Week 5 · Chapter

Chapter 5 — Categorical Reasoning

Critical Thinking Free sample edition

The module's primary reading · ~35–45 minutes · Objective 3 (with Objective 2 carried forward) · Read it first this week; the lecture, the slides, the tutorial, the quiz, and the Argument Workshop all build on it.

In plain language

This week is about claims that sort things into groups, such as "all members may vote" or "some desserts contain nuts." You will learn four sentence patterns for these claims, the categorical forms (the four ways of saying that a group is inside, outside, or partly inside another group). You will put two such claims together to reach a third, in an argument called a categorical syllogism (a three-line argument about three groups). Then you will test that argument with a Venn diagram (a drawing of overlapping circles, one for each group). Reading "all" and "some" carefully keeps you from flipping a rule around and drawing a conclusion it never gave you.

Why you already care about this

A student club posts its rules on a corkboard: "Only members may vote at the spring meeting." A first-year student who joined last week reads it and says, "Great, I'm a member, so I can vote." A second student 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.

You read sentences about groups like that one all the time. A store says no sale items can be returned. A recipe card says some of the desserts contain nuts. A clinic says all visitors must sign in at the front desk. Each sentence draws a line around one group and says where another group sits. People misread them every day, most often by flipping them around.

Last week you tested arguments built on "if" sentences. This week you test arguments built on "all", "no", and "some", and you get a new tool for it: three overlapping circles, read one region at a time.

By the end of this chapter you will be able to:

write a claim in each of the four categorical forms and say what each one does and does not say;
rewrite an everyday claim, including an "only" claim, into one of the four forms;
name the minor, major, and middle terms of a categorical syllogism;
test a categorical syllogism with a three-circle Venn diagram described in words, and state the verdict in one plain sentence;
explain why a Venn diagram shows only empty and occupied regions, and why the course reads "all" without an existence claim.

Carry this line with you: All S are P is not All P are S.

Where we are

Already covered: Weeks 1 and 2 taught you to find an argument and write it in standard form. Week 3 taught you validity and the counterexample method. Week 4 taught you the valid "if" forms and the two formal fallacies.
This week: the four categorical forms, rewriting everyday claims into them, the categorical syllogism and its three terms, the Venn test, and the modern reading of "all".
How they connect: a Venn diagram is a way of searching for a counterexample. An open region the premises leave unsettled is room for a case with true premises and a false conclusion.
Where we're going next: Week 5 closes the deductive unit, and Week 6, Language and Meaning, turns to the words themselves: definitions, vagueness, and ambiguity.

Week 1 gave you the habit of finding a conclusion and its reasons. Week 2 gave you standard form, with each premise on its own numbered line. Week 3 gave you the definition that matters most this week: an argument is valid when it is impossible for all its premises to be true and its conclusion false. Week 4 showed that a pattern can be judged apart from its topic, and that flipping an "if" sentence around changes what it says. This week the same lesson returns with groups in place of "if" sentences.

1. Read the four forms: all, no, some, and some-not

Think about a coffee shop's allergy card. "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." Four sentences, one group of muffins, four different promises. A customer with an egg allergy, a peanut allergy, or a walnut allergy needs to know exactly what each sentence says.

You can write sentences like these in four fixed patterns, with letters standing in for the groups.

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.

Each sentence you read this way has two groups in it.

The subject term is the group a categorical claim is about, written S in the four forms; in All voters are members, the subject term is voters.

The predicate term is the group a categorical claim says the subject group is in or out of, written P in the four forms; in All voters are members, the predicate term is members.

Notice a change from Week 4. There, P and Q stood for whole claims, such as "it rained overnight." This week, S and P stand for groups of things, such as muffins or members. The letters are the same kind of shorthand, but they name different things.

Each of the four forms has a name.

The universal affirmative is the categorical form All S are P, which says that nothing is in the S group without also being in the P group.

The universal negative is the categorical form No S are P, which says that nothing is in both the S group and the P group.

The particular affirmative is the categorical form Some S are P, which says that at least one thing is in both the S group and the P group.

The particular negative is the categorical form Some S are not P, which says that at least one thing is in the S group and outside the P group.

The word to watch, for you, is "some." In this course, "some" means "at least one." It does not mean "a few," and it does not mean "some but not all."

Worked example — the allergy card (illustrative). Write each line of the card in its form and say what it does not say.

