowledge according to St. Thomas is a quality of the living mind and not a quantity of memorized information.

Let us delve a little deeper in the matter and see how the teacher will assist the student to acquire knowledge. We will take the example of geometry. (Geometry is very good for training the mind to think in a logical fashion.) This science uses deductive reasoning to establish a body of demonstrated conclusions from a few accepted premises. It is the teacher’s job to provide the student with a good understanding of the first principles of geometry. He will also help the student to sharpen his skill in solving problems by explaining to him the basic process when the pupil has a difficulty, he will give him examples, comparisons, schemas, etc….which will be so many "pointers" leading the pupil to grasp this or that particular truth. When you are in front of a wall too high for you to leap, you may need the assistance of someone to point out a ladder which you had not seen. The teacher’s art consists in employing language so skillfully that his student will go beyond the words to the realities they signify.

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Source: Fr. Herve de la Tour, "St. Thomas On Teaching," Edocere, accessed March 2, 2014, http://www.edocere.org/articles/st_thomas_on_teaching.htm.

St. Thomas Aquinas's Philosophy of Teaching in a Nutshell

Just as a person may be cured in a twofold manner, through the operation of nature alone or through nature with the aid of medicine, so there is a twofold manner of acquiring knowledge, the one when the natural reason of itself comes to a knowledge of the unknown, which is called "discovery," the other when someone extrinsically gives aid to the natural reason, which is called "instruction." Now, in those things which are done by nature and art, art works in the same way and by the same means that nature does, for just as nature in one suffering from cold induces health by warming him, so does the doctor. Hence, art is said to imitate nature. Similarly, it happens in the acquisition of knowledge that the one teaching leads another to a knowledge of the unknown in the same way as he (the learner) would lead himself to a cognition of an unknown in discovery. Now, the process of reason in one who arrives at a cognition of an unknown in discovery is the application of general, self-evident principles to definite matters, and proceeding from them to particular conclusions and from these to others. Hence, and according to this, one man is said to teach another because the teacher proposes to another by means of symbols the discursive process which he himself goes through by natural reason, and thus the natural reason of the pupil comes to a cognition of the unknown through the aid of what is proposed to him as with the aid of instruments. As, then, a doctor is said to cause health in a sick person through the operation of nature, so man is said to cause knowledge in another through the operation of the learner's natural reason—and this is to teach.

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Source: St. Thomas Aquinas, De Magistro, trans. by Mary Helen Mayer, a. 1.

Fr. Antonio Royo Marin's Commentary on the Hail Mary, pt. 6: "And Blessed Is The Fruit of Thy Womb"

And blessed is the fruit of thy womb

1. Exegesis. These words of St. Elizabeth reveal their full significance in the words that immediately follow: "And from where is it that the Mother of my Lord should visit me?" (Lk 1:43). Through the internal inspiration of the Holy Ghost (cf. v. 41), St. Elizabeth knew clearly that she was in the presence of the Mother of the Lord, or the Messiah announced through the prophets, which Mary carried within her womb. Hence she humbled herself before Mary, wondering exceedingly that Mary, who was not Elizabeth's neighbor but the "Mother of the Lord," should visit her. With good reason then has an excellent contemporary exegete said that "because the Holy Ghost at this moment reveals to Elizabeth the mystery of Nazareth, it seems that she did not discover this mystery only partly but was, rather, instructed about the messianic and divine prerogatives of the Son of Mary."

2. Theology. Here is the theological commentary of St. Thomas [(Expositio salutationis angelicæ, article 3: "Blessed is the fruit of your womb"; trans. by Joseph B. Collins (New York 1939); available here: http://dhspriory.org/thomas/AveMaria.htm)]:

The sinner often seeks for something which he does not find; but to the just man it is given to find what he seeks: “The substance of the sinner is kept for the just” [Prov 13:22]. Thus, Eve sought the fruit of the tree (of good and evil), but she did not find in it that which she sought. Everything Eve desired, however, was given to the Blessed Virgin. 

Eve sought that which the devil falsely promised her, namely, that she and Adam would be as gods, knowing good and evil. “You shall be,” says this liar, “as gods” [Gen 3:5]. But he lied, because “he is a liar and the father of lies” [Jn 8:44]. Eve was not made like God after having eaten of the fruit, but rather she was unlike God in that by her sin she withdrew from God and was driven out of paradise. The Blessed Virgin, however, and all Christians found in the Fruit of her womb Him whereby we are all united to God and are made like to Him: “When He shall appear, we shall be like to Him, because we shall see Him as He is” [1 Jn 3:2]. 

Eve looked for pleasure in the fruit of the tree because it was good to eat. But she did not find this pleasure in it, and, on the contrary, she at once discovered she was naked and was stricken with sorrow. In the Fruit of the Blessed Virgin we find sweetness and salvation: “He who eats My flesh... has eternal life” [Jn 6:55]. 

The fruit which Eve desired was beautiful to look upon [Gen 3:6], but that Fruit of the Blessed Virgin is far more beautiful, for the Angels desire to look upon Him: “You are beautiful above the sons of men” [Ps 44:3]. He is the splendor of the glory of the Father. 

Eve, therefore, looked in vain for that which she sought in the fruit of the tree, just as the sinner is disappointed in his sins. We must seek in the Fruit of the womb of the Virgin Mary whatsoever we desire. This is He who is the Fruit blessed by God, who has filled Him with every grace, which in turn is poured out upon us who adore Him: “Blessed be God and the Father of our Lord Jesus Christ, who has blessed us with spiritual blessings in Christ” [Eph 1:3]. He, too, is revered by the Angels: “Benediction and glory and wisdom and thanksgiving, honor and power and strength, to our God” [Rev 7:12]. And He is glorified by men: “Every tongue should confess that the Lord Jesus Christ is in the glory of God the Father” [Phil 2:11]. The Blessed Virgin is indeed blessed, but far more blessed is the Fruit of her womb: “Blessed is He who comes in the name of the Lord” [Ps 117:26].

