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May 22nd, 2017 (Permalink)

Lesson in Logic 18: Categorical Syllogisms

Finally, what I've been leading up to and you've been waiting for is here: how to use Venn diagrams to evaluate categorical syllogisms. In the previous lesson1, you learned how to diagram the premisses of such an argument on a pretzel2, so you're almost there. All that's left is to learn how to evaluate such a diagram for validity.

First, a definition: a categorical syllogism is an argument with two premisses and a conclusion, all of which are categorical statements. Furthermore, the statements making up a categorical syllogism have exactly three categorical terms among them. Also, one of the three categorical terms3 occurs in both premisses but not in the conclusion. The other two terms occur once each in the conclusion and once in one of the premisses. Here's an example of a categorical syllogism:

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

In this example, 1 and 2 are the premisses and 3 is the conclusion, as indicated by the word "therefore". The three terms are: woodpeckers, birds, and sapsuckers. "Woodpeckers" is the middle term because it occurs in both premisses but not in the conclusion. "Birds" occurs once in the conclusion and once in the first premiss, and "sapsuckers" also occurs once in the conclusion and once in the second premiss. So, this argument fits the definition of a categorical syllogism.

To test an argument of this type for validity, do the following:

  1. Diagram the premisses on a pretzel.
  2. Do not diagram the conclusion.
  3. Answer the question: Does the diagram show the conclusion of the syllogism to be true?
    • If so, then the syllogism is valid.
    • If not, then the syllogism is invalid.
Categorical syllogism

Here's how this works on the example, above. The diagram to the right shows the first premiss in red and the second in blue. Does the diagram show the conclusion to be true? In order for the conclusion to be shown true, the area of the diagram representing all sapsuckers that are not birds, which is outlined in yellow, would have to be empty. The diagram does show that area as empty. Therefore, the argument is valid, as should be intuitively obvious.

The only tricky part of evaluating syllogisms with Venn diagrams comes when a premiss and the conclusion of the argument are particular statements4. It may happen that the diagram of a syllogism of this type does not look exactly the way you would diagram the conclusion. In particular, the "X" from the particular premiss may not be on a line. However, the point to remember is that if the diagram of the premisses shows the conclusion to be true, then the argument is valid. This is best shown with an example: Categorical syllogism

  1. All sapsuckers are woodpeckers.
  2. Some birds are not woodpeckers.
  3. Therefore, some birds are not sapsuckers.

If you diagram the first premiss―shown in red―before the second, as you learned to do in the previous lesson1, then you will not put the "X" on a line, since the area outlined in yellow is empty. Given the first premiss, birds that are not woodpeckers must be in the area marked by the "X".

To evaluate the example for validity, look at the diagram to see whether it shows the conclusion to be true. In this case, look to see whether the diagram shows that there are birds which are not sapsuckers. The diagram does indeed show that there is at least one bird that is outside of the circle representing sapsuckers. Therefore, the argument is valid.

You'll learn more from practice than just by reading, so try the following exercises.

Exercises: Use a Venn diagram to evaluate the following arguments. Indicate whether each is valid or invalid.

  1. No woodpeckers are raptors. All sapsuckers are woodpeckers. Therefore, no sapsuckers are raptors.
  2. All flickers are woodpeckers. All sapsuckers are woodpeckers. Therefore, all sapsuckers are flickers.
  3. No sapsuckers are raptors. Some woodpeckers are sapsuckers. Therefore, some woodpeckers are not raptors.
  4. No woodpeckers are raptors. All sapsuckers are raptors. Therefore, no sapsuckers are raptors.
  5. Some woodpeckers are sapsuckers. All sapsuckers are raptors. Therefore, some raptors are not woodpeckers.

Answers to the Exercises

Next Lesson: Sorites!


  1. Lesson in Logic 17: Pretzel Logic, 4/28/2017
  2. A three-circle Venn diagram.
  3. Called "the middle term".
  4. That is, I and O statements. If both premisses are particular, then the syllogism will be invalid. Similarly, if only one premiss is particular, but the conclusion is not. For now, take my word for it, but when you've become skilled in testing syllogisms for validity you can prove these facts for yourself.