"All of our muffins contain eggs." Form: All S are P, with S "muffins" and P "foods that contain eggs." It does not say that every food with eggs is a muffin; the shop's quiche contains eggs too.
"None of our muffins contain peanuts." Form: No S are P, with S "muffins" and P "foods that contain peanuts." It also says, with the same strength, that no food with peanuts is a muffin, because "No S are P" is about the overlap, and the overlap is empty either way.
"Some of our muffins contain walnuts." Form: Some S are P. It says at least one muffin has walnuts. It does not say that some muffins are walnut-free.
"Some of our muffins do not contain gluten." Form: Some S are not P. It says at least one muffin is gluten-free. It does not say that most are.

In words: each form makes one precise promise about where the muffins sit, and your safest reading keeps every promise exactly that size.

Misconception check — the flip. Students often think "All S are P" means the same as "All P are S." That is tempting, because the sentence uses the same two groups either way. But "all muffins contain eggs" is true while "all foods with eggs are muffins" is false. Flipping an "all" claim changes what it says, the same way flipping an "if" sentence did in Week 4.

Misconception check — "some" is not "some but not all". Students often hear "some muffins contain walnuts" and conclude that some muffins do not. That is tempting, because in conversation people say "some" when they mean "not all." But "some" in logic means "at least one." If every muffin had walnuts, "some muffins contain walnuts" would still be true.

Memory hook: "some" means at least one; "all" never flips.

2. Rewrite the claim: everyday sentences into the four forms

The people around you rarely talk in the four forms. They say "only", "none of", "every", "not all", and "a few". Before you can test an argument about groups, you rewrite each claim into one of the four forms, the way you rewrote "only if" before testing it in Week 4.

The rewrite has three steps. Find the two groups. Decide whether the claim covers the whole subject group ("all" or "no") or at least one member ("some"). Then make sure both terms name groups, adding a word such as "people" or "items" when a term is an action or a description.

Worked example — the club, the store, and the recipe card (illustrative). Rewrite each sentence into a categorical form.

The club's rule. "Only members may vote" becomes "All voters are members". The word "only" puts the group after it in the predicate spot: everyone who votes is a member, and the rule promises nothing about every member voting.
The store's policy. "Sale items cannot be returned" becomes "No sale items are returnable items". The verb "be returned" turns into the group "returnable items," so both terms name groups.
The recipe card. "A few of the desserts contain nuts" becomes "Some desserts are dishes that contain nuts". "A few" means at least one, so the form is "Some S are P."
The store's refunds. "Not every return is refunded in cash" becomes "Some returns are not cash refunds". "Not every" denies an "all" claim, which leaves at least one return outside the cash-refund group.

In words: each rewrite keeps the sentence's meaning and puts it in a form whose promise you can read exactly, so "only members may vote" becomes "All voters are members".

Two more patterns will come up for you often. "Every S is P" and "Each S is P" both become "All S are P." "Not all S are P" becomes "Some S are not P," because it says at least one S falls outside P. It does not say "No S are P."

Misconception check — "only" flips the groups. Students often rewrite "Only members may vote" as "All members are voters." That is tempting, because "members" comes first in the sentence. But "only" marks the requirement, so the group after "only" becomes the predicate. A member who never shows up breaks "All members are voters" without breaking the club's rule. The right rewrite is "All voters are members."

A caveat: a rewrite is a reading, and some everyday sentences allow more than one. "Students who work late get tired" might mean all of them or most of them. When the words leave room, choose the reading that fits what the speaker most plausibly meant, as the principle of charity from Week 2 asks, and say which reading you chose.

Memory hook: the group after "only" is the predicate — not the subject.

3. Build the argument: the categorical syllogism and its three terms

A group chat is arguing about the campus gym. Someone writes: "Everyone in the cycling club is in the student union, and everyone in the student union can use the gym, so everyone in the cycling club can use the gym." That is two claims about groups leading to a third. Arguments shaped like that have a long history.

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 do not need its details here; it is enough to know that this way of reasoning about groups is very old.

Each of the three terms gets a name, and you find it from the term's place in the conclusion.

The minor term is the subject of a categorical syllogism's conclusion, the group the conclusion is about.

The major term is the predicate of a categorical syllogism's conclusion, the group the conclusion places the minor term in or out of.

The middle term is the group that appears in both premises of a categorical syllogism and never in its conclusion, the link the premises share.

Worked example — naming the three terms (illustrative). Take the gym argument from the categorical-syllogism set and write it in standard form.

P1: All club cyclists are student-union members.

P2: All student-union members can use the gym.

So,

C: All club cyclists can use the gym.

