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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.

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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.

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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.

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:

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.

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Notes:

1. ↑ See:
   11. Class Diagrams, 6/22/2016
   12. Two-Circle Venn Diagrams, 7/16/2016
   13. Categorical Statements, 8/17/2016
   14. Equivalence, 11/15/2016
   15. Contradiction, 12/13/2016
   16. The Third Circle, 2/16/2017
   17. Pretzel Logic, 4/28/2017
   18. Categorical Syllogisms, 5/22/2017
   19. Polysyllogisms, 6/19/2025
   20. 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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August 11th, 2025 (Permalink)

A Rising Tide

It's time once again to play America's favorite game: Name that fallacy! Here's how it's played, in case you haven't played before: I will show you a passage from a written work and your task is to identify the logical fallacy committed. It's not so important whether you can actually put a name to it if you recognize the nature of the mistake. So, let's get started.

What logical fallacy is committed by the following passage?

Should it surprise anyone that alcohol abuse in America has been rising, not falling? The number of adults who either consume too much alcohol or have an outright dependency on it rose from 13.8 million in 1992 to 17.6 million in 2002, according to the National Epidemiologic Survey on Alcohol and Related Conditions (NESARC).¹

Fallacy

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Notes:

1. ↑ Steve Salerno, SHAM: How the Self-Help Movement Made America Helpless (2005), pp. 227-228.
2. ↑ "Alcohol Abuse Increases, Dependence Declines Across Decade: Young Adult Minorities Emerge As High-Risk Subgroups", National Institute on Alcohol Abuse and Alcoholism, 6/10/2004.
3. ↑ "What was the population of the United States in 1992?", Wolfram Alpha, accessed: 8/9/2025.
4. ↑ "What was the population of the United States in 2002?", Wolfram Alpha, accessed: 8/9/2025.
5. ↑ The news release cited in note 2, above, gives these percentages as 7.4% for 1992 and 8.5% for 2002, for a slightly larger increase of 1.1 percentage points. The release does not give the population data on which these percentages were figured, so I used the data cited in the previous two notes.

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August 3rd, 2025 (Permalink)

Crack the Combination X^*

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. 3 5 6 0: Two digits are correct but only one is in the right position.
2. 2 6 7 0: Two digits are correct but neither is in the right position.
3. 7 2 8 6: Three digits are correct but none is in the right position.
4. 4 0 2 5: One digit is correct and in the right position.

Can you determine the correct combination from the above clues?

Hint

Solution

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

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August 1st, 2025 (Permalink)

Photos Don't Lie, but Photographers Do

The following short video won't teach you much about how to spot fake photographs generated by so-called artificial intelligence, so it's misleadingly named. However, you might be able to use the geometric techniques briefly demonstrated. Instead, it shows why spotting phony photos and videos is becoming a task for professionals. The presenter is the author of a book―not for amateurs―on analyzing and exposing faked photography¹, which contains more detail on the techniques for analyzing perspective and lighting. By the way, I quite agree with the presenter's comments on anti-social media near the end of the video.

• Hany Farid, "How to Spot Fake AI Photos", TED, 7/18/2025

  Though fake photography and videography are becoming increasingly sophisticated, there are still many photos and videos in the media that use techniques that have been around since the beginnings of photography, such as selective editing², false captioning³, or cropping⁴. The following article describes a couple of recent cases. Warning: the photos are unpleasant to look at, which is part of their effectiveness as propaganda.

• Miriam Metzinger, "When Images Mislead: The Facts Behind 2 Viral Gaza Cases", The Media Line, 7/30/2025

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Notes:

1. ↑ Hany Farid, Fake Photos (2019). Highly technical book with a lot of mathematics, so appropriate for professionals or those with the mathematical background looking to become professionals.
2. ↑ See: How to Lie with Photographs, Part 1, 12/9/2023.
3. ↑ See: How to Lie with Photographs, Part 2, 5/11/2024.
4. ↑ See: How to Tell Half-Truths with Photographs, 2/12/2025.

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Disclaimer: I don't necessarily agree with everything in the video and article, but I think they are worth watching or reading in their entirety.

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July 29th, 2025 (Permalink)

Lesson on Logic 20: Venn Diagrams, Invalidity and Counter-Examples

In previous lessons¹, we saw how Venn diagrams are used to represent categorical statements and show the validity of arguments constructed with such statements. One advantage of such diagrams is that they also show the invalidity of invalid arguments. Since the diagrams represent the logical relations between the argument's classes, if the argument is invalid then there must be some way for its premisses to be true and its conclusion false. Also, if an argument is invalid it may be possible to find an example that shows its invalidity, and a diagram can help do that.

