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January 21st, 2017 (Permalink)

Book Club: Winning Arguments, Introduction

Let's start this year off right with a book club! For those who don't know, a book club is where I will read a book, usually at the rate of a chapter a month, and write a commentary on that chapter. Hopefully, at least some of you will read along, otherwise it's going to be a small club.

I've chosen last month's "New Book" Winning Arguments by Stanley Fish1 to read. It has some desirable qualities: First of all, it's short―there are only six chapters and an introduction of a few pages―so no one has to commit themselves to any heavy lifting. Secondly, based on past familiarity with Fish's work, I expect to disagree with at least some of it. This is important because if I agreed with it too much I wouldn't have anything to say other than "yes", "that's right", "I agree", etc., which gets tedious.

Let's start things off this month with the introduction, which is only two and a half pages long. It's available to be read online at the publisher's website2, so you can get started now even if you don't have a copy of the book. However, for later installments of the club, you'll need to get your own copy if you want to read along.

I think you'll get the most out of these entries if you first read the chapter for the month and then read my commentary on it―my comments may not make much sense to you if you haven't read the chapter. So, for this month, please go read the introduction then come back.

The full title of the introduction is: "Introduction, or What This Book Promises to Do", so it should tell us what we can expect to learn from the book, and what we should complain about if it doesn't deliver.

Fish starts by saying that he aims to do two things that might seem to be incompatible but are not:

  1. Explain the effect of context on argumentation.
  2. Make the case that argumentation is unavoidable and never-ending.

At this point, you might well wonder what this has to do with winning arguments―the title of the book―and what works and doesn't work in politics, the bedroom, the courtroom, and the classroom―the subtitle. The connection may be marketing. The titles and subtitles of books are often supplied by the publisher rather than the author, so it's possible that Harper-Collins thought that a book titled "Winning Arguments" would sell better than one entitled "The Context of Argumentation" or "The Inevitability of Argument".

However, the first of Fish's promises plausibly has some bearing on the "winning" of arguments, namely, context. Fish writes:

Say you're a politician, and you want to persuade people to vote for you. What kind of arguments can you make and what arguments might you do better to avoid? … Or say you find yourself in court. What arguments will a judge allow you to make and what arguments will be ruled inadmissible…? Or suppose you are a college teacher eager to advance in the profession by making arguments that will be compelling to your peers. … These are questions with answers, and you will find them in the pages that follow.3

In other words, context has an effect upon the effectiveness of argumentation. This certainly seems correct, but Fish is promising a lot if he's going to tell us in one slim book how to argue in politics, the law, the academic conference, etc. There are whole books on political and legal argumentation. So, this is an ambitious goal; let's remember it so that we can hold Fish to it.

In contrast, it's hard to see how achieving Fish's second aim would contribute to winning arguments. Rather, it's the sort of preliminary argument you might make if you wanted to convince a skeptical reader that this is a subject worth pursuing. If, as Fish claims, we cannot avoid argument or bring it to an end, then we'd better learn how to do it as well as we can. I would expect such an argument to occur in the introduction, rather than the body, of the book. Instead, all that we get here is a rather unclear statement of Fish's position, not an argument for it. However, it appears that this is one of the things we should expect him to argue for over the course of the book.

Here's a question to be think about: if argumentation is interminable, how is it possible to "win"? In a game such as chess, an unending game would count as a draw. If we can never reach a final conclusion that everyone will admit, how can anyone claim to have "won"? Is this what Fish was alluding to when he wrote: "This book tries to do two things that might seem to pull against each other, but in the end I don't think they do." In other words, achieving the second goal seems to undermine the possibility of achieving the first. How does Fish resolve the seeming incompatibility of his aims?

Finally, consider the following passage:

Each of us occupies a partial, time-bound perspective and none of us has access to the God's-eye view from which the "big picture" might be seen at a glance. Therefore any statement any of us makes is an argument because, as an assertion that proceeds from an angle, it can always be, and almost always will be, challenged by those whose vision is also angled, but differently so.4

This is an argument in the logician's sense of "argument", that is, a group of claims, one of which is supposed to follow from the others. Now is a good time to review how to recognize the structure of arguments if we want to be able to understand and evaluate Fish's arguments. For this purpose, I recommend the lesson on argument analysis from almost ten years ago5, but almost any introductory textbook on logic will serve.

I will leave it as an exercise for the reader to determine the structure of Fish's argument; see the next installment to check your answer.