Find the conclusion's subject. The conclusion is about club cyclists, so "club cyclists" is the minor term.
Find the conclusion's predicate. The conclusion places them among people who can use the gym, so "people who can use the gym" is the major term.
Find the shared link. "Student-union members" appears in both premises and not in the conclusion, so it is the middle term.

In words: the three terms are club cyclists (minor), people who can use the gym (major), and student-union members (middle), and the middle term is what lets the premises talk to each other.

The letters from Section 1 carry over. In the course's notation, S and P are the subject and predicate terms of a form, and M is the middle term, the group named in both premises and not in the conclusion.

Misconception check — sharing a term is not enough. Students often think that two premises sharing a term must lead somewhere. Take "All poodles are dogs. All beagles are dogs. So all poodles are beagles." Both premises mention dogs, so the argument feels connected. But both premises only place their groups somewhere inside the dog group; neither says how poodles and beagles relate to each other. The conclusion is false while both premises are true. Section 4 shows exactly where the gap sits.

An honesty note: logicians have built a large system of names and rules for syllogisms, with labels for every pattern. This course leaves that system out. The Venn test in the next section settles every syllogism you will meet, and it shows its reasons in plain sight.

Memory hook: the middle term links the premises and then leaves.

4. Test it with circles: the Venn diagram

Picture three hula hoops dropped on a gym floor so that each one overlaps the other two. The floor inside the hoops splits into separate patches. Some patches sit inside one hoop, some inside two, and one patch sits inside all three. A Venn diagram is that picture drawn on paper, with one circle for each term of a syllogism.

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 (1880), in "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" in the Philosophical Magazine, introduced these diagrams.

Two marks do all the work for you. A shaded region is marked empty, with nothing in it. An X says that at least one thing sits in that region. An X on a line means the thing is in one of the two regions the line divides, without saying which.

You run the test in three moves, always in this order:

Shade every region a premise says is empty, doing the "All" and "No" premises first.
Then place an X wherever a "Some" premise says at least one thing exists, and put the X on a line when the premises do not settle which side it belongs on.
Finally, check whether the conclusion's region is already shaded or already holds an X.

If the premises alone have already drawn the conclusion for you, the syllogism is valid. If the conclusion's region is still open, the premises leave room for a case with true premises and a false conclusion, and the syllogism is invalid.

The table below lists every region of the gym diagram and says what the premises do to it.

The 8 regions of the club-cyclist diagram and their status after both premises are drawn
Region	Cyclist circle	Union circle	Gym circle	After the premises
1	inside	inside	inside	open
2	inside	inside	outside	shaded by P2
3	inside	outside	inside	shaded by P1
4	inside	outside	outside	shaded by P1
5	outside	inside	inside	open
6	outside	inside	outside	shaded by P2
7	outside	outside	inside	open
8	outside	outside	outside	open

The conclusion, "All club cyclists can use the gym," needs every cyclist region outside the gym circle to be empty. Those are regions 2 and 4, and both are already shaded.

Worked example — the categorical-syllogism set, tested (illustrative). Test all six syllogisms with a three-circle Venn diagram, tally the verdicts, and count the diagram's regions.

The diagram. Each syllogism has 3 terms, so it gets 3 circles; each region is either inside or outside each circle, so 2 × 2 × 2 = 8 regions, counting the region outside all three circles.
Syllogism 1. "All club cyclists are student-union members. All student-union members can use the gym. So all club cyclists can use the gym." You shade the part of the cyclist circle outside the union circle, then the part of the union circle outside the gym circle. Every part of the cyclist circle outside the gym circle is already shaded, so the syllogism is valid.
Syllogism 2. "No reptiles are mammals. All snakes are reptiles. So no snakes are mammals." You shade the whole reptile-and-mammal overlap, then the part of the snake circle outside the reptile circle. The whole snake-and-mammal overlap is now shaded, so the syllogism is valid.
Syllogism 3. "All tutors passed the course. Some first-year students are tutors. So some first-year students passed the course." You shade the part of the tutor circle outside the passed circle first. The X for "some first-year students are tutors" can then go only in the region inside all three circles, which sits inside the first-year-and-passed overlap, so the syllogism is valid.
Syllogism 4. "All poodles are dogs. All beagles are dogs. So all poodles are beagles." You shade the part of the poodle circle outside the dog circle and the part of the beagle circle outside the dog circle. The region of poodles that are not beagles, inside the dog circle, is left unshaded, so the syllogism is invalid.
Syllogism 5. "Some volunteers are seniors. Some seniors are runners. So some volunteers are runners." Neither premise shades anything, so you place both X marks. Each X lands on the line of the circle the premise does not mention, so no X is forced into the volunteer-and-runner overlap, and the syllogism is invalid.
Syllogism 6. "All nurses are trained in first aid. No lifeguards are nurses. So no lifeguards are trained in first aid." You shade the part of the nurse circle outside the first-aid circle, then the whole lifeguard-and-nurse overlap. The region of lifeguards trained in first aid who are not nurses is left unshaded, so the syllogism is invalid.
The tally. 3 valid syllogisms + 3 invalid syllogisms = 6 syllogisms in the set.

