Syllogisms

Learn the four categorical statements, the parts of a syllogism, how to test validity with Venn diagrams, and how to chain if-then statements.

Advanced30 min18 problems
Practice Problems

Definition

A syllogism is a deductive argument with exactly two premises and one conclusion. A categorical syllogism talks about groups and uses exactly three terms:
  • the minor term SS: the subject of the conclusion
  • the major term PP: the predicate of the conclusion
  • the middle term MM: it appears in both premises but never in the conclusion
The premise that contains PP is the major premise, the premise that contains SS is the minor premise.
Every statement in a categorical syllogism has one of four forms (the letters A, E, I, O are the traditional codes, taken from the Latin words *affirmo*, I affirm, and *nego*, I deny: A and I affirm, E and O deny):
CodeStatementWhat it says about the groups
AAll SS are PPno member of SS lies outside PP
ENo SS are PPSS and PP have no common member
ISome SS are PPat least one member lies in both SS and PP
OSome SS are not PPat least one member of SS lies outside PP
Testing validity with a Venn diagram. Draw three overlapping circles SS, MM and PP. They make 8 regions:
RegionInsideOutside
1SSMM, PP
2MMSS, PP
3PPSS, MM
4SS, MMPP
5SS, PPMM
6MM, PPSS
7SS, MM, PPnone
8noneSS, MM, PP
First shade (mark as empty) every region that an All or No premise rules out. Then place an x for each Some premise, but only where the x is forced into one region. The syllogism is valid if the finished diagram already shows the conclusion, without drawing anything more.
Classic valid form:
All M are P,All S are M⇒All S are P\text{All } M \text{ are } P,\quad \text{All } S \text{ are } M \quad\Rightarrow\quad \text{All } S \text{ are } P
The same chain works for if-then statements. This is the hypothetical syllogism (chain rule):
p→q,q→r⇒p→rp \to q,\quad q \to r \quad\Rightarrow\quad p \to r

Try it now

All rectangles are parallelograms. All squares are rectangles. Therefore all squares are parallelograms. What is the middle term?

Worked Examples

All birds can fly. All penguins are birds. Therefore all penguins can fly. Name the three terms and the two premises. Is the argument valid? Is the conclusion true?
1

Find the minor and major term in the conclusion

Conclusion: All penguins (subject) are things that can fly (predicate) → SS = penguins, PP = things that can fly
2

Find the middle term

The term in both premises but not in the conclusion is birds → MM = birds
3

Name the premises

All birds can fly contains PP; All penguins are birds contains SS → Major premise: All birds can fly. Minor premise: All penguins are birds.
4

Write the form with letters

All MM are PP, All SS are MM, so All SS are PP → The classic valid form: the argument is valid
5

Check the truth of the premises

The major premise is false: penguins and ostriches are birds that cannot fly → Valid but not sound, and the conclusion is false

Common Mistakes

Calling an argument valid because its conclusion is true

Why it's wrong: Validity is about the form, not about the facts. "All roses are flowers. Some flowers fade quickly. So some roses fade quickly." has a believable conclusion, but the form is invalid: the flowers that fade quickly could all be tulips.

Correct: Replace the words with SS, MM, PP and test the form with the Venn diagram. Only then ask whether the premises are true.

Linking two groups through a middle term that only appears as "are MM" (undistributed middle)

Why it's wrong: All SS are MM and All PP are MM only put SS and PP somewhere inside MM. They can sit in completely different parts of MM, like cats and dogs inside mammals.

Correct: The middle term must cover its whole group at least once: it must be the subject of an All or No statement, or the predicate of a No or Some not statement.

Drawing a conclusion from two Some premises

Why it's wrong: Some SS are MM and Some MM are PP can be about different members of MM. With S={1,2}S = \{1, 2\}, M={2,3}M = \{2, 3\}, P={3,4}P = \{3, 4\} both premises are true, but no member of SS is in PP.

Correct: At least one premise must be universal (All or No). Two particular premises, or two negative premises, never give a conclusion.

Reversing an All statement

Why it's wrong: All squares are rectangles does not mean all rectangles are squares. In the diagram, All SS are PP only empties the part of SS outside PP; the part of PP outside SS can still have members.

Correct: No and Some can be reversed (No SS are PP means No PP are SS; Some SS are PP means Some PP are SS), but All and Some not cannot.

Interactive Visual

Venn Diagram

Premises

  • ENo fish are mammals
  • AAll sharks are fish
sharksfishmammalsU14257638
= empty (no elements)x= at least one element

Conclusion

E∴ No sharks are mammals

Tap a region to shade it (shaded means empty).

Venn Diagram

Premises

  • AAll squares are parallelograms
  • AAll rectangles are parallelograms
rectanglesparallelogramssquaresU14257638
= empty (no elements)x= at least one element

Conclusion

A∴ All rectangles are squares

Tap a region to shade it (shaded means empty).

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Practice Problems

18 problems
Problem 1 of 18
Easy
All rectangles are parallelograms. All squares are rectangles. Therefore all squares are parallelograms. What is the middle term?

Why It Matters

Syllogisms are the smallest building blocks of every logical proof, and they hide in many everyday arguments:
  • Geometry proofs: every square is a rectangle and every rectangle is a parallelogram, so every square is a parallelogram. Long proofs are chains of such steps.
  • Search filters and databases: if every item tagged waterproof is also tagged outdoor, a search for outdoor items already contains all waterproof items.
  • Rules and laws: a rule (all visitors under 12 need an adult) plus a fact (Mia is a visitor under 12) gives a conclusion about Mia.
  • Programming: if every Square object is a Shape, then everything you can do with any Shape you can also do with a Square.
  • Spotting bad arguments: ads and debates often use invalid forms such as "all healthy snacks are low in fat, our cookie is low in fat, so our cookie is healthy". Knowing the forms lets you see the gap at once.
A Venn diagram turns a question about words into a question about shaded regions, so you can check an argument instead of trusting how it sounds.

