Lecture: Categorical Reasoning
Lecture script · about 15–20 minutes spoken · Week 5
Opening
Picture a corkboard outside a student club's meeting room. A rule is pinned to it: "Only members may vote at the spring meeting." A student who joined last week reads it and says, "Great, I'm a member, so I can vote." Then a friend reads the next line aloud. Voting also needs a ballot request filed two weeks ahead. The rule said who is shut out. It never said who gets in.
Sentences like that one sort the world into groups. A nursing student checks which patients a safety rule covers. A business major reads a store's return policy. A club officer reads who may vote. In every case, knowing exactly what "all", "no", and "some" claim about a group keeps you from drawing a conclusion the rule never gave you.
In Week 1, you learned to find an argument. In Week 2, you wrote it in standard form. In Week 3, you learned validity and the counterexample method. In Week 4, you met the valid "if" forms and the two formal fallacies. In this lecture, I'll show you four sentence patterns for claims about groups, how to rewrite everyday sentences into them, and how to build an argument from them. Then we'll test those arguments with three overlapping circles, and finish with what the circles do not say. After that, in Week 6, the course turns to Language and Meaning.
1. Four ways to talk about groups
Let's start at a coffee shop counter. There's an allergy card by the register. "All of our muffins contain eggs." "None of our muffins contain peanuts." "Some of our muffins contain walnuts." "Some of our muffins do not contain gluten." That's one group of muffins and four different promises. If you have an egg allergy or a walnut allergy, you need to know exactly what each line says.
Here's the term. A categorical form is one of the four sentence patterns for claims about groups: All S are P, No S are P, Some S are P, and Some S are not P, where S and P name groups of things. The subject term is the group the claim is about, and the predicate term is the group it puts the subject in or out of. In "All voters are members," the subject term is voters and the predicate term is members.
Notice a change from Week 4. There, P and Q stood for whole claims, like "it rained overnight." This week, S and P stand for groups of things, like muffins or members. Same kind of shorthand, different job.
Each form has a name. "All S are P" is the universal affirmative. It says nothing is in the S group without also being in the P group. "No S are P" is the universal negative. It says nothing is in both groups. "Some S are P" is the particular affirmative, and "Some S are not P" is the particular negative.
Now the word to watch. In logic, "some" means "at least one." It doesn't mean "a few," and it doesn't mean "some but not all." If every muffin on the tray had walnuts, "some muffins contain walnuts" would still be true.
And "all" never flips on its own. "All muffins contain eggs" is true. "All foods with eggs are muffins" is false, because the shop's quiche has eggs too. Flipping an "all" claim changes what it says, the same way flipping an "if" sentence did in Week 4.
Pause and try this. The card says, "Some of our muffins contain walnuts." Does that sentence tell you that some muffins are walnut-free?
Think about what "some" promises, and what it leaves open.
Here's the answer. No, it doesn't. "Some" means at least one, so the line stays true even if every muffin has walnuts.
"Some" means at least one; "all" never flips.
2. Rewriting everyday sentences
People around you rarely talk in the four forms. They say "only," "none of," "every," "not all," and "a few." So before you test an argument about groups, you rewrite each claim into one of the four forms. It's the same move you made with "only if" in Week 4.
The rewrite takes three steps. Find the two groups. Decide whether the claim covers the whole subject group, with "all" or "no," or at least one member, with "some." Then make sure both terms name groups, adding a word like "items" or "people" when a term is an action.
Let's run it on the club's rule. "Only members may vote" becomes "All voters are members." Think of "only" as a bouncer at the door. Everyone inside has a wristband. That doesn't mean everyone with a wristband is inside. The group after "only" goes in the predicate, and the rule promises nothing about every member voting.
Now a store's policy. "Sale items cannot be returned" becomes "No sale items are returnable items." The action "be returned" turns into the group "returnable items," so both terms name groups.
A recipe card says, "A few of the desserts contain nuts." That becomes "Some desserts are dishes that contain nuts." A few means at least one, so the form is "Some S are P."
