Syllogisms

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** $S$: the subject of the conclusion - the **major term** $P$: the predicate of the conclusion - the **middle term** $M$: it appears in both premises but never in the conclusion The premise that contains $P$ is the **major premise**, the premise that contains $S$ 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): | Code | Statement | What it says about the groups | |---|---|---| | A | All $S$ are $P$ | no member of $S$ lies outside $P$ | | E | No $S$ are $P$ | $S$ and $P$ have no common member | | I | Some $S$ are $P$ | at least one member lies in both $S$ and $P$ | | O | Some $S$ are not $P$ | at least one member of $S$ lies outside $P$ | **Testing validity with a Venn diagram.** Draw three overlapping circles $S$, $M$ and $P$. They make 8 regions: | Region | Inside | Outside | |---|---|---| | 1 | $S$ | $M$, $P$ | | 2 | $M$ | $S$, $P$ | | 3 | $P$ | $S$, $M$ | | 4 | $S$, $M$ | $P$ | | 5 | $S$, $P$ | $M$ | | 6 | $M$, $P$ | $S$ | | 7 | $S$, $M$, $P$ | none | | 8 | none | $S$, $M$, $P$ | 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:** $$\text{All } M \text{ are } P,\quad \text{All } S \text{ are } M \quad\Rightarrow\quad \text{All } S \text{ are } P$$ The same chain works for if-then statements. This is the **hypothetical syllogism** (chain rule): $$p \to q,\quad q \to r \quad\Rightarrow\quad p \to r$$

Parts of a Syllogism, and Valid Is Not the Same as True

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?

Find the minor and major term in the conclusion: Conclusion: All penguins (subject) are things that can fly (predicate) = $S$ = penguins, $P$ = things that can fly

Find the middle term: The term in both premises but not in the conclusion is birds = $M$ = birds

Name the premises: All birds can fly contains $P$; All penguins are birds contains $S$ = Major premise: All birds can fly. Minor premise: All penguins are birds.

Write the form with letters: All $M$ are $P$, All $S$ are $M$, so All $S$ are $P$ = The classic valid form: the argument is valid

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

Answer: $S$ = penguins, $P$ = things that can fly, $M$ = birds. The argument is valid (correct form), but the major premise is false, so it is not sound and its conclusion is false.

Checking a Valid Form with the Venn Diagram

No fish are mammals. All sharks are fish. Therefore no sharks are mammals. Test the argument with the 8 region diagram.

Assign the letters: Conclusion: No sharks are mammals = $S$ = sharks, $P$ = mammals, $M$ = fish

Shade for No $M$ are $P$: The overlap of $M$ and $P$ is regions 6 and 7 = Regions 6 and 7 are empty

Shade for All $S$ are $M$: The part of $S$ outside $M$ is regions 1 and 5 = Regions 1 and 5 are empty

Read the conclusion No $S$ are $P$: The overlap of $S$ and $P$ is regions 5 and 7, and both are already shaded = The diagram shows the conclusion

Answer: The argument is valid: after shading regions 1, 5, 6 and 7, the whole overlap of sharks and mammals (regions 5 and 7) is empty.

The Undistributed Middle

All squares are parallelograms. All rectangles are parallelograms. Therefore all rectangles are squares. Find the flaw.

Assign the letters: Conclusion: All rectangles are squares = $S$ = rectangles, $P$ = squares, $M$ = parallelograms

Shade for All $P$ are $M$: The part of $P$ outside $M$ is regions 3 and 5 = Regions 3 and 5 are empty

Shade for All $S$ are $M$: The part of $S$ outside $M$ is regions 1 and 5 = Regions 1 and 5 are empty

Read the conclusion All $S$ are $P$: It needs regions 1 and 4 to be empty, but region 4 is not shaded = The diagram does not show the conclusion

Give a counterexample for region 4: A rectangle with sides $4$ and $2$ is a parallelogram but not a square = Invalid: undistributed middle

Answer: Invalid. Both premises only say that squares and rectangles lie somewhere inside the parallelograms; they never connect squares and rectangles. This flaw is called the undistributed middle.

A Syllogism with Some

All members of the chess club are good at puzzles. Some students in class 9B are members of the chess club. Which conclusion follows: all students in 9B are good at puzzles, or some students in 9B are good at puzzles?

