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July 4th, 2025 (Permalink)

The Fourth of July Picnic

At the Logicians' Club July Fourth picnic, hot dogs were served. Three condiments were available for the frankfurters: mustard, ketchup, and pickle relish. Of the forty-one people who attended the picnic―logicians and their families―all used at least one condiment on their hot dogs, except for one little boy who ate his plain. Twenty-six attendees put mustard on their hot dogs, sixteen used ketchup, and eighteen added relish. Five picnickers put both mustard and ketchup on their frankfurters, six had both ketchup and relish, while three had all three condiments.

Question: How many people had both mustard and relish but no ketchup on their hot dogs?

Disclaimer: The above puzzle is a work of fiction: no one would ever put both mustard and ketchup on a hot dog.

Proof
July 2nd, 2025 (Permalink)

New Book: Proof

Quote: "Life is full of situations that can reveal remarkably large gaps in our understanding of what is true and why it's true. This is a book about those gaps. It is the story of the ideas that have helped scientists and societies discern between truth and falsehoods, improving decision-making and reducing dangerous errors. From medieval juries to modern scientific revolutions, it is about the methods people have used to accumulate evidence, negotiate uncertainty, and converge on proof. And, crucially, what happens when those methods fail."1

Title: Proof

Subtitles: This book has different subtitles for its United Kingdom and United States editions:

  • UK: The Uncertain Science of Certainty2

    Comment: This is certainly an intriguing subtitle, but I'm uncertain what it means. The science of certainty would seem to refer to formal logic, but what's uncertain about it?

  • US: The Art and Science of Certainty

    Comment: What is meant by "the art…of certainty"? The invocation of certainty, like the word "proof", suggests the logical and mathematical, as opposed to more informal notions, of "proof". I'll have more to say about these ideas in the General Comments, below.

I wonder why it was thought advisable to have different subtitles for the US and UK. All of this, of course, may have little to do with the book or its author, since the subtitles may have been selected by the book's publisher. Was there some reason to think that the second subtitle would sell better here in the US than the first, or that the first would sell better in the UK?

Author: Adam Kucharski

Comment: Please don't hold it against him, or against me, but I'd never heard of Kucharski prior to this book. Based on the author's short biography at the end of the book, he is a mathematician and epidemiologist, and has written two previous books―The Perfect Bet and The Rules of Contagion―neither of which I've read.

Date: 2025

Summary: So far, I've read only the Introduction to the book, which lacks a summary of its structure, and the chapter titles are not very revealing, so I'm guessing as to the topics covered. Beside the Introduction, there are eight chapters. I suppose that the first chapter is an introductory one; the second appears to concern "proof" in the fullest sense, that is, in logic and mathematics; the third, in contrast, seems to deal with the weaker sense of "proof" used in the law and legal trials; the fourth would seem to concern the sense of "proof" in the statistical trials of medical and social research; the penultimate chapter may have something to do with computers; and the last may be a summing up since there is no afterword.

Given that Kucharski is an epidemiologist, I was curious whether he discusses the events of the last few years in the book. None of the chapter titles suggest a concern with the pandemic, though the fourth is a likely suspect. Thankfully, the index indicates that, unlike Marty Makary, author of a previous New Book3, Kucharski does treat the pandemic at some length.

The Blurbs: The book is positively blurbed by Tim Harford, author of The Data Detective, and Alex Bellos, who wrote Can You Solve My Problems?, both of which I've read.

General Comments: "Proof" has both a strong and a weak sense:

  • Strong: The strong sense of proof is that which occurs in deductive logical and mathematical contexts. In this sense, a proof is an argument that's valid. This means that if the premisses of the proof are true then the conclusion must also be true. It does not mean that the conclusion of the proof is true, let alone necessarily so, since one or more of the premisses might not be true. It would appear to be this sense of "proof" that's discussed in chapter four of the new book.
  • Weak: A proof in this weaker sense is one that makes its conclusion more likely to be true than false, but not necessarily so. This sense is weaker than the strong one in that the premisses of such a proof could be true and the conclusion false. For instance, the legal standard of "beyond a reasonable doubt" is not a proof in the strong sense since it does not mean beyond any doubt whatsoever, and it's possible that an innocent person might be convicted of a crime based on evidence that meets the standard. Similarly, the evidence supporting a theory in empirical science never amounts to a proof in the strong sense as there is always a possibility that new evidence will refute it. This sense would seem to be the subject of chapters three and four.

