Chain Reasoning and Hypothetical Syllogism

Single conditionals are powerful. But real exam problems — and real arguments — rarely stop at one step. They build chains: A implies B, B implies C, therefore A implies C. This is hypothetical syllogism, and mastering it is the key to solving the multi-step logic problems that appear most frequently on Korean public administration, civil service, and legislative exams.

The Core Rule

If A → B and B → C, then we can conclude A → C.

This seems almost too simple — and it is, at its core. The difficulty comes from:

Recognizing when to apply it across multiple premises
Combining it with contrapositive reasoning
Keeping track of which direction the arrows run

Why It Works: Transitivity of Implication

Think of each conditional as a one-way door. If room A leads to room B, and room B leads to room C, then anyone who starts in room A will eventually reach room C — even if they cannot jump from A directly to C without passing through B.

Formally:

Step	Statement
Premise 1	A → B
Premise 2	B → C
Conclusion	A → C

Building Longer Chains

The pattern extends to any length:

A → B → C → D → E

From this we can derive: A → C, A → D, A → E, B → D, B → E, C → E — any pair where the left side appears earlier in the chain.

Example:

If it rains, the field becomes muddy.

If the field is muddy, the match is postponed.

If the match is postponed, refunds are issued.

Chain: Rain → Muddy → Postponed → Refunds Derived: Rain → Refunds (direct conclusion skipping intermediate steps)

Combining Chains with Contrapositives

This is where exam problems get interesting. The premises may be given in contrapositive form, requiring you to “reverse” them before chaining.

Key: If A → B, then ¬B → ¬A (contrapositive).

Example:

Premise 1: A → B Premise 2: ¬B → C

Can we chain these? Not directly — Premise 2 starts with ¬B, while Premise 1 ends with B.

Convert Premise 2 to its contrapositive: ¬C → B Now we have: A → B and ¬C → B

These share the consequent B, not a middle link. We cannot chain them into a single line.

But suppose:

Premise 1: A → B Premise 2: C → ¬B

Contrapositive of Premise 2: B → ¬C Chain: A → B → ¬C, therefore A → ¬C.

The “Broken Chain” Problem

Exam questions often present premises that almost form a chain but have a gap or a misdirection. The goal is to identify whether a clean chain exists.

Problem Set-Up:

Premise 1: P → Q Premise 2: R → ¬Q Premise 3: S → R

Can we chain all four? Let us trace:

Premise 3 + Premise 2: S → R → ¬Q (chain works here)
Contrapositive of Premise 1: ¬Q → ¬P
Full chain: S → R → ¬Q → ¬P

Conclusion: S → ¬P. “If S is true, then P must be false.”

The skill is consistently converting to contrapositive when needed to find the connecting middle term.

Standard Exam Technique: Arrow Diagram

Draw each premise as a directed arrow:

P ──→ Q       (Premise 1)
R ──→ ¬Q      (Premise 2)
S ──→ R       (Premise 3)

Convert Premise 1 contrapositive: ¬Q → ¬P

Now connect:

S → R → ¬Q → ¬P

Read from left: “S leads all the way to ¬P.”

This visual method prevents errors in longer chains (5+ premises).

Practice Problems

Problem 1:

All members who attend training receive a certificate. (T → C)

All members who receive a certificate are eligible for promotion. (C → P)

Jiyeon attended training.

What can you conclude about Jiyeon?

Solution

Chain: T → C → P. Jiyeon attended training (T is true), so T → C → P applies. Conclusion: Jiyeon received a certificate and is eligible for promotion.

Problem 2:

If the server is overloaded, the system slows down. (O → S)

If the system slows down, users complain. (S → U)

Users did not complain. (¬U)

What can you conclude?

Solution

Chain: O → S → U. Contrapositive of chain: ¬U → ¬S → ¬O. ¬U is given, so: the system did not slow down, and the server was not overloaded.

Problem 3:

A → ¬B

C → B

D → C

Does D → ¬A follow?

Solution

Chain Premises 3+2: D → C → B. Contrapositive of Premise 1: B → ¬A. Full chain: D → C → B → ¬A. Yes, D → ¬A follows.

Problem 4 (government exam style):

A policy committee has these rules:

If measure X is passed, measure Y is also passed.

If measure Y is passed, measure Z is blocked.

If measure Z is blocked, the director must resign.

Measure X was passed.

What are the consequences?

Solution

Premise 4 triggers the chain: X (given) → Y (Premise 1) → ¬Z / Z blocked (Premise 2) → Director resigns (Premise 3). Consequences: Y is passed, Z is blocked, the director resigns.

Common Mistakes

Mistake 1: Affirming the Consequent Given A → B and B is true, concluding A is true. Invalid. B might be true for many reasons besides A.

Mistake 2: Reversing the Chain Given A → B → C, concluding C → A. Invalid. The chain is one-directional unless you have the full contrapositive (¬A → ¬B → ¬C… wait, that is also wrong — the contrapositive of A→C is ¬C → ¬A, which skips B).

Mistake 3: Breaking at a Negation Given A → ¬B and B → C, many students think these cannot be chained. They can — take the contrapositive of B → C to get ¬C → ¬B, and then: A → ¬B ← ¬C… still not a chain. In this case, you actually cannot chain them in a single direction, and that is the correct answer: no chain conclusion is available.

Chain reasoning rewards patience: write out every premise as an arrow, convert contrapositives where needed, and follow the connective links until you reach the conclusion. With practice, you will see the full chain in seconds.

Oiyo