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March 12th, 2019 (Permalink)

Three-Dice Monty

Last month, "Three-Card" Monty*, the carnival con artist, shamelessly attempted to swindle a young couple on Valentine's Day using a pair of tricky dice! Now, he's back in his booth on the midway, but this time he has three strange dice.

Imagine that you approach Monty's booth and see on the counter before him three cubical dice: one colored red, the second white, and the third blue.

"As you can plainly see," Monty says to you, "these are three unusual dice. As you probably know, a standard die has one through six dots on its six faces. The number of dots on these dice range from one through nine, instead. Moreover, each of these nine numbers appears twice, both times on the same die. The numbers are distributed in such a way that each die has different numbers on its faces than the other two, so no number is shared among them. This means that when two of the dice are rolled against one another, they never tie. In every other way these dice are the same as standard dice: they're not loaded or gimmicked in any way.

"The first die has the numbers two, four, and nine, twice each on its six red faces. The second, white die has, instead, the numbers one, six, and eight. Finally, the last, blue die has the numbers three, five, and seven. The numbers on each die total thirty.

"Wouldn't you agree that they are evenly matched? I offer you an even money bet that the die I select will beat yours. That is, each of us will put up a dollar, each will roll his die, and the die that comes up with the highest number wins. Moreover, to be perfectly fair to you I offer you the first choice: you may select any one of the three dice and I will take one of the remaining two. What could be fairer than that?

"So, my friend," Monty finished, "which die do you choose?"

Is the bet that Monty is offering you fair? Should you accept or reject his wager? Assuming that you accept it, which die should you choose?

* In case you don't know Monty: he is a trickster, but he always speaks the exact truth. However, this does not mean that he necessarily tells the whole truth. Also, while he is a sharpster, he prides himself in not doing any sleight-of-hand. For previous puzzles involving Monty, see:

March 8th, 2019 (Permalink)

Rule of Argumentation 41: Be as definite as possible!

Before proceeding to this month's rule, I want to mention something that I should have explained earlier in this series, probably in the introduction: Each of these rules is a heuristic or "rule of thumb". In other words, there are exceptions to all of them, that is, situations in which you should not follow them, but such situations are exceptional.

Previous rules in this series were rules that governed the overall process of arguing: appealing to reason, acknowledging one's own fallibility, and focusing on arguments themselves rather than arguers. This is the first rule that deals with the content of argumentation. It says that the claims and arguments you make should be as definite as possible.

In order to be as definite as possible myself, I will explain what I mean by both "definite" and "possible":

Next Month: Rule 5


  1. Previous entries in this series:
  2. Unfortunately, there doesn't appear to be a synonym for "unambiguous" lacking a negative prefix, so I choose the ambiguous "definite" so as to remain as affirmative as possible in this rule.
  3. For more on the types of ambiguity, including subtypes and examples, see the entry for the fallacy of Ambiguity.

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February 28th, 2019 (Permalink)

Recommended Reading

February 27th, 2019 (Permalink)


Q: "I've had an idea for a web/news/facebook plugin that I think you might be interested in investigating. I've thought about this for years, but it's not something I plan on doing (even though I'm a computer science guy). It would be a plugin/app that can analyze text and detect if a logical fallacy is being used. This is a pretty difficult problem to solve from a computer science standpoint in terms of analyzing text and correctly identifying a fallacy.

"I don't think anything like this exists yet. It would be an amazing feature for message boards/etc. that could automate down-voting of nonsense based on a logical fallacy 'score'.

"I'm curious if you've ever explored this topic or found anything that actually already does this? I just want to be able to read things on the internet without constantly parsing through all the logical nonsense!"―Matt Brunelle

A: Let me begin by pointing out that my own work in artificial intelligence ended over fifteen years ago, and I haven't made an effort to keep up with developments in the field. That said, I think you're right that nothing close to an automagical fallacy-checker currently exists. In this answer to your question, I want to pursue the reasons why such a program does not yet exist, and suggest some ways that progress could be made towards it: What is it that is holding us back, and how can we take at least some baby steps forward?