Sources & Resources:

  1. New Book: Winning Arguments, 12/22/2016
  2. Stanley Fish, "Introduction", Winning Arguments
  3. Pages 1-2 in the hardcover edition.
  4. Page 2 in the hardcover edition.
  5. Lesson in Logic 7: Argument Analysis, 7/31/2007. Also see previous lessons if this one isn't understandable on its own.

Next Month: Chapter 1. Living in a World of Argument


January 18th, 2017 (Permalink)

Using Venn Diagrams to Solve Puzzles

Take a break this month from the lessons in logic and learn how to use Venn diagrams to solve some types of puzzles. Of course, you can use such diagrams to solve logic puzzles based on categorical syllogisms or other types of reasoning with categorical statements―I've posted a few puzzles of that type here. What I have in mind now is a different type of puzzle that uses numbers or percentages, which means that it's not the sort of puzzle you might think of using a Venn diagram to solve. Here's an example:

A poll of British tea drinkers found that half take their tea with milk, only 15% drink it straight, but 60% do not use sugar. What percentage use both sugar and milk?

Now, you can solve this type of puzzle algebraically, but I think it's easier and more intuitive to use a diagram. Let's see whether you agree with me.

Here's how. If you consider the universe of this puzzle to be British tea drinkers, there are two classes involved: those who put milk in their tea and those who use sugar. There are, of course, subclasses mentioned, such as those who use neither milk nor sugar, and the question that you are asked is about one such subclass: those who take both sugar and milk. These are the subclasses determined by a two-circle Venn diagram. Here's how to solve such a puzzle using such a diagram. The first step is, of course, to set up the diagram, labelling the circles "Milk" and "Sugar", so: Venn Diagram for the tea drinkers puzzle

The puzzle gives you three pieces of information:

  1. 50% of the tea drinkers use milk.
  2. 15% use neither milk nor sugar.
  3. 60% don't use sugar.

You can't use clues 1 or 3 to start with, since you only know generally where the percentages belong. For instance, the first clue tells you that half of the tea drinkers are contained in the "Milk" circle, but you don't yet know what percentage is in the left-hand crescent-shaped section, and what in the lens-shaped overlap section. However, the second clue tells you that 15% of the tea drinkers are outside of either of the two circles, so add this information to the diagram by writing "15%" in the space outside of both circles, like so: 15% of tea drinkers use neither milk nor sugar.

From the first clue, you know that half of tea drinkers do not use milk, of which 15% drink tea straight. This means that 50-15=35% use only sugar. The part of the diagram that represents the subclass of those who use only sugar is the right-hand crescent. Add this information to the diagram: 35% of tea drinkers use only sugar.

Now, from clue 3 you know that 60% don't use sugar, and those who use neither milk nor sugar are a subclass of those who don't use sugar. The other part of the class of those who don't use sugar are those who take only milk, so that subclass must contain 60-15=45% of the tea drinkers. The subclass of those who take only milk is represented by the left crescent area of the diagram, so add the information you have just gained: 45% of tea drinkers use only milk.

You've used up all three clues and you're almost done! The only part of the diagram that is unmarked is the part you need to know about, namely, those who put both milk and sugar in their tea, which is represented by the lens-shaped area in the middle of the diagram. So, 100-45-35-15=5% of the tea drinkers use both milk and sugar. Add this information to the puzzle and you're finished: 35% of tea drinkers use only sugar.

So, the answer to the puzzle is: 5%. Wasn't that easy?

Here's another one to practice on:

Half of incoming freshman students at Midwestern College do not take logic, but 43% take philosophy. 76% take either philosophy or logic. What percentage of incoming freshmen take logic but not philosophy?

Solution


January 16th, 2017 (Permalink)

New Book: How to Win an Argument

Cassius: Did Cicero say any thing?

Casca: Ay, he spoke Greek.

Cassius: To what effect?

Casca: Nay, an I tell you that, Ill ne'er look you i' the face again: but those that understood him smiled at one another and shook their heads; but, for mine own part, it was Greek to me.5

Last month's new book was one on winning arguments called―what else?―Winning Arguments1. I pointed out then that this is a perennially popular topic for books, since everyone wants to "win" arguments, whatever that means. As evidence for my claim, I pointed out that there are five different books in The Fallacy Files library on how to win an argument, including Michael Gilbert's aptly titled How to Win an Argument. This month we have yet another book of that name! So, for those of you keeping score at home, that makes at least seven how-to books on winning arguments.