May 5th, 2017 (Permalink)

A Word Puzzle for This Friday

Consider the following list of words:

educated, ill-fated, notice, unite, decisive, branch

Which of the following words comes next in the list?

copper, truck, tenuous, neat, hum, explain


May 4th, 2017 (Permalink)

Poll Watch: No Margin for Error

Here's a fake news headline:

Hispanics give Trump higher approval rating than rest of U.S.1

According to the first two sentences of the article beneath the headline:

Hispanic support for Donald Trump has surged since Election Day, and now tops that of the president's overall approval rating. In its latest survey, Zogby Analytics said that Hispanic support has hit 45 percent, two points higher than the president's generic approval.1

Two points? The standard margin of error (MoE) for national polls is plus-or-minus (±) three percentage points, so this is within the MoE. Moreover, the statistic refers to the support of hispanics, a subgroup within the general population, and thus likely to be a subset of the sample taken for the poll. The smaller the sample, the wider the confidence interval that determines the MoE will be. So, the MoE for the hispanic subsample in this poll is likely to be greater than ±3 points2.

The reporting of this poll is worse than is usual for American newspapers, since most will at least give the MoE at the foot of the article even if they ignore it in the body. However, this Washington Examiner story doesn't even bother to mention it, which seems to violate American journalistic standards. At least the article links to Zogby's own report on the poll3, but the only thing Zogby has to say about Trump's support among hispanics is the following:

The biggest surprise in this new poll is Trump's approval among Hispanic voters, which is at 45% approval/51% disapproval. In February the numbers were less among Hispanics at 39% approval/53% disapproval.3

So, Zogby doesn't make a big deal about hispanic support for Trump exceeding support among the general population of likely voters, probably because they realize that it isn't statistically significant. Its report links to a tabulation of the results4, where you can finally find out that the MoE for the entire sample is ±3.3 points, so the alleged higher hispanic approval rating for Trump is within that margin. However, you can also read the following at the bottoms of the pages: "Subsets have a larger margin of error than the whole data set."4

In fact, the number of hispanics included in the poll was only 965, which means that the MoE at the 95% confidence level is ±10 percentage points6. So, not only is two percentage points far from significant, even the six point increase in hispanic approval of Trump is well within the MoE. As far you can tell from this poll, the supposed "surge" in Trump support among hispanics is just random sampling error.

That's why the Washington Examiner article is fake news: subtract the innumeracy and there's zero story left. Do hispanics support Trump more, less, or the same as the rest of us? Has hispanic approval of Trump increased in the last few months? Who knows? This poll certainly doesn't provide the answer.


  1. Paul Bedard, "Hispanics give Trump higher approval rating than rest of U.S.", Washington Examiner, 5/3/2017
  2. How to Read a Poll: Margin of Error Errors
  3. "Zogby Analytics Online Survey of Likely Voters", Zogby Analytics, 5/3/2017
  4. "The Zogby Poll: Trump overall approval down, but up among Hispanics", Zogby Analytics, 5/2/2017
  5. The last table, p. 3.
  6. Courtney Taylor, "How to Calculate the Margin of Error", Thought Co., 12/16/2014

Update (5/11/2017): In case you thought that the Washington Examiner's bad poll reporting was just a fluke, it's back a few days later with even worse reporting. Once again, the MoE is not reported, but that's the least of its problems. Here's how it originally began:

Couples are fighting over President Trump more than ever, and many are turning to divorce court to get out of their politically ravaged marriages. New data from Wakefield Research found that one in 10 couples, married and not, have ended their relationships in a battle over Trump. For younger millennials, it's 22 percent.1

Of course, the claim that ten percent of couples have broken up because of a difference of opinion over Trump is wildly implausible. Here's how Wakefield itself reported its results:

The results revealed that more than 1 in 10 Americans (11%) have ended a romantic relationship over political differences. This number jumps notably among the younger generation, with 22% of Millennials having broken up with someone over political differences.2

So, the 10% result refers to Americans who have had a romance end over politics, not just about Trump, and presumably occurring at any point in the person's lifetime, not just in the last year or two. The same applies to the 22% result for the so-called Millennials. These results really don't reveal anything about Trump's effect, if any, on romantic relationships, since it's possible that all of these break-ups happened before his foray into politics.