And a store's refund desk says, "Not every return is refunded in cash." That becomes "Some returns are not cash refunds." Notice what it does not become. It isn't "No returns are cash refunds." Saying "not every" leaves at least one return outside the cash group, and that's all it says.
One caveat. A rewrite is a reading, and some sentences allow more than one. When the words leave room, choose the reading that fits what the speaker most plausibly meant, the way the principle of charity asks in Week 2, and say which reading you chose.
Pause and try this. Someone rewrites "Only members may vote" as "All members are voters." What went wrong?
Ask which group the word "only" sits in front of.
Here's the answer. The groups got flipped. "Only" marks the requirement, so the right rewrite is "All voters are members," and a member who never shows up doesn't break the rule.
The group after "only" is the predicate, not the subject.
3. Building a syllogism
Now picture a group chat arguing about the campus gym. Someone writes, "Everyone in the cycling club is in the student union. Everyone in the student union can use the gym. So everyone in the cycling club can use the gym." That's two claims about groups leading to a third.
Here's the term. A categorical syllogism is a deductive argument with two premises and a conclusion, each written in one of the four categorical forms, that uses exactly three terms between them. The study of the syllogism begins with Aristotle's Prior Analytics. You don't need its details. It's enough to know this way of reasoning about groups is very old.
Each of the three terms gets a name from its place in the conclusion. The minor term is the subject of the conclusion, the group the conclusion is about. The major term is the predicate of the conclusion. The middle term is the group that appears in both premises and never in the conclusion, the link the premises share.
In the gym argument, the conclusion is about club cyclists, so that's the minor term. It places them among people who can use the gym, so that's the major term. Student-union members show up in both premises and then leave, so that's the middle term. Think of the middle term as a mutual friend who introduces two people and then steps out of the photo.
Now here's a trap. "All poodles are dogs. All beagles are dogs. So all poodles are beagles." Both premises mention dogs, so it feels connected. But each premise only puts its group somewhere inside the dog group. Neither says how poodles and beagles relate to each other. The conclusion is false while both premises are true. Sharing a term is not enough. We'll see exactly where the gap sits in the next part.
Pause and try this. In "All tutors passed the course. Some first-year students are tutors. So some first-year students passed the course," which group is the middle term?
Find the group that shows up in both premises but not in the conclusion.
Here's the answer. Tutors is the middle term, because it appears in both premises and drops out of the conclusion.
The middle term links the premises and then leaves.
4. Testing with three circles
Picture three hula hoops dropped on a gym floor so each one overlaps the other two. The floor inside them splits into patches. Some patches sit inside one hoop, some inside two, and one sits inside all three. A Venn diagram is that picture on paper, with one circle for each term.
Here's the term. A Venn diagram is a drawing of overlapping circles, one for each term, used to test a categorical syllogism: a shaded region is empty, and an X sits in a region that holds at least one thing. Venn introduced these diagrams in 1880.
Count the patches first. Each syllogism has 3 terms, so it gets 3 circles. Every patch is either inside or outside each circle, so 2 times 2 times 2 gives 8 regions, counting the one outside all three circles.
The test runs in three moves, always in this order. First, shade every region an "All" or "No" premise says is empty. Second, place an X where a "Some" premise says at least one thing exists, and put the X on a line if the premises don't settle which side it belongs on. Third, check whether the conclusion's region is already shaded or already holds an X. If the premises have already drawn the conclusion, the syllogism is valid. If the conclusion's region is still open, it's invalid.
Let's run the gym argument. The first premise says all club cyclists are union members, so you shade the part of the cyclist circle outside the union circle. The second says all union members can use the gym, so you shade the part of the union circle outside the gym circle. Now look at the cyclist circle. Every part of it outside the gym circle is already shaded. Valid.
The reptile argument works the same way. "No reptiles are mammals" shades the whole reptile-and-mammal overlap. "All snakes are reptiles" shades the snake circle outside the reptile circle. The whole snake-and-mammal overlap ends up shaded. Valid.