This new book is authored by a very old author―in fact, a very dead one―Marcus Tullius Cicero4. He is most commonly known as simply Cicero, though for some obscure reason philosophers like to call him "Tully"―sort of a nickname, I guess. Cicero is, of course, a famous Roman orator, politician, and lawyer of the first century B.C.E. He also has a bit part in Shakespeare's Julius Caesar, a play famous for the oratory of Brutus and Mark Antony's funeral oration, though Cicero himself only talks about the weather.

Cicero never actually wrote a book called How to Win an Argument, and not merely because he didn't write English; neither did he write a book whose Latin title might reasonably be translated as "How to Win an Argument". Instead, this new tome is a compilation of selections from Cicero's writings that have some bearing on the title topic, as selected, edited, and translated by James M. May, a rhetorician and classics professor. The book is thus subtitled: "An Ancient Guide to the Art of Persuasion".

Its main sources seem to be Cicero's De Inventione2, which means "On Invention"―in rhetoric, "invention" refers to the process of inventing arguments―and De Oratore3, "On the Orator". I've never read either book, so I've no idea how much light they cast on the subject of winning arguments, but I look forward to reading the new one and finding out.

Sources & Resources:

  1. New Book: Winning Arguments, 12/22/2016
  2. Cicero, "De Inventione", Classic Persuasion
  3. Cicero, "De Oratore", Internet Archive
  4. John Ferguson & John P. V. Dacre Balsdon, "Marcus Tullius Cicero", Encyclopaedia Britannica, 5/12/2015
  5. William Shakespeare, Julius Caesar, I. ii.

January 8th, 2017 (Permalink)

Charts & Graphs

Longtime friend of The Fallacy Files Lawrence Mayes sends in the following unusual graph* that he found:

Most popular Erasmus destinations

I've never seen anything like it before, which makes it difficult to interpret. However, this difficulty might be justified if it were a genuinely new way to present information in a graphical form, since any new type of chart will take some getting used to. But, it's not. Lawrence asks:

What parameter is used to illustrate the figures? Line length or angle? The answer is line length.

This is because this graph is actually our old familiar friend the bar chart, only it has been turned on its side―which is alright, or at least it's been done before. In a bar chart, it's the length of the bars―or narrow, line-like bars in this case―that convey data. In a pie chart, it's the angle of a slice of the pie that carries information, which is usually a fraction of a whole.

Here, the curving length of the thin bars represents the number of students who travelled to the countries listed down the middle. However, the bars are so long that they seem to have drooped down in a curve from their own weight. Either that, or this is a bar graph that really wants to be a pie chart. Lawrence explains why this is problematic:

The eye is likely to use the angle as the measure and this is where an error may arise. It's almost an optical illusion―the smaller number of students lie on the circumferences of smaller circles―and a smaller length goes further around the circle. Thus, for example, Turkey attracts about one fifth of those students attracted by Germany but it looks like it's nearer half (45 degrees vs 90 degrees).

Another instance of the same illusion is the five countries Czech Republic, Denmark, Ireland, Austria, and Turkey. The curving lines of these nations all appear to be the same length, yet the number of students visiting each ranges from a low of 6,145 for Turkey to 6,437 for the Czech Republic. The whole point of a bar chart is to make it possible to visually compare the sizes of different sets without having to look at the numbers, but curving the bars in this way defeats that purpose.

As if that weren't bad enough, Lawrence points out:

The case of Spain is really bizarre―it looks like it's gone round the circle by over 280 degrees but actually what they've done is to break off the line at 90 degrees and stick the bit they broke off back on the diagram at the left.

I doubt that the chartmaker intended that anyone think that Spain had over three times as many student visitors as Germany, though that would be a natural impression. Rather, I suspect that the maker was just trying to come up with a chart that was novel and visually-striking, and succeeded. It's a pretty picture, but as far as communicating data goes, you'd be better off ignoring the curvy colored lines and just read the numbers down the middle.