I plan to read this book and may have more to say about it in the near future.

Disclaimer: I haven't read this book yet, so can't review or recommend it, but its topic is right in the wheelhouse of my bailiwick, so I'm interested in it and thought that readers might be as well.

Notes:

  1. ↑ "Introduction", p. 3.
  2. ↑ For the UK cover with subtitle, see: "Proof: The Uncertain Science of Certainty", Adam Kucharski, accessed: 7/1/2025.
  3. ↑ New Book, 3/27/2025.
Innumerate Poster
June 27th, 2025 (Permalink)

Innumeracy in the News

During a congressional oversight hearing on the Department of Government Efficiency (DOGE), the large poster was displayed that you can see in the above screenshot taken from video of the meeting1. As you can see, the percentages shown for the three categories, which appear to be mutually exclusive, add up to 110%. Of course, all percentages of mutually exclusive categories out of some whole should total no more than 100%2, since 100% of the whole is all of it. So, something went wrong with the numbers.

What went wrong? According to the Representative who displayed the poster, Democrat Melanie Stansbury3, the percentages on the placard were taken from a poll conducted by Quinnipiac University two and a half weeks ago. However, according the poll itself4, 57% of those responding rated DOGE's work as either "not so good"―12%―or "poor"―45%. So, the extra ten percentage points were not from Quinnipiac but, presumably, from someone in Stansbury's office. Moreover, at the very least, someone on her staff should have noticed the discrepancy and checked it with the poll before displaying it.

I mention the party of the representative not to further humiliate it or her, but because it reveals an underlying bias. The Democratic Party opposes the efforts of DOGE to cut government spending, so that pointing out the apparent unpopularity of those efforts supports its position. It's unlikely that such a mistake in the opposite direction―that is, one that undermined their position―would have been displayed on a large placard during a public meeting, since someone would have caught such an error beforehand. The actual poll numbers support Stansbury's claim that DOGE is unpopular but, by falsely exaggerating them, Stansbury undermined her own position and made herself and her party look foolish.

This is just a reminder that one should be even more careful not to make mistakes that support one's own position as those that undermine it, not just because we want to get at the truth, but also because the result of carelessness can hurt one's cause.

Notes:

  1. ↑ See: Breaking News: Marjorie Taylor Greene Battles Democrats In Fiery DOGE Hearing, Forbes, 6/24/2025.
  2. ↑ The one exception is due to rounding: if the percentages are rounded, the total may add up to slightly more than 100%, but this should be noted somewhere.
  3. ↑ Joe DePaolo, Watch: Democrat Trots Out Placard in House Hearing With a Glaring Math Error―Promptly Gets Dunked On By Marjorie Taylor Greene, Mediaite, 6/24/2025.
  4. ↑ See: Majority Of Voters Oppose GOP Budget Bill, With Just 67% Of Republicans In Support, Quinnipiac University National Poll Finds; Trump Job Approval: 38%, His Handling Of Russia-Ukraine War Lowest Among List Of Issues, Quinnipiac University, 6/11/2025.
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June 22nd, 2025 (Permalink)

A Chess Tournament at the Logicians' Club

Tonight is the Logicians' Club annual chess tournament. I'm not much of a chess player so I won't be participating, but I got stuck with some of the preparations. Eight members of the club have entered the tournament: Andy, Bernie, Cathy, Debby, Ellie, Freddy, Gertie, and Hank.

You might wonder how so many members of such a small club were available for the tournament. The answer is that three branches of the club from nearby towns are included: A-town, B-ville, and C-city. The players from A-town are Andy, Ellie and Hank; from B-ville they are Bernie, Cathy and Freddy; and Debby and Gertie are from C-city. Since they have many opportunities to play against one another, no players from the same town are matched against each other in the tournament.

The eight players have been assigned to three teams named Cantor, Frege, and Russell. Cantor consists of Cathy and Hank; Frege of Andy, Debby and Freddy; and Russell of Bernie, Ellie and Gertie. In the tournament, players from the same team will not be matched against each other.

