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Syllogisms: All, Some, None and What Follows

Categorical statements come in four forms. The square of opposition, a circle-drawing method that settles validity, and the invalid forms that feel right.

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Key takeaways

  • Categorical statements come in exactly four forms, traditionally labelled A, E, I and O: all S are P, no S is P, some S are P, and some S are not P.
  • A and O contradict each other, as do E and I, so in each of those pairs exactly one member is true — which gives you a free move whenever a premise is negated.
  • Two overlapping circles settle a single statement and three settle a syllogism: shade what the universal premises rule out first, then place a mark for any particular premise.
  • Drawing the universal premises before the particular one is the practical rule, because shading first often leaves only one place the mark can go.
  • The three classic failures are an undistributed middle, converting an 'all' statement, and drawing a conclusion from two particular premises.
  • Whether 'all S are P' licenses 'some S are P' depends on whether anything is an S, which is the existential import question the traditional and modern treatments answer differently.

A categorical syllogism is an argument with two premises and a conclusion, each of which says something about how two groups overlap. There are only four ways a categorical statement can be built, and validity can be settled mechanically by drawing circles — which means this is one of the few reasoning tasks where you never have to rely on whether an argument sounds right.

The four forms carry traditional labels. Writing S for the subject group and P for the predicate group, the Stanford Encyclopedia of Philosophy's entry on the traditional square of opposition sets them out as follows.

Label Form Name What it asserts
A Every S is P Universal affirmative The whole of S sits inside P
E No S is P Universal negative S and P do not overlap at all
I Some S is P Particular affirmative At least one thing is both
O Some S is not P Particular negative At least one S is outside P

"Some" in this tradition means at least one, not some but not all. That single convention removes a large fraction of the confusion people bring to the subject, because ordinary English usually implies the second meaning.

The square of opposition, and the free moves it gives you

Arrange the four forms in a square and three relationships appear, each of which is a shortcut you can use without drawing anything.

Contradictories. A and O cannot both be true and cannot both be false; the same holds for E and I. So if you learn that "every S is P" is false, you immediately have "some S is not P", and vice versa. This is the most useful of the three, because negating a universal is something people routinely get wrong: the denial of "all swans are white" is not "no swans are white" but "at least one swan is not white".

Contraries. A and E cannot both be true, though they can both be false. "Every S is P" and "no S is P" cannot hold together, but if S is split down the middle, neither holds.

Subalternation. If A is true then I is true, and if E is true then O is true — on the traditional reading. That reading depends on affirmative statements carrying existential import, which the entry describes as the traditional position: affirmatives required their subjects to have instances, negatives did not. Modern treatments generally drop that assumption, which is why "all unicorns are white" does not license "some unicorn is white" in a contemporary logic course. Neither convention is wrong; they answer different questions, and the only error is switching between them mid-argument.

Drawing it: the method that settles validity

Three circles, one per term, overlapping so that every combination of memberships has its own region. Then two rules:

  • A universal premise is drawn by shading the region it says is empty. "No S is P" shades the whole overlap of S and P. "Every S is P" shades the part of S that lies outside P.
  • A particular premise is drawn by placing a mark in a region it says is occupied.

The practical rule is to draw every universal premise first. Shading frequently removes all but one of the regions where a mark could go, which settles where the mark belongs without guesswork. If you place the mark first, you often find you have to move it, and moving it is where mistakes creep in.

Then read the diagram. If the conclusion is already true of every diagram consistent with the premises, the argument is valid. If you can draw a consistent diagram in which the conclusion fails, it is not.

Four valid arguments, worked

1. Every member of the choir is a volunteer. Every volunteer is insured. Therefore every member of the choir is insured.

Two universals. Shading puts the choir circle inside volunteers and volunteers inside insured, so the choir circle ends up inside insured. Valid.

2. No member of the choir is under sixteen. Every soprano is a member of the choir. Therefore no soprano is under sixteen.

Shade the overlap of choir and under-sixteens, then place sopranos inside choir. The soprano circle now lies entirely in shaded-free territory outside under-sixteens. Valid.

3. Every cellist is a string player. Some volunteers are cellists. Therefore some volunteers are string players.

Universal first: shade the part of the cellist circle outside string players. Now the mark for "some volunteers are cellists" has only one legal home — inside cellists and inside string players. The conclusion is forced. Valid.

4. No cellist is a drummer. Some volunteers are cellists. Therefore some volunteers are not drummers.

Shade the cellist–drummer overlap, then place the mark in the volunteer–cellist region, which is now necessarily outside drummers. Valid.