In words: syllogisms 1, 2, and 3 are valid because the premises alone already shade or mark the conclusion's region. Syllogisms 4, 5, and 6 are invalid because the premises leave a region open that could hold a case with true premises and a false conclusion.

An open region is a counterexample waiting for you to describe it. In syllogism 6, picture a lifeguard who took a first-aid course and is not a nurse. Both premises stay true, and "no lifeguards are trained in first aid" is false. That person lives in the region the premises left open.

Misconception check — the order of the moves. Students often place the X first, as soon as they read a "Some" premise. That is tempting, because the premise is right there. But in syllogism 3, an X placed early could land in the part of the tutor circle outside the passed circle, a region the "All" premise says is empty. Shade first, then place the X, and the X has only one place to go.

Misconception check — a valid verdict is not a true conclusion. Students sometimes read "valid" on syllogism 2 as "snakes really are not mammals." That conclusion happens to be true, but validity only says the premises would guarantee it. Week 3's lesson still holds: a valid syllogism with a false premise can have a false conclusion.

A limitation worth naming: the Venn method is a classroom tool for organizing your judgment about one argument at a time. It settles arguments that really are in the four forms. It cannot tell you whether you rewrote an everyday sentence fairly in the first place.

Memory hook: shade first, then X, then look.

5. Read only what is drawn: the modern reading

A finished Venn diagram looks like a map, and maps tempt you to count. Resist that urge when you read one. 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 ten thousand.

There is a second thing your diagram does not say. Suppose a community garden posts a sign (illustrative): "All trespassers after dark will be fined." Is the sign true on a night when nobody trespasses? On the reading this course uses, it is. The sign only rules out a trespasser who escapes the fine. It does not claim that any trespasser exists.

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 is why the Venn test shades for an "All" premise and never adds an X for it. Shading says "empty"; it does not say that anything is left in the rest of the circle.

Older logic, which you may meet in other books, read the forms 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.

Here is one such case for you to discuss, never to be graded (illustrative): "All trespassers after dark will be fined. All trespassers after dark will be reported. So some people who will be reported will be fined." On the traditional reading, the premises say trespassers exist, so at least one person is both reported and fined, and the argument is valid. On the modern reading, both premises are true on a quiet night with no trespassers, while the conclusion can be false, so the argument is invalid. In the diagram, the two "All" premises shade regions but place no X, and the conclusion needs an X.

You will discuss cases like this one in words. None of them is graded, because the verdict depends on which reading you start from, and the course keys only verdicts that do not.

Misconception check — a diagram is not a head count. Students often read a region with an X as "one thing" and a larger-looking region as "more things." That is tempting, because pictures usually show size. But an X means at least one, which could be one student or the whole first-year class, and the size of a circle on the page means nothing. Read each region for exactly one fact: empty, occupied, or unsettled.

Misconception check — "all" does not promise anyone exists. Students often think "All trespassers will be fined" proves someone trespassed. That is tempting, because people rarely make rules about empty groups. But on the modern reading, an "All" claim only rules things out. If you need a "some" conclusion, a "Some" premise has to supply the X.

An honesty note: the modern reading is a choice the course makes so that every keyed verdict comes from the diagram alone. The traditional reading is not a mistake. It reflects how people often talk about groups they know are real.

Memory hook: shading says empty; only an X says someone is there.

