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September 17th, 2025 (Permalink)

Lesson in Logic 22: Polysyllogisms and Euler Diagrams*

As discussed in lesson 20, the technique of turning a polysyllogism into a chain of categorical syllogisms can show the validity of an argument that a single standard Venn diagram could not handle. However, the circles of Leonhard Euler, introduced in the previous lesson, can be used to show validity in a single diagram. To see how this method works, let's apply it to the polysyllogism used as an example in lesson 19: All sapsuckers are woodpeckers

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

There are three premisses, all of which are A-type statements, that need to be represented in our diagram. Recall from the previous lesson that Euler represented such statements by drawing a circle for the subject class inside a circle for the predicate class. In this case, it doesn't matter which premiss you start with, so lets begin at the beginning with the first premiss. "Sapsuckers" is the subject class and "woodpeckers" is the predicate class, so we represent the first premiss as shown above. Euler diagram 1

Turning now to the second premiss, remember that in diagramming arguments the premisses are all represented on a single diagram, whether Venn or Euler. So, we need to show on the same diagram that the class of woodpeckers is contained within the class of birds. Since we already have a circle for woodpeckers, all that we need is a new circle for birds, and the former should be inside the latter as shown above.

Notice that the second diagram shows that the class of sapsuckers is a subclass of the class of birds―in other words, all sapsuckers are birds―which was the intermediate conclusion in the chain argument given to show this polysyllogism valid in lesson 19. The final step is to diagram the third and last premiss, which means placing the "Birds" circle within a circle representing all animals, as shown above.

The finished diagram clearly shows the logical relationships between the four classes, and you can see that the conclusion is true and, therefore, the argument is valid. In my opinion, this is far easier and more perspicuous than the chain argument of lesson 19. However, that's just one example, so let's look at another example from that lesson, this one including an E-type statement: Euler diagram 2

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

Since we already know how to diagram A statements, let's consider premisses 1 and 3 first. The result of diagramming both will look like the second diagram above but with "flickers" in place of "sapsuckers". Now, to diagram the second premiss, we must add a circle representing mammals that is disjoint from the circle for birds; the result looks as shown. Again, you can see from the diagram that the conclusion of the argument is true―that no flickers are mammals―and, thus, that the argument is valid, since it shows that the classes of flickers and mammals are disjoint.

As I mentioned in the previous lesson, Euler's diagram's for particular statements―that is, I- and O-type statements―are what led to Venn's different approach to using circles to represent classes. In the next lesson, we'll see how to combine Venn's technique with Euler's to diagram polysyllogisms with particular premisses. In the meantime, here's a polysyllogism to practice diagramming:

Exercise: Use an Euler diagram to show the following polysyllogism valid:

  1. All rimshaks are sleestaks.
  2. No sleestaks are triktraks.
  3. All flipjaks are triktraks.
  4. Therefore, no flipjaks are rimshaks.

* ↑ For previous lessons in this series, see the navigation panel to your right.

Puzzle
September 6th, 2025 (Permalink)

Crack the Combination XI*

The combination of a lock is four digits long and each digit is unique, that is, each occurs only once in the combination. The following are some incorrect combinations.

  1. 8 1 7 4: One digit is correct but is in the wrong position.
  2. 5 4 9 6: No digits are correct.
  3. 7 4 3 1: Two digits are correct but neither is in the right position.
  4. 4 1 9 0: One digit is correct and in the right position.

Can you determine the correct combination from the above clues?

* ↑ Previous "Crack the Combination" puzzles: I, II, III, IV, V, VI, VII, VIII, IX, X

September 4th, 2025 (Permalink)

What's New?

The Fallacy Files Taxonomy of logical fallacies is―that is, there's a brand new version of it: just click on "Taxonomy" to your upper right. In case you're interested, the old versions are still available from the following page, where you can also read about how you might make use of the taxonomy: The History of the Taxonomy. Check it out!

Recommended Reading
September 1st, 2025 (Permalink)

To err is human but to really foul things up requires artificial intelligence

Notes:

  1. ↑ See: "Today we're releasing Claude Opus 4.1, an upgrade to Claude Opus 4 on agentic tasks, real-world coding, and reasoning.", Anthropic, 8/5/2025.
  2. ↑ See: The Gee-Whiz Bar Graph, 4/4/2013.
  3. ↑ See: Half a Graph, 11/23/2024.
  4. ↑ See:

Disclaimer: I don't necessarily agree with everything in the above articles, but I think they are worth reading. I have sometimes suppressed paragraphing or rearranged the paragraphs in the excerpts to make a point.

August 26th, 2025 (Permalink)

The Jacksonville Fallacy?

A couple of years ago, Governor Ron DeSantis claimed that crime in his state of Florida was at a fifty-year low while "major" crime in New York City had increased by 23% the previous year1. Now, this is not a fact check but a logic check, so I'm just going to assume that the statistics given by DeSantis and others quoted in this entry are factually correct. Instead of fact-checking these statistics, the question I'm addressing is: What if anything do they prove?