Let us listen now to the commentary of St. Bernard (Hom. 3 in laud. Virg. Mat., n. 5-6):

"Blessed are you among women." To these words, which St. Elizabeth spoke, we ought to add: "And blessed is the fruit of your womb." It is not because you are blessed that the fruit of your womb is blessed as well, but because He has come to you with the blessings of sweetness, you are therefore blessed. Truly is the fruit of your womb blessed, for in Him shall all nations be blessed; and from His fullness you also have received as we have as well but you in a way far more excellent than we. Thus not without reason are you blessed, blessed among women; further blessed is He, though not among men, nor angels precisely, but as the Apostle says, "above all things, God blessed for ever" (Rom 9:6). We usually speak of blessed men, blessed women, blessed earth, and to speak of other creatures which we know to have received a blessing, but singularly blessed is the fruit of your womb, being He Who is above all things, God blessed for ever. 

Blessed, then, is the fruit of your womb. Blessed in His fragrance, blessed in His savor, blessed in His beauty. 

a) It is the fragrance of this sweet-smelling fruit of which it has been said [by Isaac]: "This is the odor of my son as of a field blessed to the full by the Lord" (Gen 27:27). Is He not truly blessed to whom the Lord has bestowed His blessings? 

b) Of the savor of this fruit, one who has tasted it would say: "Taste and see how sweet is the Lord" (Ps 33:9); and in another place: "How great, O Lord, is the abundance of Your sweetness, which you have hidden and reserved for those who fear You!" (Ps 30:20). And in another place as well: "If you could only taste and see how sweet the Lord is" (1 Pet 2:3). The Fruit Himself spoke of Himself, inviting all: "Those who eat Me shall hunger even more, and those who drink Me shall thirst even more" (Sir 24:29). He was speaking of the sweetness of His savor, which when tasted excites the appetite even further. This good fruit is food and drink in this life for souls who hunger and thirst for righteousness and sanctity. 

c) You have heard of His odor, of his savor. Listen also regarding His beauty. For if it is true that the fruit of death was not only sweet to eat but also, according to the testimony of Scripture, agreeable to behold, how much more ought to be the vivifying beauty of this living fruit, upon which, as Scripture equally testifies, the angels desired to look? Contemplating this beauty in spirit and desiring to see the same in the flesh the psalmist has said: "Out of Sion comes the splendor of His beauty" (Ps 49:2). And so that you may not think that he was praising some merely ordinary beauty, remember what you have heard written in another psalm: "You surpass the beauty of all the sons of men; grace is poured upon your lips; thus God has blessed you for ever" (Ps 44:3).

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Source: Fr. Antonio Royo Marín, La Virgen María, trans. by R. Grablin (Madrid, Spain: BAC, 1996), 453-455.

John Deely on Romanticizing Nature

MT: I guess part of this semioethical realization concerns what real-world consequences all our human actions have, and to what extent the human world practically – ehm – messes with a natural world that is non-human.

JD [John Deely]: Did you say ‘messes’?

MT: In search of a better word – ...or interferes in a non-human world. Isn’t part of the realization that there is not a sharp distinction between what concerns humans and what concerns others than humans?

JD: Yes, but when you start to speak about the human world as ‘messing with’ the world of nature, you’re coming from an almost Rousseauean, romantic idea of nature: that nature is what we need to just leave alone. And that’s completely naive, because we’re part of nature. They’re talking about – you know, it’s so criminal that we would eat the flesh of animals, et cetera. I like the cartoon – I think it was in The New Yorker, where there are these three characters who are very ecologically friendly, and there’s a couple of lions that are trying to eat them. And they’re up in a tree, and then two of them say to the third one: "Would you please explain to this guy [the closest lion] how much we’ve done for him?"

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Source: John Deely, interviewed by Morten Tønnessen, "Tell Me, Where Is Morality Bred? The Semioethics Interviews I: John Deely," Hortus Semioticus 4 (April 7, 2009): 56, accessed March 1, 2014, http://www.ut.ee/hortussemioticus/4_2009/pdf/hortussemioticus_4_deely.pdf.

The Unity of the Speculative and Practical Intellect

[St. Thomas Aquinas] made the – really quite brilliant – [observation that] what distinguishes human understanding is precisely the ability to come to know things as they are. And it’s from the knowledge of the way the things are that practical knowledge follows. So there’s not just a split between speculative and practical, there’s a sequence: The human intellect comes to understand things, and on the basis of what it understands becomes able to exercise control over what is understood. [1]

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Footnotes:

1. Aquinas’ phrase that “the speculative intellect by extension becomes practical” (“intellectus speculativus per extensionem fit practicus”) is from Summa theologiae, Prima Pars, Quaestio 79, Articulus 11, sed contra, cf. Aquinas c1252-73, and is a paraphrase of Aristotle’s statement (De anima, II, 10 433a 15) that “the practical intellect [...] differs from the speculative intellect by the end at which it aims” (“(intellectus) speculativus differt a practico, fine”), cf. Aristotle c350 AC.

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Source: John Deely, interviewed by Morten Tønnessen, "Tell Me, Where Is Morality Bred? The Semioethics Interviews I: John Deely," Hortus Semioticus 4 (April 7, 2009): 54, accessed March 1, 2014, http://www.ut.ee/hortussemioticus/4_2009/pdf/hortussemioticus_4_deely.pdf.