For some mysterious reason, Professor Knight*, who is also not playing in the tournament, was given the task of drawing up the seating chart for the competition. There are four tables in the large meeting room where the tournament is to be held―known simply as tables one, two, three, and four―and each pair of players were assigned a table at which they would play their match.

Unsurprisingly, Professor Knight managed to lose the seating chart, so I asked him what he remembered about the seating.

"Let's see," he said, scratching his beard and looking up at the ceiling, "I seem to remember that Debby was at table one―or was it three? Well, it was one of those two. And I recall that Cathy was definitely not at the fourth table, but I can't narrow it down any further. Andy was also not at table four, but he wasn't at the first table, either. Hank was at table three or maybe table one, I'm not sure which, but he wasn't at the same table as Debby. Ellie was also at the third table, or perhaps it was the second one. Hmm. Anyway, that's all I remember, but I'm sure you can figure it out." He smiled and walked away. I just sighed.

Is the professor right that the seating arrangements can be reconstructed from his fragmentary memories? Can you help figure out at which table each tournament participant is supposed to sit?

* ↑ If you don't know Professor Knight, see:

Disclaimer: This puzzle is a work of fiction, Professor Knight is not a real person, and there is no Logicians' Club though there ought to be.

June 19th, 2025 (Permalink)

Lesson in Logic 19: Polysyllogisms

In previous lessons, I discussed how circles can represent classes, how two-circle Venn diagrams can represent categorical statements and be used to test such statements for equivalence and contradiction. Of course, two-circle Venn diagrams can only represent logical relations between two classes; so, to handle reasoning involving a third class, you must add a third circle. Three-circle Venn diagrams―or "pretzels", as I like to call them―can be used to represent the logical relations between three classes, and test arguments involving three classes, notably categorical syllogisms, for validity.1

If you've studied the lessons covering the topics mentioned in the previous paragraph, or in some other way learned how to use Venn diagrams to evaluate categorical syllogisms2, you have every right to take pride but, before you get too arrogant, what would you do with the following argument?

  1. All sapsuckers are woodpeckers.
  2. All woodpeckers are birds.
  3. All birds are animals.
  4. Therefore, all sapsuckers are animals.

Now, you might say, quite correctly: "That's not a syllogism! It has three premisses instead of two. It has four class terms rather than three. I don't know how to diagram four class terms on a three-circle diagram!" Yet, it's an obviously valid argument and all of the statements making it up are categorical statements, so it seems a shame not to know what to do with it.

Your first idea might be to add another circle to the Venn diagram, but you can't do that. You may remember from a previous lesson3 that adding a circle to a Venn diagram should double the number of areas that the circles divide the space into so that every logically possible subclass will be represented on the diagram. So, a Venn diagram with four circles should have sixteen areas for subclasses. Unfortunately, it's not possible to add a fourth circle to the diagram in such a way as to yield sixteen areas4.

However, it is possible to create diagrams with the right number of subdivisions for any number of class terms5, but using shapes other than, or in addition to, circles. However, the more classes involved, the harder it is to correctly draw such a diagram freehand. Moreover, though the diagramming process is an extension of that for three-circle Venn diagrams, the more complex and tangled the diagrams get, the more difficult it becomes to diagram the premisses correctly. So, if you wish to use the diagrammatic method to evaluate arguments such as the above example, it will help to make copies of the appropriate diagrams6.

There's another way to show that arguments such as the example above that are really two syllogisms in one are valid using only pretzels. Such arguments are called "polysyllogisms", meaning "many"―"poly-", as in "polytheism"―syllogisms. They're also sometimes called "sorites"7―which means "heap" in ancient Greek―because they're a heap of syllogisms.

More specifically, you can reconstruct the example as a chain of two syllogisms where each syllogism is a link in the chain. What links the two syllogisms is the missing conclusion: All sapsuckers are birds. Adding that conclusion to the argument gives:

  1. All sapsuckers are woodpeckers.
  2. All woodpeckers are birds.
  3. Therefore, all sapsuckers are birds.
  4. All birds are animals.
  5. Therefore, all sapsuckers are animals.