Some critics of DeSantis replied that the homicide rate in Jacksonville, Florida was actually three times greater than that in the Big Apple2: specifically, that the homicide rate per 100K in 2022 was 16.7 in Jacksonville but only 4.8 in New York City. Of course, both of these sets of statistics can be correct: it's quite possible that crime was decreasing in Florida and increasing in New York as DeSantis claimed, but was worse in Florida than in New York as his critics claimed. But even if the statistics are correct, the governor could rightfully be criticized for cherry-picking the ones that made his state look good.

A defender of DeSantis rebutted the critics by citing the number of murders per square mile in 2022 in Jacksonville: 0.19, and New York City: 1.383. This is a statistic of dubious value in comparing the amount of murder in two places since it's affected by population density: the higher the density, the more murders per square mile. New York no doubt has much greater population density than Jacksonville. Moreover, this particular comparison is affected by a piece of trivia appropriate for a Ripley's cartoon4 or the Guinness book of world records.

What is the largest city in area in the contiguous United States, that is, the "lower 48"? This is a trivia question rather than a logic puzzle, so you either know the answer or you can look it up, but you can't figure it out. You might guess that it's Los Angeles, a notoriously spread-out city, but that's wrong. Do you give up? The answer is Jacksonville, Florida5.

So, even if it made sense to compare cities on the basis of murders per mile², it wouldn't be fair to compare New York City to Jacksonville, given that the latter is the largest city in area in the lower forty-eight, but only the eleventh in population size6.

Despite the title, I'm not ready to add an entry to the files for statistical fallacies that take advantage of Jacksonville's trivial status as the lower 48's biggest city in area. However, I've now come across two examples and if I find one more, I may just do so.

Notes:

  1. ↑ Ron DeSantis, "I visited Staten Island to talk about how law & order has been central to FL's success.", X, 2/20/2023.
  2. ↑ Mark D. Levine, "Homicides per 100k residents in 2022…", X, 2/20/2023.
  3. ↑ See: Heather Kofke-Egger, "Murders Per Square Mile", Data Behind the Data, 2/22/2023.
  4. ↑ See: The Talented Mr. Ripley, 5/19/2025.
  5. ↑ See: May I Puzzle You?, 5/20/2005.
  6. ↑ "Largest US Cities by Population 2025", World Population Review, accessed: 8/25/2025.
August 20th, 2025 (Permalink)

Lesson on Logic 21: Euler Diagrams

In previous lessons1, we saw how Venn diagrams are used to represent logical relations between classes. However, as pointed out previously, Venn's diagrams are limited to representing the relations between three classes. There are extensions of Venn's diagrams but they become increasingly awkward with increasing numbers of class terms. When faced with polysyllogisms―that is, categorical arguments involving four or more class terms―one way to work around this problem was explained in Lesson 19, namely, breaking such arguments down into a chain of categorical syllogisms.

As I mentioned in the previous lesson, the technique of turning a complex argument into a chain of simpler ones can show that the argument is valid but not that it's invalid. This is because that technique is a method of proof, and it's a general fact that a given argument's failure to prove its conclusion doesn't mean that no other would do so. In contrast, a Venn diagram either shows an argument valid or invalid. For this reason, it would be nice to have such a diagrammatic technique for polysyllogisms.

Prior to John Venn, Leonhard Euler used circles to represent the logical relationships between classes2. In my opinion, Euler's diagrams for the universal statements of categorical logic are more intuitive than those of Venn, but unfortunately those for the particular statements were neither intuitive nor useful. This problem led Venn to keep the circles but take a different approach to representing all types of categorical statement, which is a shame given the limitations of his approach both in intuitiveness and in number of terms diagrammable.

In this lesson, I will simply introduce Euler's diagrams and show how they are used to represent the logical content of universal statements but, in a future lesson, we'll see how to evaluate categorical arguments. All A is B

Euler did not have anything corresponding to Venn's primary diagrams3, which divide up all of the logical space of the diagram into every possible subclass of two or three classes. Instead, of using shading to show that certain classes were empty, Euler used the spatial relationship between the circles themselves to indicate such relationships. So, here's how Euler represented universal affirmative statements―that is, A statements:
No A is B

Similarly, to represent universal negative statements―that is, E statements―Euler drew the circles so that they did not overlap. In my view, these diagrams are more intuitive representations of these categorical relationships than the corresponding Venn diagrams, since you can see that one class is contained within another or that the two classes are disjoint.

Since Euler's methods of representing particular statements were inadequate and not as intuitive as those for universal ones, we can adopt the convention of placing a mark inside a class or subclass to indicate that it is non-empty.

In the next lesson, we'll start looking at how to use this combination of Euler and Venn diagrams to evaluate categorical arguments.

Notes:

  1. ↑ See:
    1. Class Diagrams, 6/22/2016
    2. Two-Circle Venn Diagrams, 7/16/2016
    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
    9. Polysyllogisms, 6/19/2025
    10. Venn Diagrams, Invalidity and Counter-Examples, 7/29/2025
  2. ↑ Leonhard Euler, Lettres à une Princesse D'Allemagne (1843), Lettre xxxiv.
  3. ↑ See lessons 12 & 16.

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