What is a counter-example to an argument? It is an example that shows an argument to be invalid by making the premisses true and the conclusion false. To see how counter-examples work, let's look at a simple invalid argument, namely, an illicit conversion²: All dogs are mammals, therefore all mammals are dogs. This is such an intuitively invalid argument that a counter-example isn't necessary, but let's create one as an example of the process.

A counter-example to this argument is an example that shows the premiss―all dogs are mammals is true―but the conclusion―all mammals are dogs―is false. We know that the premiss is true from basic biology, so it's really the conclusion that we need to show false. Showing the conclusion to be false is equivalent to showing that its negation―not all mammals are dogs―is true. The negation of the conclusion is equivalent to: some mammals are not dogs. Thus, to give a counter-example to the argument we need to find an example of a mammal that is not a dog; Shadow the cat, for instance.

Now, let's see how to use a Venn diagram to construct a counter-example. Here's the diagram representing the premiss of the argument as true by showing the part of the diagram representing non-mammal dogs―shaded in blue―is empty:

To find a counter-example, we look to see what would make the conclusion false. The diagram does not show the crescent-shaped area on the right, representing non-canine mammals, as empty. Therefore, a counter-example will be any mammal that is not a dog, such as the aforementioned Shadow. This counter-example is shown by the red "X" in the second diagram.

Having seen the basic technique in a simple example, let's look at a more complicated one involving three classes. Consider the categorical syllogism:

All cats are mammals.
All dogs are mammals.
Therefore, all dogs are cats.

Here's a Venn diagram showing the premisses as true, with the blue shading representing the first premiss and the yellow the second. The conclusion is not shown to be true because the area with a red question mark is not shaded. It's in this area that we need to find a counter-example, that is, a dog that is not a cat, which is easy enough as any dog will do.

While counter-examples are useful for showing invalidity, they are not always available. The previous example was so obviously invalid that a counter-example was scarcely necessary, and it's common knowledge that the conclusion is false. However, not every argument of the same form will be obviously invalid. For instance, consider the argument:

All oranges are citrus fruits.
All mandarins are citrus fruits.
Therefore, all mandarins are oranges.

This argument sounds much more plausible, since its conclusion is true, yet it has the same form as the previous implausible one. As a result, the same diagram serves to represent it with the circles relabeled with "oranges" for "cats", "mandarins" for "dogs", and "citrus fruits" for "mammals". So, despite its plausibility it's invalid. However, a counter-example would have to be a mandarin that is not an orange. Despite the lack of a counter-example, the diagram shows the argument to be invalid since the premisses leave open the possibility that there are non-orange mandarins. Thus, even without a counter-example the diagram shows the argument's invalidity. By the way, both of the above syllogisms commit the fallacy of undistributed middle term³.

In addition to cases where the conclusion of an argument is true for reasons irrelevant to its premisses, counter-examples are not possible when the conclusion is a particular statement, that is, an I or O statement. However, when diagramming such an argument it should be obvious by inspection that the premisses do not make the particular conclusion true, so a counter-example will be unnecessary.

Now, let's practice on a couple of examples. Construct diagrams for the following arguments and counter-examples if possible.

1. No bats are birds.
   Therefore, no non-birds are non-bats.

   Solution

   

   Counter-Example: The conclusion of this argument claims that the class of things outside of both the birds and bats circles is empty. So, anything that is neither a bird nor a bat, such as you, is a counter-example. This argument commits the fallacy of illicit contraposition⁴.

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2. No fascists are communists.
   All communists are socialists.
   Therefore, no fascists are socialists.

   Solution

   

   Counter-Example: The Nazis were national socialists, so they fall in the area of the diagram that would have to be empty for the conclusion to be true. This syllogism commits the fallacy of illicit distribution of the minor term⁵.

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Notes:

1. ↑ See:
   11. Class Diagrams, 6/22/2016
   12. Two-Circle Venn Diagrams, 6/22/2016
   13. Categorical Statements, 8/17/2016
   14. Equivalence, 11/15/2016
   15. Contradiction, 12/13/2016
   16. The Third Circle, 2/16/2017
   17. Pretzel Logic, 4/28/2017
   18. Categorical Syllogisms, 5/22/2017
   19. Polysyllogisms, 6/19/2025
2. ↑ See: Illicit Conversion
3. ↑ See: Undistributed Middle Term
4. ↑ See: Illicit Contraposition
5. ↑ See: Illicit Minor

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