Fallacy detection, whether done by human or by computer, is not a single simple task, but a series of tasks. This suggests that one way in which progress can be made towards the goal is divide and conquer. Let's break down the task of checking natural language1 arguments for fallacies into its subtasks2:

  1. Identifying Arguments: Fallacious arguments are, of course, arguments, so the first subtask is identifying arguments. I'm not aware of any current program that does this in natural language, but I think it should be possible. By using argument indicator words and phrases, a useful though by no means perfect accuracy could be achieved without the need for any major breakthroughs.
  2. Identifying Premisses and Conclusions: Having identified an argument, the next step is to analyze its internal structure at the level of premisses and conclusion. After all, the difference between a valid modus ponens and a fallacious affirming the consequent is a difference between what is the premiss and what the conclusion. By dividing argument indicators into premiss- and conclusion-indicators, I think this could also be done today with an imperfect but helpful level of accuracy.

    At this point, let me note that mechanizing fallacy detection will be much easier for formal fallacies than for informal ones since, as suggested by their name, formal ones can be identified on the basis of logical form alone. As indicated above, the difference between formally-valid arguments and formal fallacies is often a difference between what play the roles of premiss and conclusion. So, the ability to identify premisses and conclusions of arguments puts us within striking distance of identifying formal fallacies.

    For the purposes of this answer, I will continue to discuss here only the easier task of identifying formal fallacies, since informal ones open up a Pandora's box of additional problems. Moreover, there is a problem that enters at this point which will make even the detection of formal fallacies difficult to accomplish:

  3. Identifying Enthymemes: In natural language, arguments often occur as enthymemes, that is, arguments in which either a premiss or the conclusion is left unstated. Thus, to automate fallacy detection, we would need to accomplish an additional task of identifying enthymemes. This is a difficult task even for people to do since there are no enthymeme indicator words and phrases to help.

    This is a stumbling block: A fallacy-checker that fails to recognize that arguments have suppressed premisses would identify such arguments as formally-invalid since they are missing a premiss. My own field research in identifying and collecting specimens of formal arguments shows that most are enthymemes; rarely are they fully stated. As a result, a formal fallacy checker without an enthymeme-recognition module would be subject to more false alarms than true ones, and thus would probably be worse than useless.

  4. Identifying Suppressed Premisses and Conclusions: Identifying that an argument is missing a premiss or conclusion is only a preliminary step to this one, namely, supplying the missing part. However, it is no longer enough to simply recognize the forms of arguments; the fallacy-checker must understand what the premisses and conclusion are about, that is, their content. So, such a program must understand natural language.

    In addition, it requires common sense knowledge, and the ability to identify shared contextual knowledge. The reason premisses are left unstated is often that they are obvious because they are either common knowledge, or knowledge shared in the context of argumentation. So, a fully-functioning fallacy-detector would be able to understand natural language, and share our common sense and contextual knowledge. In other words, it would be a fully-functioning artificial intelligence.

If the above analysis is correct, the task of automating fallacy-checking amounts to that of creating a full-fledged AI. No wonder we're not there yet! However, that's no reason not to take steps 1 and 2 above; in fact, it's an additional reason to pursue them. Moreover, an automated argument-analysis tool would make parsing through all the logical nonsense we are confronted with a little bit easier for us humans.


  1. Natural languages, such as English, are those that arose naturally in human social groups communicating, as opposed to artificial languages, such as Esperanto, that are created intentionally by an individual or group for some specific purpose, such as international communication.
  2. This is the same series of tasks that a person needs to do to check natural language for fallacies.

February 14th, 2019 (Permalink)

A Valentine's Day Puzzle

"Three card" Monty1, that mind-manipulating mountebank, is back on the midway. A young couple, arm-in-arm and obviously deeply in love, were walking past his booth. "Step right up, sir, and win a prize for your young lady," he called out to the couple, who approached him.

"I have here a pair of dice," Monty said, holding out his hands, palms up, for the couple to see lying on each palm a plastic cube with numbers on its six faces. "As I'm sure you know, a standard die has the numbers 1 through 6 on its faces, for an average2 value of three-and-a-half points a roll. The dice that I am holding are non-standard, with different numbers on their faces such that none are shared by both dice, making ties impossible when they play against one another. Even though these dice are numbered differently from standard ones, they are in every other respect fair dice; they're not loaded or rigged in any way.