Real World Applications

School Clubs and Surveys

When a school counts club members, statements like "all members of the robotics club are in the math club" tell you which regions of the Venn diagram are empty, so you can count without double counting anyone.

Example:

No drama club member is in the math club, and all robotics club members are in the math club. By the syllogism (No MM are PP, All SS are MM), no robotics club member is in the drama club.

1Try It Yourself

A class has 5050 students. All 1212 members of the robotics club are in the math club, which has 3030 members. No member of the drama club (99 students) is in the math club.

How many students are in none of the three clubs?

Step 1: Write the mathematical expression

Write the number of students in no club as one expression:

Search Filters in Online Shops

Shop categories form groups inside groups. If every waterproof item is also an outdoor item, the filter outdoor already contains every waterproof item, which is the syllogism All $S$ are $M$, All $M$ are $P$ at work.

Example:

All rain jackets in the shop are waterproof items, and all waterproof items are outdoor items, so all rain jackets appear in the outdoor filter.

2Try It Yourself

A shop has 120120 outdoor items. All 4545 waterproof items are outdoor items. 1818 of the waterproof items are on sale, and 1010 outdoor items that are not waterproof are on sale.

How many outdoor items are on sale, and how many outdoor items are neither waterproof nor on sale?

Step 1: Write the mathematical expression

Write the number of outdoor items on sale as an addition:

Email Filters and Rule Chains

Email programs follow chains of if-then rules. The hypothetical syllogism ($p \to q$, $q \to r$, so $p \to r$) tells you where a message ends up without following every rule one by one.

Example:

If an email has a link from an unknown sender, it is flagged. If an email is flagged, it goes to quarantine. So every email with a link from an unknown sender goes to quarantine.

3Try It Yourself

A school receives 200200 emails in one day. 3535 of them have a link from an unknown sender, and every such email is flagged. 1515 other emails are flagged because of a suspicious attachment. Every flagged email goes to quarantine, and only flagged emails go there.

How many emails reach the inbox?

Step 1: Write the mathematical expression

Write the number of emails that reach the inbox as one expression:

Key Takeaways

  • 1A syllogism has two premises and one conclusion; a categorical syllogism uses three terms: minor term SS, major term PP and middle term MM
  • 2The four statement types are A (All SS are PP), E (No SS are PP), I (Some SS are PP) and O (Some SS are not PP)
  • 3To test validity, shade the regions that All and No premises empty, place an x for Some premises, then check whether the diagram already shows the conclusion
  • 4All MM are PP, All SS are MM, so All SS are PP is valid; All SS are MM, All PP are MM, so All SS are PP is invalid (undistributed middle)
  • 5Valid means the conclusion must be true if the premises are true; sound means valid with true premises
  • 6The hypothetical syllogism (chain rule) links if-then statements: p→qp \to q and q→rq \to r give p→rp \to r

Frequently Asked Questions

Yes, if at least one premise is false. "All birds can fly, all penguins are birds, so all penguins can fly" is valid but has a false conclusion. What is impossible is a valid argument with true premises and a false conclusion.
Yes, if at least one premise is false. "All birds can fly, all penguins are birds, so all penguins can fly" is valid but has a false conclusion. What is impossible is a valid argument with true premises and a false conclusion.
No. In logic, Some SS are PP means at least one SS is a PP, and it is still true if all SS are PP. Some SS are PP does not imply Some SS are not PP.
Only if you know that SS has at least one member. In modern logic All SS are PP does not claim that any SS exists (All unicorns are animals is true even though there are no unicorns). The Venn diagram shows this: the premises shade regions but place no x.
Each of the three statements has one of the four types (4×4×4=644 \times 4 \times 4 = 64 combinations), and the middle term can stand in four different positions, which gives 64×4=25664 \times 4 = 256 forms. Only 1515 of them are valid in modern logic (2424 if you assume that every group has at least one member). That is why testing with a diagram is safer than trusting how an argument sounds.

Glossary

Syllogism
A deductive argument with exactly two premises and one conclusion
Categorical statement
A statement of the form All SS are PP, No SS are PP, Some SS are PP or Some SS are not PP
Middle term
The term that appears in both premises but not in the conclusion; it links the other two terms
Major and minor premise
The major premise contains the predicate PP of the conclusion, the minor premise contains its subject SS
Undistributed middle
An invalid form in which the middle term never covers its whole group, for example All SS are MM, All PP are MM
Hypothetical syllogism
The chain rule for if-then statements: from p→qp \to q and q→rq \to r it follows that p→rp \to r

Formula Card

Classic valid form

All MM are PP, All SS are MM ⇒\Rightarrow All SS are PP

The middle term links the two groups like a chain

Valid form with No

No MM are PP, All SS are MM ⇒\Rightarrow No SS are PP

If $S$ lies inside $M$ and $M$ shares nothing with $P$, then $S$ shares nothing with $P$

Valid form with Some

All MM are PP, Some SS are MM ⇒\Rightarrow Some SS are PP

The x for Some is pushed into the region inside $S$, $M$ and $P$

Undistributed middle (invalid)

All SS are MM, All PP are MM ⇏\not\Rightarrow All SS are PP

Being in the same big group does not connect two smaller groups

Hypothetical syllogism

p→q, q→r⇒p→rp \to q,\ q \to r \Rightarrow p \to r

Chains of if-then statements; with the contrapositive also $\neg r \to \neg p$

Today's challengeOne problem, three tries

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