All, no, some: checking a syllogism in two minutes

Someone at work says this to you in the lift: "Everyone who gets promoted here has done the leadership course. Everyone on my team has done the course. So everyone on my team is going to get promoted."

Put it in standard form, the way lesson 1 taught you:

1. All people who get promoted here are people who have done the course.
2. All people on my team are people who have done the course.
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C: All people on my team are people who get promoted here.

Something is wrong, and you can probably feel it. But feeling it isn't checking it, and if the conclusion had been one you liked, you might not have felt it at all. Lesson 2 gave you the counterexample method: imagine a case where both premises are true and the conclusion false. That works, and it takes imagination, which runs out when you're tired or when the argument is about something you care about. This lesson gives you the same test done with a pencil. You draw three circles, draw the premises, and look. If the conclusion is already on the page, the argument is valid. If it isn't, the gap on the page is your counterexample.

This is still step 4 of the six-step check. Find the conclusion, find the premises, supply what's missing, test the link, test the premises, and only then consult your opinion of the conclusion. Lesson 3 gave you four fast shapes for arguments built from "if". This lesson gives you one fast method for arguments built from "all", "no", and "some". Between them they cover most of the deductive arguments you'll meet, and the rest of the course is about the arguments that aren't deductive at all.

The four sentences

Everything here is built from four sentence shapes. A and B stand for any two categories: dogs, mammals, people on my team, things that flood.

• All A are B.
• No A are B.
• Some A are B.
• Some A are not B.

Aristotle worked out the logic of these four in the Prior Analytics, in the fourth century BC, and the method you're about to learn is a much later picture of his system, drawn with circles (Smith, 2022).¹ It's old because it works.

Everyday sentences don't arrive in these shapes, so the first skill is translation. Here's what people get wrong.

Bare plurals mean "all". "Dogs are mammals" is "All dogs are mammals." "Ground-floor flats flood" is a claim about all of them, and the speaker will retreat to "well, most of them" the moment you find one that didn't. That retreat matters: "most" isn't one of the four forms, and an argument built on "most" is an inductive argument, which is lesson 5's business.

"Not all" is not "none". "Not all the trains were late" means "Some trains were not late." It does not mean "No trains were late." This one starts more bad rebuttals than any other, and there's a section on it below.

"All A are not B" is ambiguous in English. "All the shops are not open" usually means "Not all the shops are open" (some are shut), but sometimes means "No shops are open." Ask which. Don't guess.

"Only" reverses the direction. "Only members may park here" is "All people who may park here are members." It is not "All members may park here." This is the same reversal as "only if" in lesson 3: "only" tells you what's necessary, not what's sufficient.

"Some" means "at least one". "A few of the applicants are qualified", "there are applicants who are qualified", "at least one applicant is qualified": all of these are "Some applicants are qualified." And, as you'll see, so is "all of the applicants are qualified", which surprises people.

What "some" means

In logic, "Some A are B" means: at least one A is B. That's all. It says nothing about the rest of the As.

So "Some of the students passed" is true if one student passed, true if half did, and true if every single one did. Read that last clause again, because it's the one people fight. If all the students passed, then certainly at least one did, so "some passed" is true.

Why does it feel wrong? Because in conversation, if you knew that everyone passed, you'd say so. Saying "some" when you could have said "all" invites the listener to conclude that you couldn't say "all". Linguists call that a conversational implicature: something the listener reasonably infers from your choice of words, not something the words assert. Implicatures are real and useful in conversation. They're also cancellable ("some passed; in fact all of them did" is not a contradiction), and that's the test that shows they aren't part of the sentence's meaning.

For checking arguments, you use the logical reading. If an argument only works when "some" is read as "some but not all", it's leaning on something the premise never said.

How to deny each one

Every argument you check will eventually be argued about, and most of those arguments go wrong in the first minute because someone denies the wrong thing. So learn exactly what contradicts each of the four sentences.

Two sentences are contradictories when exactly one of them is true: they can't both be true and they can't both be false (Smith, 2022).¹

• "All A are B" is contradicted by "Some A are not B."
• "No A are B" is contradicted by "Some A are B."

That's the whole table, read in both directions. "Some A are B" is contradicted by "No A are B", and "Some A are not B" by "All A are B".

Notice what is not on the list. "All A are B" and "No A are B" are not contradictories. They can both be false at once, which happens whenever some As are B and some aren't. Logicians call that pair contraries. So if someone says "all the trains were late" and you reply "no they weren't, none of them were", you've taken on a much bigger claim than you needed, and you're probably wrong too.