Notice how much work the ordering did in the third and fourth arguments. That is the whole technique.

Three failures that feel valid

Undistributed middle. Every accountant is numerate. Every actuary is numerate. Therefore every accountant is an actuary.

Draw a large "numerate" circle and put two separate smaller circles inside it. Both premises are satisfied and the conclusion is false. The shared term is the larger category in both premises, so it never connects the two groups.

Converting an A statement. Every poodle is a dog, therefore every dog is a poodle.

The A form is not symmetric. E and I are — "no S is P" and "no P is S" say the same thing, as do the two I forms — but A and O are not, and treating them as if they were is a common slip in longer arguments.

Two particular premises. Some cyclists are commuters. Some commuters are lawyers. Therefore some cyclists are lawyers.

Two marks, placed independently, with nothing forcing them into the same region. No conclusion follows from two particular premises, ever.

Why a believable conclusion feels like a valid one

There is a persistent interference between content and structure here. A meta-analysis by Dries Trippas and colleagues, published in Psychonomic Bulletin and Review in 2018, pooled 22 studies covering 993 participants and reported that syllogisms with believable conclusions are endorsed more often than structurally identical ones with unbelievable conclusions. Their analysis attributed the effect to a shift in people's general willingness to endorse arguments rather than to a change in their ability to tell valid from invalid.

The countermeasure is the diagram. Circles are indifferent to whether a conclusion is agreeable, which is the point of using them on any argument where you have a stake in the answer. Note also that validity says nothing about whether the premises are true; where those premises come from is a separate question, and our guide to primary and secondary sources covers how to weigh them.

Load, practice, and where our app fits

Holding two premises, a conclusion and a part-drawn diagram in mind at once is a real capacity problem rather than a failure of character — which is why writing the diagram down beats reasoning in your head, and why working memory and training is worth reading alongside this.

On the product, plainly: IQ Test & Brain Training Games, our own app, has logic games covering analogies, sequences, conditional reasoning and deduction, and an IQ-style test that includes logic. Its listing does not say whether any of them uses categorical syllogisms of the kind above, and it describes test results as estimates based on app activities, not clinical or professional assessments. If you practise deduction there, the diagram habit still applies: draw the circles on paper rather than accepting a conclusion because it sounds believable. How the app describes its games and tests is on the IQ Test & Brain Training Games app page. For the underlying concepts of validity and soundness, start with deductive reasoning; for the conditional version of the same machinery, see if-then reasoning; and for rule-finding rather than argument-testing, the entry point is number sequence puzzles. The complete set is in the logic and pattern puzzles hub and the education and brain training section. The open textbook forall x: Calgary is a good free next step for the formal treatment.

Good to know

Frequently asked questions

What are the four categorical statement forms?
Traditionally labelled A, E, I and O. The A form is universal affirmative, every S is P. The E form is universal negative, no S is P. The I form is particular affirmative, some S are P. The O form is particular negative, some S are not P. Every categorical syllogism is built from three of these, two as premises and one as a conclusion, and the labels let you name the shape of an argument compactly.
How do you test a syllogism with circles?
Draw three overlapping circles, one for each term. Represent each universal premise by shading the regions it rules out, and each particular premise by putting a mark in a region it requires to be occupied. Do the universals first, because the shading often leaves only one place the mark can legitimately go. Then read the conclusion off the diagram: if it is already forced, the argument is valid.
Why is 'all A are C, all B are C, so all A are B' invalid?
Because the shared term C is the wider category in both premises, so the two groups can sit inside it without touching each other. All accountants are numerate and all actuaries are numerate, but nothing there connects accountants to actuaries. This failure is called an undistributed middle, and the quickest way to expose it is to draw C as a large circle with two separate smaller circles inside.
Does 'all' imply 'some'?
Only if the subject class has members. The traditional square of opposition treats affirmative statements as carrying existential import, so all S are P licenses some S are P. Modern treatments usually do not, because that inference breaks for empty categories: all unicorns are white is arguably true by vacuity, while some unicorn is white is false. Say which convention you are using and the puzzle disappears.

Sources

  1. The Traditional Square of Opposition (opens in a new tab)Stanford Encyclopedia of Philosophy — accessed
  2. forall x: Calgary. A Free and Open Introduction to Formal Logic (opens in a new tab)Open Logic Project, University of Calgary — accessed
  3. Characterizing belief bias in syllogistic reasoning: A hierarchical Bayesian meta-analysis of ROC data (opens in a new tab)Psychonomic Bulletin and Review, via PubMed Central — accessed

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