In addition to the poor reporting, the current version of the story on the Examiner's website3 has been silently edited to fix the worst misstatements of the poll's results. Of course, it's good that it's been corrected, but there ought to be some acknowledgment that changes were made. Otherwise, it looks as though the Examiner is trying to conceal its mistakes. However, Fox News reprinted the first half of the original article, which contained its worst misstatements, and still shows the unedited version so you can check it for yourself1.


  1. Paul Bedard, "Fights over Trump drive couples, especially millennials, to split up", Fox News, 5/8/2017
  2. "New Wakefield Research Study: The Trump Effect on American Relationships", Wakefield, 5/10/2017
  3. Paul Bedard, "Fights over Trump drive couples, especially millennials, to split up", Washington Examiner, 5/8/2017.

Via: Eugene Volokh, "No, a survey doesn’t show that 1 in 10 couples have broken up over their views on Trump", The Volokh Conspiracy, 5/10/2017

April 28th, 2017 (Permalink)

Lesson in Logic 17: Pretzel Logic

"So, other than solving puzzles1, what is a three-circle Venn diagram2 good for?", you may ask. I'm glad you asked that question!

In previous lessons, you learned how to use the two-circle Venn diagram to tell when two categorical statements are logically equivalent3 and contradictory4. In the next lesson, you'll discover how to use the three-circle diagram―or "pretzel"5―to test categorical syllogisms for validity. This one is preparatory for the next; in this one, you'll learn how to diagram categorical statements onto a pretzel.

If you remember from a previous lesson6 how to diagram the four types of categorical statement onto a two-circle diagram, then doing so on a three-circle one should be fairly easy7. It's mostly a matter of concentrating on the two relevant circles and ignoring the third. However, there are some subtleties that you need to learn, especially involving the I and O statements, or you'll end up making mistakes. It's those subtleties that you should learn from this lesson, as well as getting some practice at diagramming on three circles. All As are Bs.

As you should remember, the four types of categorical statement, A, E, I, and O, relate two class terms. A categorical syllogism contains three such statements with a total of three class terms, which is why it can be diagrammed on a three-circle diagram, with each circle representing one of the three classes. However, each statement only has two class terms, so when you diagram it on the three circles you should pretend that the third circle8 does not exist. For example, suppose that you wish to diagram the A statement "All As are Bs" on a pretzel with classes A, B, and C. Simply ignore the C-circle and diagram the statement on the two circles as shown in red. No Bs are Cs.

Similarly, if you want to diagram the E statement "No Bs are Cs" on the same diagram you should ignore the A circle, as shown in blue. We've now diagrammed two categorical statements―"All As are Bs" and "No Bs are Cs"―on a single pretzel, which is what you need to do to diagram a syllogism. So far, this should be easy as long as you remember how to diagram the A and E statements.
All As are Bs and no Cs are As.

A slight complication arises if the areas that you need to shade when diagramming the two statements overlap. For instance, supposing you wish to diagram both "All As are Bs" and "No Cs are As" on a single pretzel. Diagramming just "All As are Bs" will, of course, produce the first diagram above. However, diagramming "No Cs are As" on the same diagram will involve shading the same area twice, as shown in purple. The purple area is shaded by both the red and blue shading, hence its color.

Now, remember that shading an area just means that the area is empty. So, it's not necessary to use different colors to mark an area as shaded twice, since a doubly-shaded area is not doubly empty. I just do that to show the area of overlap, and because it's pretty.

That's all there is to diagramming A and E statements, but Is and Os are tricky. Unfortunately, you cannot simply ignore the third circle when diagramming an I or O statement. For example, suppose that you want to diagram "Some As are Bs" on a pretzel. To do so, you need to place an X in the area of overlap of the two circles for A and B. However, that area is now divided in two by the circle for C. Where should you put the X: above or below the curve of the C circle that divides the overlap area in two?