The practical rule: to knock down an "all", find one exception. To knock down a "no", find one case. To knock down a "some", you have to show there are none at all, which is why "some" claims are so hard to refute and so easy to make.

The Venn check

Now the method. The idea is that a Venn diagram is a picture of every possible case at once, and that makes it a machine for finding counterexamples.

Start with two circles that overlap, one labelled A and one labelled B. Together they mark out three regions: A outside B, the overlap, and B outside A. Every possible thing sits in one of those regions or outside both.

Each of the four sentences is a mark on that picture, and Knachel's chapter is the place to see many more of them drawn (Knachel).²

• "All A are B" says the region "A outside B" is empty. Shade it. Shading means empty.
• "No A are B" says the overlap is empty. Shade the overlap.
• "Some A are B" says at least one thing is in the overlap. Put an X there. An X means at least one thing is here.
• "Some A are not B" says at least one thing is in "A outside B". Put an X there.

Here are all four, drawn:

Two conventions to fix in your head: shading means nothing lives here; an X means something does. Shading is a strong claim about a whole region. An X is a weak claim about a single occupant.

A syllogism has two premises and three terms. Each premise mentions two of the terms, the conclusion mentions the two that appear only once, and the third term, the one that appears in both premises but not the conclusion, is called the middle term. It's the hinge. In the promotion argument the middle term is "people who have done the course".

So draw three overlapping circles, one per term, and label them. Then:

1. Draw the premises, and only the premises. Never draw the conclusion.
2. Shade the "all" and "no" premises first.
3. Then place the X for any "some" premise. The region it belongs in will be cut in two by the third circle. If one half is shaded, the X goes in the other half. If neither half is shaded, the X goes on the line between them, which means "somewhere in here, and the premises don't say where".
4. Now read the conclusion off the page. If it's already drawn (the region it needs shaded is shaded; the X it needs is definitely inside the right region), the argument is valid. If it needs something that isn't there, the argument is invalid, and the unshaded region, or the other side of the line, is your counterexample.

Why does this work? Because a valid argument is one with no counterexample: no possible case where the premises hold and the conclusion fails. The diagram shows every possible case. When you draw the premises you rule out the cases they rule out, and no others. If the conclusion's requirement is still unmet after that, there's a case the premises allow in which the conclusion is false, and there's your counterexample. This is lesson 2's method, with the imagining done by the page.

Why shade before placing the X? Because the shading tells you where the X can't go. Place the X first and you may put it somewhere a later premise empties, and then you'll have to move it.

Worked example 1: dogs and cats

Here is the argument every logic teacher uses, because it catches everyone.

1. All dogs are mammals.
2. All cats are mammals.
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C: All dogs are cats.

Draw three circles: Dogs, Cats, Mammals, with all three overlapping in the middle. Now draw the premises.

Premise 1, "All dogs are mammals", says the part of the Dogs circle outside the Mammals circle is empty. Shade it. That's two small regions: dogs that are neither cats nor mammals, and dogs that are cats but not mammals.

Premise 2, "All cats are mammals", says the part of the Cats circle outside Mammals is empty. Shade that too.

Now stop drawing and look at the conclusion. "All dogs are cats" would need the part of the Dogs circle outside the Cats circle to be shaded. Is it? Half of it is: you shaded "dogs, not cats, not mammals" under premise 1. But the other half, "dogs, not cats, but mammals", is untouched. It's sitting there, open.

That open region is the counterexample. It's a dog that's a mammal and isn't a cat. Which is to say, a dog. The premises allow it, the conclusion forbids it, and so the argument is invalid. Here is the finished page; the region the leader line points at is the one the conclusion needed shaded and the premises left open.

Textbooks call this the fallacy of the undistributed middle: the middle term, "mammals", is never used to say something about all mammals, so it can't connect the other two terms. The IEP's list of fallacies uses this very argument, in dogs, cats and animals, as its example of the mistake (Dowden).³ You don't need the name. You need the picture, because the picture also works when the content isn't absurd.

Go back to the lift. Everyone promoted did the course; everyone on my team did the course; so everyone on my team gets promoted. Same three circles, same two shadings, same open region: a person who did the course and isn't promoted. The argument is invalid in exactly the way the dog argument is, and it took you thirty seconds. Your colleague may still get promoted. That's step 6, and it's a different question.

Worked example 2: a "some" premise, and the X on the line

Now an argument with a "some" premise, which is where the method earns its keep, because the X has to be placed carefully.

1. Some of the applicants are qualified.
2. All the qualified applicants have degrees.
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C: Some of the applicants have degrees.