Assign the letters: $S$ = students in 9B, $P$ = good at puzzles, $M$ = chess club members = Premises: All $M$ are $P$, Some $S$ are $M$

Shade the All premise first: All $M$ are $P$: the part of $M$ outside $P$ is regions 2 and 4 = Regions 2 and 4 are empty

Place the x for Some $S$ are $M$: The overlap of $S$ and $M$ is regions 4 and 7, and region 4 is shaded = The x must go in region 7

Test Some $S$ are $P$: Region 7 lies inside both $S$ and $P$ = Some $S$ are $P$ is shown

Test All $S$ are $P$: It needs regions 1 and 4 empty, but region 1 is not shaded = All $S$ are $P$ is not shown

Answer: Only "some students in 9B are good at puzzles" follows. The premises say nothing about the 9B students outside the chess club.

The Chain Rule for If-Then Statements

Premises: If $n$ is divisible by $12$, then $n$ is divisible by $6$. If $n$ is divisible by $6$, then $n$ is divisible by $3$. If $n$ is divisible by $3$, then the digit sum of $n$ is divisible by $3$. What follows, and what does it say about $n = 84$ and about $n = 98$?

Name the statements: $p$: divisible by $12$, $q$: divisible by $6$, $r$: divisible by $3$, $s$: digit sum divisible by $3$ = $p \to q$, $q \to r$, $r \to s$

Link the first two: $p \to q$ and $q \to r$ = $p \to r$

Link the result with the third premise: $p \to r$ and $r \to s$ = $p \to s$

Check $n = 84$: $84 = 12 \times 7$, and the digit sum is $8 + 4 = 12 = 3 \times 4$ = As predicted, the digit sum is divisible by $3$

Use the contrapositive for $n = 98$: Digit sum $9 + 8 = 17$ is not divisible by $3$, and $\neg s \to \neg p$ = $98$ is not divisible by $12$

Answer: If $n$ is divisible by $12$, then the digit sum of $n$ is divisible by $3$ ($p \to s$). For $84$ this holds ($8 + 4 = 12$). Since $98$ has digit sum $17$, the contrapositive shows that $98$ is not divisible by $12$.

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.

Mistake: Calling an argument valid because its conclusion is true

Why: 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 $S$, $M$, $P$ and test the form with the Venn diagram. Only then ask whether the premises are true.

Mistake: Linking two groups through a middle term that only appears as "are $M$" (undistributed middle)

Why: All $S$ are $M$ and All $P$ are $M$ only put $S$ and $P$ somewhere inside $M$. They can sit in completely different parts of $M$, 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.

Mistake: Drawing a conclusion from two Some premises

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

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

Mistake: Reversing an All statement

Why: All squares are rectangles does not mean all rectangles are squares. In the diagram, All $S$ are $P$ only empties the part of $S$ outside $P$; the part of $P$ outside $S$ can still have members.

Correct: No and Some can be reversed (No $S$ are $P$ means No $P$ are $S$; Some $S$ are $P$ means Some $P$ are $S$), but All and Some not cannot.

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.

No drama club member is in the math club, and all robotics club members are in the math club. By the syllogism (No $M$ are $P$, All $S$ are $M$), no robotics club member is in the drama club.

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.

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.

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.

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.

A syllogism has two premises and one conclusion; a categorical syllogism uses three terms: minor term $S$, major term $P$ and middle term $M$

The four statement types are A (All $S$ are $P$), E (No $S$ are $P$), I (Some $S$ are $P$) and O (Some $S$ are not $P$)

To 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

All $M$ are $P$, All $S$ are $M$, so All $S$ are $P$ is valid; All $S$ are $M$, All $P$ are $M$, so All $S$ are $P$ is invalid (undistributed middle)

Valid means the conclusion must be true if the premises are true; sound means valid with true premises

The hypothetical syllogism (chain rule) links if-then statements: $p \to q$ and $q \to r$ give $p \to r$

Q: Can a valid syllogism have a false conclusion?

A: 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.

Q: Does "some" mean "not all"?

A: No. In logic, Some $S$ are $P$ means at least one $S$ is a $P$, and it is still true if all $S$ are $P$. Some $S$ are $P$ does not imply Some $S$ are not $P$.

Q: From All $M$ are $P$ and All $S$ are $M$, may I conclude Some $S$ are $P$?

A: Only if you know that $S$ has at least one member. In modern logic All $S$ are $P$ does not claim that any $S$ 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.

Q: How many valid categorical syllogisms are there?

A: Each of the three statements has one of the four types ($4 \times 4 \times 4 = 64$ combinations), and the middle term can stand in four different positions, which gives $64 \times 4 = 256$ forms. Only $15$ of them are valid in modern logic ($24$ 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.

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Syllogisms

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Logic & Sets / Logical Reasoning

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 min

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