Source: "270 000 students benefitted from EU grants to study or train abroad", Ibercampus, 11/7/2014

December 31st, 2016 (Permalink)

Book Review: Standard Deviations

Title: Standard Deviations

Sub-Title: Flawed Assumptions, Tortured Data, and Other Ways to Lie with Statistics

Author: Gary Smith

Publisher: Overlook

Date of Publication: 2014

Quote: "Sometimes, the unscrupulous deliberately try to mislead us. Other times, the well-intentioned are blissfully unaware of the mischief they are committing. My intention in writing this book is to help protect us from errors―both external and self-inflicted. You will learn simple guidelines for recognizing bull when you see it―or say it. Not only do others use data to fool us, we often fool ourselves." (P. 5)

Review: Not long ago, I remarked that there seems to have been a spate of books in the last few years about how to detect misinformation on the internet and elsewhere1. This is one of those I mentioned, and it's a good one. With the current concern about the influence of fake news, it's as timely as when it first appeared a couple of years ago.

If you want to learn statistics, I can recommend a few good textbooks, but this is not one of them. That's because it's not a textbook at all, but a book on statistics for us non-statisticians. We may not be statisticians, but we are consumers of statistics. Politicians, activists, advertisers, and others, use statistics to influence us to vote for them, support their causes, or buy their products. Unfortunately, many of these statistics are impostors, and most of us don't know how to separate the statistical sheep from the wolves-in-sheeps'-clothing. If you're not a statistician, then this book's for you.

The author of this book, Gary Smith, is an economist, but he's also authored statistics textbooks. I haven't read any of them, but if they're as good as this non-textbook, I would recommend them as well.

I actually learned some things from this book. I write that with some surprise because I've read several books that cover much the same territory, and some of which were excellent. Not only that, but I've written entries on some of the fallacies discussed by Smith, so that most of what is covered here is familiar ground to me. Here is a selective outline of topics covered in this book:

Lest I seem unduly positive in this review, I will mention one mistake that I think Smith makes: In chapter 6, he discusses a well-known problem in probability theory sometimes known as the "boy-girl paradox" (pp. 93-96). If you're unfamiliar with it, I based a puzzle last Christmas on this very paradox10. While I'm not an expert, I gave as the solution to that puzzle what Smith disparagingly refers to as the "expert's" answer. At first, I was impressed by Smith's arguments but, counterintuitive as it may seem, in this case I think that the experts were right. Oh, well, nobody bats 1.000.

Recommendation: Highly recommended for non-statisticians and other non-experts.

Sources & Resources:

  1. New Book: A Field Guide to Lies, 10/19/2016
  2. Check 'Em Out, 1/21/2010
  3. Michael Shermer, "Patternicity: Finding Meaningful Patterns in Meaningless Noise", Scientific American, 12/1/2008
  4. New Book: Standard Deviations, 8/30/2014
  5. Charts & Graphs: The YY Graph, 11/19/2013. See the previous entries listed at the bottom of this entry.
  6. The Gambler's Fallacy
  7. The Hot Hand Fallacy
  8. The Regression Fallacy
  9. The Texas Sharpshooter Fallacy
  10. A Puzzle for Christmas, 12/25/2015

December 25th, 2016 (Permalink)

A Christmas Puzzle Present

The Smith family has four children in all, each a year apart in age. Santa Claus brought each child a different type of puzzle for Christmas, including a book of logic puzzles. Each present was wrapped in a different color ribbon, one of which was gold, and tied with a bow on top. The four children took turns opening their presents on Christmas morning.

Annie was the first to open her present, the thirteen-year-old sibling was second, a book of acrostics was the third present opened, and the present topped with a green ribbon―which was not the jigsaw puzzle―was the last to be opened, though not by Charlie.

Bonnie, who is not the oldest sibling, received a present sporting a shiny silver ribbon.

Donnie, who did not receive the book of crossword puzzles as his gift, is a year younger than the sibling who received the jigsaw puzzle.

The gift with a red ribbon was unwrapped immediately after the twelve-year-old's present was opened.

From the above information, can you determine the order in which each child opened his or her present, the age of each child, the type of puzzle each received as a gift, and the color of ribbon in which the puzzle was wrapped?

Solution


December 22nd, 2016 (Permalink)

New Book: Winning Arguments

Stanley Fish―yes, that Stanley Fish―has a new book out titled Winning Arguments: What Works and Doesn't Work in Politics, the Bedroom, the Courtroom, and the Classroom. This is a popular topic for books, since everybody wants to "win" arguments, and the Fallacy Files library contains at least five books with similar titles:

  1. Madsen Pirie's How to Win Every Argument 1, the updated and expanded edition of the classic Book of the Fallacy2.
  2. An earlier book with the same title by Nicholas Capaldi, though like Pirie's book it originally had a different title: The Art of Deception3.
  3. Lawyer Gerry Spence's How to Argue and Win Every Time4.
  4. How to Win Arguments, by the late publisher of National Review, William Rusher, with the more modest subtitle: "More Often Than Not". So, a more accurate title for the book would have been: How to Win Most Arguments.
  5. Finally, the most modestly titled book is How to Win an Argument5, by philosopher Michael Gilbert. Apparently, Gilbert is not claiming that his book will teach you how to win every argument, or even most, and if you win just one you'll have no cause for complaint.