Three circles: Applicants, Qualified, Degrees. Shade first. Premise 2 says the part of Qualified outside Degrees is empty. Shade it: that's "qualified, not an applicant, no degree" and "qualified applicant, no degree".

Now place the X for premise 1, "some applicants are qualified". It belongs in the overlap of Applicants and Qualified. That overlap is cut in two by the Degrees circle: "applicant, qualified, has a degree" and "applicant, qualified, no degree". Look at the second half. You've just shaded it. So the X can't go there, and it goes in the first half: applicant, qualified, with a degree.

Now do the next step yourself before reading on. The conclusion is "Some applicants have degrees." Where would an X have to be for that to be true? Is there one there?

Here's the wrinkle. Swap two of the terms around:

1. Some of the applicants have degrees.
2. All the qualified applicants have degrees.
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C: Some of the applicants are qualified.

It reads nearly the same, and a hiring manager in a hurry would nod along. Draw it. Shade premise 2 as before: Qualified outside Degrees is empty. Now the X for "some applicants have degrees" belongs in the overlap of Applicants and Degrees. That overlap is cut in two by the Qualified circle: "applicant, degree, qualified" and "applicant, degree, not qualified". Is either half shaded? No. Premise 2 shaded parts of the Qualified circle outside Degrees; both of these regions are inside Degrees. So the X goes on the line between them.

The conclusion wants an X definitely inside Applicants and Qualified. Yours is on the line, which means "this applicant might be qualified or might not; the premises don't say". Not forced. Invalid.

The counterexample is the other side of the line: an applicant with a degree who isn't qualified, because the degree is in the wrong subject, say. Both premises are true of that person, and the conclusion is false. Notice it's the dog-and-cat pattern again, with a "some" instead of an "all": both premises attach things to "degrees", and nothing says that "degrees" reaches only the qualified.

The two finished diagrams are below, and the difference between them is the whole method: in the first, the shading leaves the X only one place to go; in the second, the shading never touches the X's home, so it sits on the line.

One more thing about that first, valid version. "Some of the applicants are qualified" told you nothing about any applicant being unqualified. If someone hears that argument and says "so some of them aren't qualified, then", they've added a premise. Maybe it's true. It wasn't stated, and it isn't something the diagram will ever show, because an X in one region says nothing about the others.

Why the course stops here

You could spend a semester on this. A full logic course goes on from here to propositional logic, truth tables, and formal proof; forall x is a free, openly licensed textbook that takes that route properly (Magnus et al.).⁵ This course gives categorical logic one lesson and then moves on, and you deserve to know why, because it's a real argument among people who teach reasoning.

One side, call it the formal-first view, holds that formal logic gives a precise and general account of what "follows from" means, that working through it trains habits of rigour, and that it is worth knowing on its own terms, the way mathematics is. On this view a card puzzle is a poor measure of what a logic course teaches: nobody tests a mathematics degree with one brainteaser, and a logic course is not a training programme for puzzles but a body of knowledge with its own questions. That's the tradition behind the standard logic sequence and behind forall x.⁵

The other side is the informal-logic movement, which grew up in the 1970s around Ralph Johnson and Anthony Blair at Windsor, with Howard Kahane's 1971 Logic and Contemporary Rhetoric as an early landmark, and a family of textbooks has carried the programme on since. The field studies reasoning and inference "as they occur in public discussion and debate, educational and intellectual pursuits, interpersonal exchanges, and law, medicine and other 'real-life' contexts" (Groarke).⁶ Its founding complaint, in Johnson's words, was "dissatisfaction with formal logic as the vehicle for teaching skill in argument evaluation and argument formation": real arguments, on this view, usually fail on a premise that shouldn't be accepted or a reason that isn't relevant, not on an invalid form.⁶

There is some evidence on the transfer question, and it's worth reading with care. Cheng, Holyoak, Nisbett and Oliver tested 53 students at the start and end of two introductory logic courses at the University of Michigan, about 40 hours of lectures each, taught from the standard logic textbooks of the day and covering the conditional forms of lesson 3 (Cheng et al., 1986).⁷ The test was Wason's selection task. Before the course, 75% of students made at least one error on the conditional problems; after it, 72%. The authors' summary was that "the only apparent influence of a one-semester logic course was a small decrease in the tendency to make the error corresponding to affirming the consequent".⁷ After the semester, the same students solved only 11% of the arbitrary problems correctly, when before any training they'd solved 62% of the problems that were phrased as permissions.⁷