Remember what the X represents: it tells us that the area it occupies is non-empty, that is, that there is something in that area. For this reason, you cannot simply place it in either of the two sub-areas of the overlap region. If you put it in the area inside the C circle, that would mean that some As that are also Cs are Bs; whereas, if you put it in the area outside the C circle, that would mean that some As that are not Cs are Bs. Moreover, if you put two Xs in the overlap area, one in each sub-area, that would mean that there are at least two As that are Bs, one of which is also C and one that is not. However, all that the I statement says is that some As are Bs; it does not say whether or not they are also Cs.

For this reason, you need to learn two new diagramming conventions for I and O statements: Some As are Bs.

  1. When you need to place an X in an area of the diagram that is divided into two sub-areas, place the X on the line dividing the area in two. So, the way to diagram the I statement "Some As are Bs" on the three circles is as shown.
  2. When diagramming two categorical statements on a pretzel, one of which is an A or E and the other is an I or O, always diagram the A or E statement first. In other words, always shade before Xing.
All As are Bs and some Cs are As.

The reason for the second convention can best be seen via an example: Suppose that you want to diagram both "All As are Bs" and "Some Cs are As" at the same time. If you diagram the A statement first, you'll get the first diagram above. Now, when you want to diagram the I statement, you will be forced to put it into the central overlap section, as shown.

The reason you put the "X" in the central overlap is because you know from the A statement that the area representing As that are Cs but not Bs is empty. Xs and shading are like matter and anti-matter: they don't go together. If you put an X in a shaded area, that would say that the area was empty but had something in it, which is contradictory. Moreover, you shouldn't put an "X" on a line between a shaded area and an unshaded one, since the shaded area is empty and the object that the "X" represents must be in the unshaded area.

So, when diagramming two categorical statements on a pretzel, do the following:

  1. Diagram A and E statements;
  2. Do so by ignoring the third circle8;
  3. Diagram I and O statements;
  4. If you must place an "X" in an area divided into two sub-areas, place it on the dividing line between the sub-areas.

That's it! Now you're ready to start diagramming categorical syllogisms, which you will learn how to evaluate in the next lesson. Check whether you're ready for it with the exercises below.

Exercises: Diagram the following pairs of statements on a pretzel.

  1. All As are Cs; all Bs are Cs.
  2. Some As are Bs; some Bs are not Cs.
  3. All As are Cs; some Bs are not Cs.
  4. All As are Cs; some Cs are not Bs.

Answers to the Exercises

Next Lesson: Learn how to use pretzels to test the validity of categorical syllogisms.


  1. Using Venn Diagrams to Solve Puzzles, Part 2, 3/7/2017
  2. Lesson in Logic 16: The Third Circle, 2/16/2017
  3. Lesson in Logic 14: Equivalence, 11/15/2016
  4. Lesson in Logic 15: Contradiction, 12/13/2016
  5. I call the three-circle Venn diagram a "pretzel" because it looks a little bit like one, and because "pretzel" is shorter.
  6. Lesson in Logic 13: Categorical Statements, 8/17/2016
  7. If you don't remember, you'd better go back and review.
  8. That is, the one not related by the two class terms of the statement.

April 22nd, 2017 (Permalink)

The ABC Murder

Detective David Davidson was assigned to investigate the murder of notorious racketeer Victor Timm. It should have been an open-and-shut case since there were four eyewitnesses to the crime. Unfortunately for Davidson, the witnesses all disagreed.

The ABC Gang was so called because it consisted of three criminals: Adam Adamson, Brad Bradford, and Curt Curtis. It was known that the ABC Gang and Timm had had a falling out, but who was the triggerman? He had no doubt that the shooter was one of the ABC gangsters, but which one?

