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:
- All woodpeckers are birds.
- All sapsuckers are woodpeckers.
- 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:
- Diagram the premisses on a pretzel.
- Do not diagram the conclusion.
- 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.
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:
- All sapsuckers are woodpeckers.
- Some birds are not woodpeckers.
- 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.
- No woodpeckers are raptors. All sapsuckers are woodpeckers. Therefore, no sapsuckers are raptors.
- All flickers are woodpeckers. All sapsuckers are woodpeckers. Therefore, all sapsuckers are flickers.
- No sapsuckers are raptors. Some woodpeckers are sapsuckers. Therefore, some woodpeckers are not raptors.
- No woodpeckers are raptors. All sapsuckers are raptors. Therefore, no sapsuckers are raptors.
- Some woodpeckers are sapsuckers. All sapsuckers are raptors. Therefore, some raptors are not woodpeckers.
Next Lesson: Sorites!
- Lesson in Logic 17: Pretzel Logic, 4/28/2017
- A three-circle Venn diagram.
- Called "the middle term".
- 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.
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
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.
- Paul Bedard, "Hispanics give Trump higher approval rating than rest of U.S.", Washington Examiner, 5/3/2017
- How to Read a Poll: Margin of Error Errors
- "Zogby Analytics Online Survey of Likely Voters", Zogby Analytics, 5/3/2017
- "The Zogby Poll: Trump overall approval down, but up among Hispanics", Zogby Analytics, 5/2/2017
- The last table, p. 3.
- 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.
- Paul Bedard, "Fights over Trump drive couples, especially millennials, to split up", Fox News, 5/8/2017
- "New Wakefield Research Study: The Trump Effect on American Relationships", Wakefield, 5/10/2017
- 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
- Invalid. I included this one to see if you're paying attention: it's not even a categorical syllogism! This is because there is no middle term, since the term that occurs in both premisses―"raptors"―also occurs in the conclusion. Moreover, "woodpeckers" occurs only once in the argument, specifically, in the first premiss, whereas each term in a categorical syllogism must occur exactly twice. Nonetheless, so long as the premisses are categorical statements containing three terms, you can use a pretzel to evaluate it.
Can you see why? The words in the list are in reverse alphabetical order, that is, reverse the order of letters in each word, place the list obtained in alphabetical order, then put the letters back in the correct order. So, "ill-fated" comes before "notice" because "d" comes before "e" in the alphabet, and "decisive" comes after "unite" because "v" comes after "t". Therefore, since the last word in the list ends in "h", the only word from the six choices given that could come next in the list is "truck", since "k" comes after "h", but before "m", "n", "r", "s", and "t".