Resources:

  1. New Book, 12/2/2007
  2. Good News, 9/27/2004
  3. In the Mail, 10/23/2012
  4. Check it Out, 2/24/2007
  5. New Book: Arguing with People, 10/21/2014

December 19th, 2016 (Permalink)

Fake News Headline

Fake news is now all the rage, or at least worrying about it is. You'd think from all the current handwringing that people have just discovered that there is fake news on the internet. However, fake news is older than the internet; in fact, it's as old as news itself. Here's a headline from just a few years ago:

Scientific study reveals conspiracy theorists the most sane of all

This is one of a few similar headlines each found on various different fake news sites, all referring to the same study. Why is this fake news? The study had nothing to do with the sanity or lack thereof of conspiracy theorists (CTists). Don't believe me? Would you believe a co-author of the study?

This study has recently been linked to as a demonstration that people who believe 9/11 conspiracy theories are “more sane” than people who don’t. The study has no bearing on mental health, and this claim about “sanity” relies on wishful misinterpretation of the results.6

Don't believe a co-author of the study? Would you believe your own lying eyes? If so, see the study itself 5. As you can easily see for yourself, the words "sane", "insane", "sanity" and "insanity" do not even appear in it.

If the study itself had nothing to do with the comparative sanity of CTists, where did the authors of these articles get the idea that it did? It appears that they all trace back to a single source, namely, an article by Kevin Barrett on the PressTV website1.

What is PressTV? It is a TV network funded and controlled by the government of Iran, that is, it's a propaganda site2. Also, the president of Iran at the time the article was written was a notorious CTist who has called the Holocaust a "myth"4, and Iran has hosted conferences questioning its reality3.

Who is Kevin Barrett? Well, he's a CTist himself, specifically of the 9/11 variety. This fact, of course, was not revealed in the little biographical blurb at the bottom of his fake news article. In addition to writing fake news for PressTV, Barrett has also appeared on the network.

What was the "wishful misinterpretation" that led Barrett to claim that the study showed that CTists are "saner" than others?

…[A]mong people who comment on news articles, those who discount official government accounts of events like the 9/11 attacks and the assassination of John F. Kennedy outnumber believers by more than two-to-one. That means the pro-conspiracy commenters are those who are now expressing what is considered conventional wisdom, while the anti-conspiracy commenters represent a small, beleaguered minority that is often scoffed at and shunned.1

So, apparently Barrett thinks that the fact that CTists outnumbered non-CTists on the specific comment threads to the news articles examined in the study somehow shows that they are now in the majority, and that "sanity" is simply whatever the majority believes. This, of course, is wrong on both counts.

Even if the study had been about sanity, it could not have correctly concluded anything about CTists and non-CTists in general. Here's the co-author's description of the study design: "In this study…we collected over 2000 comments from online news stories about 9/11―the ones that tried to persuade people one way or another regarding whether the attacks were the result of a government conspiracy."6 So, the study was based on a sample of online comments, which is not a representative sample of people in general, but only of those who are inclined to leave online comments to news stories. It is very likely that such people are at least more strongly motivated about the subject of the story than those who choose not to comment, and so they are not representative of CTists in general, let alone people in general.

Putting aside the issue of comparative sanity, this example is consistent with a point I've made before about conspiracy theorists―or, perhaps it's more a point about those who pander to conspiracy theorists for gain. While conspiracy theorists are quick to claim that we are all being lied to by the powers that be, one of the most conspicuous traits of those who write articles promoting such theories is their mendacity or at least lack of concern about the truth. It's not hard to find out that the study in question has nothing to do with sanity or insanity, yet there are still many versions of uncorrected articles with the same or similar headlines. Admittedly, the sites that host such articles are disreputable ones―and would be disreputable for no other reason than hosting the articles. I won't link to any of them since I don't want to help the Google ranking of fake news sites, but if you want to check them for yourself―which is a practice that I generally recommend―you can do so by searching on the headline given above, or related keywords.