"As you can plainly see," Monty continued, "the die in my left hand is numbered with even numbers: sixes on four of its faces and zeroes3 on two for an average of four points per roll. The right-hand one has odd numbers: ones on four of its faces, but sevens on the remaining two, for an average of only three points a roll.

"To play the game, you will select a die and I will take the other, then we will roll against each other and the one who rolls the highest number of points wins.

"Before you choose," Monty continued, "I offer you an even money bet that my die roll beats yours. That is, I will bet you dollar to dollar that the number on my die will be higher than that on yours: if it is, I win a dollar; if not, you win one. Does that seem fair?"

The young man smiled broadly and said: "Very fair."

Can you help the young couple decide which die to choose?



  1. In case you don't know Monty: he is a trickster, but he always speaks the exact truth. So, what Monty said about the dice is the truth and nothing but. However, this does not mean that it's necessarily the whole truth. Also, while he is a sharpster, Monty prides himself in not doing any sleight-of-hand, which means that he would not switch the dice or engage in any other legerdemain. For previous puzzles involving Monty, see:
  2. Throughout this puzzle, "average" refers to the mean value, that is, the total number of points on the die's faces divided by six.
  3. Zero is not usually referred to as an even number, but it fits the usual definition of an integer evenly divisible by two.

February 13th, 2019 (Permalink)

Rule of Argumentation 31: Focus on claims and arguments!

The way I usually like to put this rule is: "Keep your eye on the ball!". Unfortunately, most people won't know what "the ball" is and others may not understand the sports metaphor.

As I pointed out in the Introduction to this series, the most general rule of relevance in argumentation is simply: Be relevant! However, it doesn't help much to tell someone to be relevant if they don't know what's relevant. In argumentation, what's logically relevant relates to what I'm calling here "the ball".

Imagine that an argument is like a tennis game2. Your goal is to hit the ball over the net, not to hit the other player with it, let alone to hit him with your racket. Another good sports-related way to put this rule is: "Play the ball, not the man!" This means to keep your eye on the ball and don't get distracted by the other player.

What is "the ball" in the analogy of argumentation to a tennis game? It is the topic or subject being debated, that is, the claim or claims that the arguers think they disagree about3. In formal debates, there is usually an explicit proposition or resolution that is the topic of the debate, which is "the ball" to keep your eye on. However, the kind of informal debates that most of us engage in most of the time lack a specific topic, which is a major source of difficulty.

Before you can keep your eye on it, you first have to spot the ball. Whenever an argument begins, you should ask yourself: What are we arguing about? If you're not clear about this―and I suspect that much of the time you won't be―how can you expect the other player to be? Often, the players involved don't even agree on what they're arguing about, which is like trying to play a game of tennis with two balls, with each player trying to hit a different ball. Even if you think you know what it is, you should ask the player on the other side what you disagree about. Many an argument is resolved at this stage when the arguers discover that they really don't disagree. However, if you skip this stage, it's possible that the argument may continue indefinitely, with both arguers arguing past each other. Such arguments are frustrating and can easily lead to bad feelings between the arguers, even though they don't actually disagree!

One underappreciated achievement of argumentation is the discovery that you don't substantively disagree with the other side; rather, you just express the same view in different words. However, the only way you will discover this is if you play the ball―the claim or claims you seem to disagree about―and not the other player. If instead of aiming at the ball, you try to hit the other player, you will only make the disagreement between you worse.

It's hard to resist the temptation to play the man instead of the ball when the other player is trying to play you. You may naturally feel that you have to defend yourself by replying in kind. Unfortunately, if the other player won't argue cooperatively, there may not be much that you can do except to refuse to play with such a person. This is another way in which cooperative argumentation is like playing tennis: it takes two willing partners.

In future entries, we'll look in more detail at ways you can lose sight of the ball and end up playing the man, instead. For now, keep in mind that the goal of this game is not to defeat your opponent, but to use arguments to put claims to the test.

Next Month: Rule 4


  1. Previous entries in this series:
  2. Many games with a ball and two players or teams would work, so feel free to substitute your favorite.
  3. At least for the sake of the argument: some arguers play Devil's advocate.

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