Two cautions. It's one classroom study, on one unusually hard task. And a separate laboratory experiment in the same paper found that abstract training did work when the rules were taught together with examples of the problems they applied to: 39% of the group trained with rules and examples made at least one error, against 75% of an untrained control group, though the authors say the confidence intervals on those lab groups were wide.⁷ That second finding is the one this course is built on. The rules are here; so are the examples, and the practice on your own material. A meta-analysis of critical-thinking instruction more broadly found it does help, with a weighted mean effect of g+ = 0.30, and the strategies associated with the larger effects were dialogue, authentic problems, and mentoring (Abrami et al., 2015).⁸

What would settle the question is a study that trained one group formally and one informally and tested both on real arguments after a delay. I don't know of a decisive one. So here is the course's line, stated openly: the four conditional shapes, the Venn check, and the counterexample method are the formal logic a general reader uses; the rest of this course is about premises, relevance, strength, and the ways people go wrong in practice. If you find you like the formal side, forall x is free and good.

What people get wrong

"Some" implies "not all". It doesn't, in logic. It's an implicature in conversation, and it's cancellable. An argument that needs "not all" must state it, and then you can check it.

"No A are B" is the opposite of "All A are B". They're contraries, not contradictories, and both can be false. To deny "all", produce one exception. Taking on "none" is taking on a claim you don't need and often can't defend.

A believable conclusion means a valid argument. Lesson 2's belief bias, again. Evans, Barston and Pollard found that people accept invalid syllogisms with believable conclusions and reject valid ones with unbelievable conclusions, and that the effect was larger on the invalid ones (Evans, Barston & Pollard, 1983).⁹ The dog argument is easy because the conclusion is absurd. The promotion argument has the same form and a conclusion someone wants. Draw both.

Placing the X before shading. Then it lands in a region a later premise empties, and you either move it or, worse, leave it and read off a wrong verdict. Shade first, always.

Treating existential import as a live dispute. It's a difference in convention with a known list of cases where it matters. State your reading and move on.

"All A are not B" read without asking. English lets that sentence mean two different things. If the argument turns on it, get the arguer to pick one.

Practice

Six arguments. Two are conditionals from lesson 3, so your first job on each is to decide which tool fits: the four shapes, or three circles. Then give the verdict and, for every invalid one, the counterexample. Write your answers down before opening the check.

1. No birds are mammals. All bats are birds. So no bats are mammals.
2. All the flats that flooded last winter were ground-floor flats. All the flats in Block B are ground-floor flats. So all the flats in Block B flooded last winter.
3. All tomatoes are vegetables. Some tomatoes are fruits. So some vegetables are fruits.
4. If the boiler's pilot light is out, there's no hot water. There's no hot water. So the pilot light is out.
5. Some of the hotel's guests ordered a newspaper. All the conference delegates are the hotel's guests. So some of the conference delegates ordered a newspaper.
6. If the meeting had run over, the room would still have been booked at three. The room wasn't booked at three. So the meeting didn't run over.

From memory. Close the page and write down: the four sentence forms and the exact sentence that contradicts each; what shading means and what an X means; why you shade before you place the X; what it means when the X lands on a line; and the one region the dog argument left open. Then check what you missed, and reread only those parts. Do this before the quiz; it's the part of the practice that does the most for you.

Connections

This closes the deductive unit. You now have two fast tools for step 4 of the check: the four conditional shapes from lesson 3, for arguments built from "if", and the Venn check, for arguments built from "all", "no", and "some". Both are the counterexample method from lesson 2 made quick, and both test only the link. The premises still need evidence, which no diagram supplies.

• Lesson 2's counterexample is what a Venn gap is: an unshaded region, or the far side of a line, is a case the premises permit and the conclusion forbids.
• Lesson 3's modus tollens is the shape of "one on-time train refutes 'all late'": the "all" says if train then late, and the one exception is the not-Q.
• Lesson 5 opens the other kind of link. There, "some" and "all" turn up inside claims about samples, "most" gets its proper treatment, and a counterexample doesn't sink the argument; it only weakens it. That difference is the whole subject of the next lesson.

Go deeper

• Knachel, Fundamental Methods of Logic, chapter 3 (free, CC BY 4.0, on LibreTexts): the clearest short treatment of categorical logic, with many more Venn diagrams drawn out than this lesson has room for; do the exercises in 3.6 if the X-on-the-line cases still feel slippery.
• forall x: Calgary, Parts I to III (free, CC BY 4.0): the symbolic route, with truth tables, for readers who found this lesson too easy and want the formal-first version done properly.
• Smith, "Aristotle's Logic", Stanford Encyclopedia of Philosophy: where the four sentences, the three figures, and the assumption about non-empty terms come from, in Aristotle's own order.
• Lagerlund, "Medieval Theories of the Syllogism", Stanford Encyclopedia of Philosophy: where the 24 traditional forms and their singsong medieval names (Barbara, Celarent) come from, if you want to see the counting done.