The tutor argument needs the order of the moves. You shade the part of the tutor circle outside the passed circle first. Then the X for "some first-year students are tutors" has only one place left to go: the region inside all three circles. That region sits inside the first-year-and-passed overlap. Valid. If you'd placed the X first, you might have dropped it in a region the "All" premise says is empty.
Now the poodles. You shade the poodle circle outside the dog circle, and the beagle circle outside the dog circle. The region of poodles that aren't beagles, inside the dog circle, stays open. Invalid.
The volunteers argument says some volunteers are seniors and some seniors are runners. Neither premise shades anything. Each X lands on the line of the circle its premise doesn't mention, so no X is forced into the volunteer-and-runner overlap. Invalid.
And the nurses. "All nurses are trained in first aid" shades the nurse circle outside the first-aid circle. "No lifeguards are nurses" shades the lifeguard-and-nurse overlap. The region of lifeguards trained in first aid who aren't nurses stays open. Picture a lifeguard who took a first-aid course. Both premises stay true, and the conclusion is false. Invalid.
So the tally is 3 valid syllogisms plus 3 invalid syllogisms, which makes 6 in the set. Syllogisms 1, 2, and 3 are valid because the premises alone shade or mark the conclusion's region. Syllogisms 4, 5, and 6 are invalid because the premises leave a region open for a counterexample.
Pause and try this. In the poodle syllogism, which region has to be empty for "all poodles are beagles" to follow, and is it?
Ask what the conclusion rules out, then check that region.
Here's the answer. The region of poodles that aren't beagles has to be empty, and it isn't, because neither premise shades the part of it inside the dog circle.
Shade first, then X, then look.
5. What the circles don't say
A finished Venn diagram looks like a map, and maps tempt you to count. Resist that. A shaded region says "empty." A region with an X says "at least one." A blank region says "nothing settled." No region tells you whether it holds one thing or a whole crowd, and the size of a circle on the page means nothing.
There's a second thing the diagram doesn't say. Suppose a community garden posts a sign: "All trespassers after dark will be fined." Is the sign true on a quiet night when nobody trespasses? On the reading this course uses, yes. The sign only rules out a trespasser who escapes the fine. It doesn't claim any trespasser exists.
Here's the term. The modern reading is the reading of the categorical forms used with Venn diagrams, on which All S are P says only that nothing is S without being P and makes no claim that any S exists. That's why an "All" premise gets shading and never an X.
Older logic read it differently. The traditional reading is the older reading of the categorical forms, on which All S are P also says that at least one S exists, which makes a few syllogisms valid that the modern reading calls invalid. Take this one. "All trespassers after dark will be fined. All trespassers after dark will be reported. So some people who will be reported will be fined." Read traditionally, trespassers exist, so someone is both reported and fined. Read the modern way, both premises hold on a quiet night with no trespassers, and the conclusion can be false.
The course discusses cases like that one in words and never grades them, because the verdict depends on which reading you start from. And here's an honest note. The traditional reading isn't a mistake. It reflects how people often talk about groups they know are real. The modern reading is a choice the course makes so that every keyed verdict comes from the diagram alone.
Pause and try this. On the modern reading, does "All trespassers after dark will be fined" tell you that anyone trespassed?
Think about what shading says, and what only an X can say.
Here's the answer. No, it doesn't. An "All" claim only rules things out, so it shades regions and places no X, and only a "Some" claim says someone is there.
Shading says empty; only an X says someone is there.
Closing
Let me pull it together. The four categorical forms are All S are P, No S are P, Some S are P, and Some S are not P, and "some" always means at least one. Everyday sentences get rewritten into those forms first, and the group after "only" goes in the predicate. A categorical syllogism has two premises, a conclusion, and three terms, with the middle term linking the premises. The Venn test shades first, places the X second, and then checks the conclusion's region, which makes syllogisms 1, 2, and 3 valid and 4, 5, and 6 invalid. And a diagram shows only empty and occupied regions, read the modern way, with no existence claim in "all".
Next in this module come the slide deck, the Readings and Resources page, the lecture tutorial, the practice set, and the quiz. Then, in Week 6, Language and Meaning, the test turns from the shape of an argument to its words.
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