Key terms

categorical form — 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. (first used: 1. Read the four forms: all, no, some, and some-not)
subject term — The group a categorical claim is about, written S in the four forms; in All voters are members, the subject term is voters. (first used: 1. Read the four forms: all, no, some, and some-not)
predicate term — The group a categorical claim says the subject group is in or out of, written P in the four forms; in All voters are members, the predicate term is members. (first used: 1. Read the four forms: all, no, some, and some-not)
universal affirmative — The categorical form All S are P, which says that nothing is in the S group without also being in the P group. (first used: 1. Read the four forms: all, no, some, and some-not)
universal negative — The categorical form No S are P, which says that nothing is in both the S group and the P group. (first used: 1. Read the four forms: all, no, some, and some-not)
particular affirmative — The categorical form Some S are P, which says that at least one thing is in both the S group and the P group. (first used: 1. Read the four forms: all, no, some, and some-not)
particular negative — The categorical form Some S are not P, which says that at least one thing is in the S group and outside the P group. (first used: 1. Read the four forms: all, no, some, and some-not)
categorical syllogism — 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. (first used: 3. Build the argument: the categorical syllogism and its three terms)
minor term — The subject of a categorical syllogism's conclusion, the group the conclusion is about. (first used: 3. Build the argument: the categorical syllogism and its three terms)
major term — The predicate of a categorical syllogism's conclusion, the group the conclusion places the minor term in or out of. (first used: 3. Build the argument: the categorical syllogism and its three terms)
middle term — The group that appears in both premises of a categorical syllogism and never in its conclusion, the link the premises share. (first used: 3. Build the argument: the categorical syllogism and its three terms)
Venn diagram — 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. (first used: 4. Test it with circles: the Venn diagram)
modern reading — 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. (first used: 5. Read only what is drawn: the modern reading)
traditional reading — 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. (first used: 5. Read only what is drawn: the modern reading)

Summary

The four categorical forms are All S are P, No S are P, Some S are P, and Some S are not P.
"Some" means at least one, never "some but not all."
"All S are P" does not mean "All P are S."
"Only members may vote" becomes "All voters are members."
A categorical syllogism has two premises, a conclusion, and exactly three terms: minor, major, and middle.
The Venn test: shade first, then place the X, then check the conclusion's region.
Syllogisms 1, 2, and 3 in the set are valid; syllogisms 4, 5, and 6 are invalid.
A Venn diagram shows empty and occupied regions, never how many things a region holds.
The course uses the modern reading, so "All S are P" makes no existence claim.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A library posts (illustrative): "Only students with a current card may borrow laptops." Rewrite it as a categorical form, and name its subject term and predicate term.
A coach says (illustrative): "Some of the players are injured, so some of the players are not injured." Explain why the conclusion does not follow from the premise.
Test this syllogism with a Venn diagram, described in words (illustrative): "No vegan dishes contain cheese. Some pizzas contain cheese. So some pizzas are not vegan dishes."
Test this syllogism with a Venn diagram, described in words, and describe a counterexample if it is invalid (illustrative): "All electricians on the crew are licensed. Some licensed workers on the crew are plumbers. So some electricians on the crew are plumbers."

Answers (worked reasoning — read after attempting)

"Only" puts the group after it in the predicate spot, so the sign becomes "All laptop borrowers are students with a current card." The subject term is "laptop borrowers" and the predicate term is "students with a current card." The sign does not say that every cardholder borrows a laptop.
"Some" means "at least one." The premise says at least one player is injured. That stays true even if every player is injured, and then "some of the players are not injured" would be false. A true premise with a possibly false conclusion means the step does not follow; the coach read "some" as "some but not all."
The three circles are vegan dishes, foods with cheese, and pizzas. Shade first: "No vegan dishes contain cheese" shades the whole vegan-and-cheese overlap. Then place the X: "Some pizzas contain cheese" puts an X in the pizza-and-cheese overlap. The part of that overlap inside the vegan circle is already shaded, so the X goes in the region of cheese pizzas outside the vegan circle. Then look: "Some pizzas are not vegan dishes" needs an X in the pizza circle outside the vegan circle, and it is there. The syllogism is valid.
The three circles are electricians, licensed workers, and plumbers. Shade first: "All electricians are licensed" shades the part of the electrician circle outside the licensed circle. Then place the X: "Some licensed workers are plumbers" puts an X in the licensed-and-plumber overlap. That overlap is split by the electrician circle, and neither part is shaded, so the X goes on the electrician circle's line. Then look: the conclusion needs an X in the electrician-and-plumber overlap, and none is forced there. The syllogism is invalid. A counterexample: a crew where every electrician is licensed, one licensed plumber is not an electrician, and no electrician is a plumber. Both premises are true and the conclusion is false.

Next up in this module: the lecture, the slide deck, the Readings & Resources page, the lecture tutorial, and the practice set. Next week, in Language and Meaning, the test turns from the shape of an argument to its words: definitions, vagueness, and ambiguity.

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