This is clearly a sequence or chain of syllogisms where the conclusion of the first syllogism―"all sapsuckers are birds"―is a premiss in the last syllogism. Such unexpressed statements in a polysyllogism that serve as both conclusion and premiss are called "intermediate conclusions". The conclusion of the last syllogism in the chain―in this case, "all sapsuckers are animals"―is called the final conclusion to distinguish it from intermediate conclusions.

Both of the arguments making up this chain are syllogisms, so that you can test them individually for validity using pretzels. Chains of arguments are like metal chains in that they are only as strong as their weakest link. In other words, if even one link in the chain is invalid, then the entire chain breaks. In contrast, if every link in a polysyllogism is valid, then the original argument is valid. Since both of the arguments in the polysyllogism above are valid, the example is itself valid.

Unfortunately, there's a confusing subtlety here: if every sub-argument in a polysyllogism is valid then the polysyllogism as a whole is valid; but it's not the case that if some sub-argument in a polysyllogism is invalid that the polysyllogism itself must be invalid―the most that you can conclude is that it has not been shown to be valid. If you want to show that it's invalid you'll have to use a diagram, for if you correctly diagram the premisses of a polysyllogism and the resulting diagram does not show the conclusion to be true, then the polysyllogism is invalid.

Example:

  1. All flickers are woodpeckers.
  2. No birds are mammals.
  3. All woodpeckers are birds.
  4. Therefore, no flickers are mammals.

The above polysyllogism should seem intuitively valid, but let's see how to show that it is by breaking it down into a chain of simple syllogisms. Since there are three premisses containing four class terms―"flickers", "woodpeckers", "birds", and "mammals"―this should be broken down into two syllogisms connected by an intermediate conclusion.

To find which premisses to combine, look for two statements with a common middle term. So, the first two premisses will not work since there are four terms between them, but the first and third premiss share the term "woodpeckers". Here's the resulting syllogism:

  • All flickers are woodpeckers.
  • All woodpeckers are birds.
  • Therefore, all flickers are birds.

This syllogism is an instance of Barbara and, therefore, valid. Now, to complete the chain we take the intermediate conclusion―"all flickers are birds"―and combine it with the remaining premiss and final conclusion:

  • All flickers are birds.
  • No birds are mammals.
  • Therefore, no flickers are mammals.

You can show that this syllogism is valid by a Venn diagram or any other technique you prefer. This shows that the original polysyllogism is valid.

Exercises: To practice proving polysyllogisms valid, use the same technique demonstrated in the above Example on the following arguments. The chains of syllogisms given in the Answers are not necessarily the only way to show the polysyllogisms valid, so yours may differ.

I.

  1. All quacks are cranks.
  2. All charlatans are grifters.
  3. No grifters are cranks.
  4. Therefore, no charlatans are quacks.

II.

  1. All twaddle is poppycock.
  2. All gobbledegook is drivel.
  3. All balderdash is twaddle.
  4. No drivel is poppycock.
  5. Therefore, no gobbledegook is balderdash.

Notes:

  1. ↑ For the Lessons in Logic that cover the topics mentioned in this paragraph, see:
    1. Class Diagrams, 6/22/2016
    2. Two-Circle Venn Diagrams, 6/22/2016
    3. Categorical Statements, 8/17/2016
    4. Equivalence, 11/15/2016
    5. Contradiction, 12/13/2016
    6. The Third Circle, 2/16/2017
    7. Pretzel Logic, 4/28/2017
    8. Categorical Syllogisms, 5/22/2017
  2. ↑ From here on, just "syllogisms" for short.
  3. ↑ Lesson 16 in note 1, above.
  4. ↑ For a proof, see Amy N. Myers, "Are Venn Diagrams Limited to Three or Fewer Sets?". Moderately mathematical; also contains blank diagrams for four and five classes that could be copied for testing arguments for validity.
  5. ↑ See: A. W. F. Edwards, Cogwheels of the Mind: The Story of Venn Diagrams (2004).
  6. ↑ See the sources in the previous two notes for such diagrams.
  7. ↑ See: W. L. Reese, Dictionary of Philosophy and Religion: Eastern and Western Thought (1980).

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