Davidson set up a line-up for the witnesses that included all three of the ABC gangmembers, together with a few fellow cops as ringers. Thankfully, none of the witnesses fingered any of the cops as the killer. Here's the gist of what each witness said:

  1. Adamson was the shooter.
  2. This witness wouldn't identify a shooter, but insisted that it wasn't Curtis.
  3. This one wasn't sure whether Adamson or Bradford was the shooter, but claimed that it was one of the two.
  4. The last witness refused to finger the shooter as Bradford but did rule out both Adamson and Curtis.

Davidson was disgusted. That didn't help at all! In fact, at least one of the witnesses had to be wrong since they contradicted each other. Luckily, Davidson received a call from a stoolie named Eddie "The Snitch" Edwards who informed him that one or more of the witnesses had been bribed or intimidated by the ABC Gang to lie to the police. At first, Davidson was still disgusted, since he already knew that at least one of the witnesses had not told the truth, but he didn't know which. However, when Eddie told him the exact number of witnesses who weren't telling the truth, Davidson smiled. Davidson now knew who the shooter was.

Assuming that one of the ABC gangsters was the shooter and that Eddie's information was correct, who shot Vic Timm?


April 9th, 2016 (Permalink)

Check it Out, Too

If you can tear yourself away from the tax forms long enough, philosopher Alan Hájek offers you a "philosophy tool kit" for thinking:

Philosophers pride themselves on thinking clearly by seeing what follows from what, exposing sophisms, spotting fallacies, and generally policing our reasoning. … But these skills are not the exclusive property of rarefied sages, accessed only with a secret handshake and insider training, as much as some philosophers wish this were so. Instead, some of these skills can be captured by generalisable, all-purpose techniques for the proper conduct of thought, whatever the topic. Many of these are easily taught and learned. As such, they can be utilised by non-philosophers too. At a time when we are bombarded more than ever with specious claims and spurious inferences, clear thinking provides a much-needed safeguard that we should all strive towards.1

It's not a full philosophical "tool kit", as it contains only a few tools. Rather, it's like the kind of small kit you might keep in your car in case of a breakdown on the road, so you might think of it as a tool kit in case of a philosophical emergency.

Hájek shows how to use the tools by applying them to some traditional philosophical problems but, as he mentions, they can be applied to many other types of intellectual, conceptual, and logical problems.

The tools in the kit are philosophical heuristics, but what is a philosophical heuristic? Well, wait: what is a heuristic? It's a rule of thumb2. Hájek gives the following example: "Here’s a good one for mathematics: if you are not making headway on a problem, modify it slightly to make it easier, and solve that one." I think this is a good rule of thumb for problem-solving in general, not just mathematics. The point of using heuristics is that, while they don't always work, they work often enough to be useful; also, even when they don't solve the problem, they may help point the way to a solution.

One reason I point you to this article is that many of the topics that Hájek discusses have been discussed here in The Fallacy Files, so you can compare his treatment of the issues to mine3:

This should keep you busy until I can get around to posting something new.


  1. Alan Hájek, "Philosophy Took Kit", Aeon, 4/3/2017
  2. And, no, the phrase "rule of thumb" does not come from a law allowing a man to beat his wife with any stick no thicker than his thumb. See: Cecil Adams, "Does 'rule of thumb' refer to an old law permitting wife beating?", The Straight Dope, 5/12/2000
  3. A useful tool that Hájek doesn't mention is "the second opinion".
  4. "A" v. "The", 7/19/2008
  5. "False Dichotomy"
  6. Puzzle it Out, 7/3/2015
  7. Q&A, 5/21/2013
  8. The Logical Problem of Evil, 4/6/2015

April 5th, 2017 (Permalink)

Check it Out

April is the cruelest month….1

…And April 15th the cruelest day. It's that time of year again, and I'm going to be rather busy for the next couple of weeks trying to figure out what the IRS wants from me. In the meantime, if you can find some time to read something other than tax instructions, here's something worth checking out2.


  1. T. S. Eliot, "The Waste Land", Poetry Foundation
  2. Eugene Volokh, "“Amid ‘Trump Effect’ fear, 40% of colleges see dip in foreign applicants”―but…", The Volokh Conspiracy, 3/28/2017

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