Sources

1. Smith, R., "Aristotle's Logic", Stanford Encyclopedia of Philosophy (rev. 2022). Aristotle's definition of a deduction; the three figures and the table of valid deductions; contradictories as pairs of which exactly one is true; Aristotle "in effect supposes that all terms in syllogisms are non-empty".
2. Knachel, M., Fundamental Methods of Logic (University of Wisconsin-Milwaukee; CC BY 4.0, read on LibreTexts), section 3.6, Categorical Syllogisms. Read at draft time: "There are 4 figures and 64 moods. That gives us 256 possible logical forms. It turns out that only 15 of these are valid" (the modern reading); the Venn diagram test, drawing the premises and then asking whether the picture entails the conclusion; universal premises depicted before particular ones; "in cases of indeterminacy, draw Xs straddling boundary lines".
3. Dowden, B., "Fallacies", Internet Encyclopedia of Philosophy. The entry's list defines the fallacy of undistributed middle as occurring "when the middle term of a syllogism is not distributed in at least one of the premises", with the example "All dogs are animals. All cats are animals. So, all dogs are cats."
4. Lagerlund, H., "Medieval Theories of the Syllogism", Stanford Encyclopedia of Philosophy (first published 2004; rev. 26 January 2026). Read at draft time, section 1: "there are 256 possible combinations of sentences. Of these, 24 have traditionally been thought to yield valid deductions", the 24 being the 19 direct moods plus five subalternate moods; the medieval mnemonic names (Barbara, Celarent) for the valid moods.
5. Magnus, P. D., Button, T., Trueman, R., Zach, R. et al., forall x: Calgary (CC BY 4.0), Part I. Validity as the absence of a counterexample; "logic doesn't care about feelings. Actually, it doesn't care about facts, either"; logic shows an argument invalid but not that a premise is false.
6. Groarke, L., "Informal Logic", Stanford Encyclopedia of Philosophy. Informal logic as the study of reasoning and inference "as they occur in public discussion and debate, educational and intellectual pursuits, interpersonal exchanges, and law, medicine and other 'real-life' contexts"; the movement's origins with Johnson and Blair at Windsor in the 1970s and Kahane's Logic and Contemporary Rhetoric (1971); Johnson's "dissatisfaction with formal logic as the vehicle for teaching skill in argument evaluation and argument formation".
7. Cheng, P. W., Holyoak, K. J., Nisbett, R. E. & Oliver, L. M., "Pragmatic versus syntactic approaches to training deductive reasoning", Cognitive Psychology 18(3), 293 to 328 (1986). Read from the full text (University of Michigan Deep Blue repository). Experiment 2: 53 students across two introductory logic classes (Ann Arbor and Dearborn), about 40 hours of lectures each, taught from two standard logic textbooks named in the paper; pretest to post-test on Wason selection problems, percentage making at least one error on conditional problems 75% before and 72% after, mean improvement "a bare 3 ± 7%"; "the only apparent influence of a one-semester logic course was a small decrease in the tendency to make the error corresponding to affirming the consequent"; after the semester students "solved only 11% of the arbitrary problems correctly, whereas the same students, prior to receiving any formal training, solved 62% of the permission problems correctly". Experiment 1, a separate laboratory study with four groups measured once: 39 ± 12% of the rules-and-examples group made at least one error, against 75 ± 12% of the control group; rules alone and examples alone did not significantly help, with the wide confidence intervals noted by the authors.
8. Abrami, P. C., Bernard, R. M., Borokhovski, E., Waddington, D. I., Wade, C. A. & Persson, T., "Strategies for teaching students to think critically: A meta-analysis", Review of Educational Research 85(2), 275 to 314 (2015). 341 effect sizes; weighted mean g+ = 0.30; dialogue, authentic problems and mentoring associated with larger effects (from the abstract).
9. Evans, J. St. B. T., Barston, J. L. & Pollard, P., "On the conflict between logic and belief in syllogistic reasoning", Memory & Cognition 11(3), 295 to 306 (1983). From the abstract: belief bias "was more marked on invalid than on valid syllogisms"; some verbal protocols were rationalisations of prejudged decisions.