Sources:

  1. Kevin Barrett, "New studies: ‘Conspiracy theorists’ sane; government dupes crazy, hostile", PressTV, 7/12/2013. This is the Wayback Machine's archive of the page from PressTV, which appears to be no longer available on PressTV's website, so linking to this page doesn't help its Google ranking.
  2. Douglas Murray, "Push off now, Press TV, and take your conspiracy theories with you", Spectator, 1/20/2012.
  3. Robert Tait, "Holocaust deniers gather in Iran for 'scientific' conference", The Guardian, 12/11/2006.
  4. Karl Vick, "Iran's President Calls Holocaust 'Myth' in Latest Assault on Jews", The Washington Post, 12/15/2005.
  5. Michael J. Wood & Karen M. Douglas, "'What about building 7?' A social psychological study of online discussion of 9/11 conspiracy theories", Frontiers in Psychology, 7/8/2013.
  6. Mike Wood, "What does online discussion tell us about the psychology of conspiracy theories?", The Psychology of Conspiracy Theories, 7/10/2013.
  7. Mike Wood, "Setting the record straight on Wood & Douglas, 2013", The Psychology of Conspiracy Theories, 7/13/2013.
  8. Mike Wood, "The Wood & Douglas (2013) commission report: Whitewash or coverup?", The Psychology of Conspiracy Theories, 7/28/2013.

Fallacy: Unrepresentative Sample


December 13th, 2016 (Permalink)

Lesson in Logic 15: Contradiction

In the previous lesson, you learned how to use Venn diagrams to tell whether or not two categorical statements are logically equivalent. In this lesson, you'll learn how to use the diagrams to tell whether two such sentences contradict one another.

As explained in the previous lesson, logically equivalent statements are ones that say the same thing, logically. Contradictory statements are ones that say the "opposite", logically. A more precise way of putting this is that contradictory statements must have opposite truth-values, that is, if one is true then the other is false, and vice versa.

An important fact about logical equivalence is that there isn't just one statement that is logically equivalent to a given statement, but many―in fact, infinitely many, though I won't bother proving so here; just think about it. Similarly, there are many statements―again, infinitely many―that contradict a given statement, all of which are logically equivalent to each other.

Whereas Venn diagrams of logically equivalent statements are either identical or mirror images of each other, diagrams of contradictory statements are like the negative and positive images of a black-and-white photograph. A negative has black where the positive image has white, and white where the positive image has black. Similarly, a Venn diagram of a statement has shading where the diagram of a contradiction has an X, and Xs where the contradictory diagram has shading. Venn Diagram for E Proposition

For example, consider the statement: No cats are pets. First off, draw a Venn diagram of this statement. Since this is an E statement, the diagram will be the typical one in which the central, overlap region is shaded. If we let A=cats and B=pets, then the generic E statement diagram to the right represents the example.

As explained above, a contradictory diagram will shade those regions that the diagram of the contradicted statement has Xs―in this case, there are no Xs in the diagram. However, it will also have Xs in those regions of the contradicted diagram that have shading. Since the diagram in this example has shading in the (American) football-shaped region, the contradictory diagram will have an X in that region―see the diagram below and to the right. Venn Diagram for I Proposition

Now, if you've been following along carefully in these lessons, you will recognize the resulting diagram as the typical one for an I statement. Thus, the categorical I statement "some cats are pets" contradicts "no cats are pets". Intuitively, this should seem correct, since if one of these two statements is true then the other will be false, and vice versa, which is the definition of contradiction.

So, if you wish to tell whether two categorical statements are contradictory, draw Venn diagrams for each statement labelling the circles the same way in each diagram. If the diagram of each statement is a negative image of the other―that is, has Xs where the other has shading and shading where the other has Xs―then the statements are contradictories; if not, not.

Assuming that you are familiar enough with Venn diagrams to read off a categorical statement from a diagram, you can also use a Venn diagram to discover a contradiction of a given statement. Just draw a Venn diagram of the statement that you wish to contradict, produce its negative image, then read off a contradictory statement from the diagram. Give it a try in the exercises below.

Exercises:

  1. What is a contradiction of "some cats are not pets"?
  2. Does "some pets are cats" contradict "some cats are not pets"?
  3. Do "no non-mammals are bats" and "some bats are non-mammals" contradict one another?
  4. What is a contradiction of "all cats are non-pets"?
  5. Are "all cats are pets" and "no pets are cats" contradictory?

Answers

Previous Lessons:

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