Logic Course — Chain Reasoning and Syllogism — How to Break a Chain of Conditional Statements

Ch3: ∧∨ Combination Conditionals covered how to handle complex antecedents and consequents. In this Ch4, we tackle situations where multiple conditionals are linked like a chain. This is the chain reasoning type frequently tested in public enterprise and civil service exams in the format: “When the following premises are all true, what is the necessarily true conclusion?“

1. Hypothetical Syllogism

The most basic and important chain reasoning:

• Premise 1: A → B
• Premise 2: B → C
• Conclusion: A → C

If A is true, B is true, and if B is true, C is true, so if A is true, C is also true.

This is the hypothetical syllogism (= chain reasoning).

Extension: Longer Chains

A → B → C → D → E

Here, if A is true, E is necessarily true, as long as no link in the chain is broken.

Using with Contrapositives

Since a contrapositive is equivalent to the original proposition, reasoning in the reverse direction is also possible:

• Original: A → B → C
• Contrapositive: ¬C → ¬B → ¬A

If ¬C, then ¬A. This is also a fully valid inference.

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2. Modus Ponens vs Modus Tollens

These are the two most frequently used basic inference rules in chain reasoning.

Modus Ponens

• Premise: A → B
• Premise: A (antecedent is true)
• Conclusion: B (consequent is also necessarily true)

Since A is true, B is also true.

Modus Tollens

• Premise: A → B
• Premise: ¬B (consequent is false)
• Conclusion: ¬A (antecedent is also necessarily false)

Since B is false, A is also false (applying the contrapositive).

2 Invalid Inferences (Test Pitfalls)

Invalid Inference	Name	Form	Why It’s Wrong
Affirming the Consequent	Affirming the Consequent	A→B, If B then A?	False: B could be true for another reason
Denying the Antecedent	Denying the Antecedent	A→B, If ¬A then ¬B?	False: B could still be true for another reason

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3. Disjunctive Syllogism

• Premise: A ∨ B
• Premise: ¬A (A is false)
• Conclusion: B

“A or B. But not A. Therefore, B.”

This is a very powerful inference tool. It is used when there is a disjunction (∨) rather than a conditional.

Note: ∨ is Inclusive OR

In logic, ∨ is basically an inclusive OR. It includes cases where both A and B are true. “A or B” allows for “A only, B only, or both”.

When it appears on a test as “Not A or not B (¬A ∨ ¬B)”:

• This means “except when both A and B are true”
• That is, equivalent to ¬(A ∧ B)

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4. Practical Chain Reasoning Problems

Problem 1 (Basic Form)

When the following premises are all true, what is the necessarily true conclusion?

① If the budget passes (A), personnel are hired (B). ② If personnel are hired (B), the project begins (C). ③ The budget has passed (A).

Solution:

Chain from ① and ②: A → B → C, which means A → C

From ③, A=T → Modus Ponens:

Conclusion: The project begins (C). ✓

Problem 2 (Reverse Application)

① If you attend the meeting (A), you submit the report (B). ② If you submit the report (B), you receive approval (C). ③ Approval was not received (¬C).

Solution:

Contrapositive chain: ¬C → ¬B → ¬A

③ ¬C is true → apply Modus Tollens sequentially:

Conclusion: You did not attend the meeting (¬A). ✓

Problem 3 (Complex Form including ∧∨)

① (A∨B) → C ② C → (D∧E) ③ B is true.

Solution:

① Equivalence learned in Ch3: (A∨B)→C ≡ (A→C)∧(B→C)

③ B=T → apply B→C from ① → C=T

② C→(D∧E), C=T → Modus Ponens: D∧E=T

D∧E=T → D=T AND E=T

Conclusion: D is true and E is true. ✓

Problem 4 (National Project Type — Advanced Practical)

① If a national project is selected (A), a budget is allocated (B) and a responsible department is established (C). ② If a responsible department is established (C), additional public officials are hired (D). ③ If a budget is allocated (B), it is reported to the Ministry of Economy and Finance (E). ④ There was no hiring of additional public officials (¬D).

Solution:

Ch3 Pattern 3 from ①: A→(B∧C) ≡ (A→B)∧(A→C)

Required chain: A→C→D

④ ¬D → Modus Tollens yields ¬C

¬C → Contrapositive of ①: (¬B∨¬C)→¬A, since ¬C is true:

Conclusion: The national project was not selected (¬A). ✓

Also: ¬A → ¬B (contrapositive), ¬B → ¬E (contrapositive of ③)

Additional conclusion: It was not reported to the Ministry of Economy and Finance (¬E). ✓

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5. Proof by Contradiction

An applied technique of chain reasoning. This is a method of proving that a conclusion is true by assuming that the conclusion is false and showing that a contradiction arises with the premises.

Structure of Proof by Contradiction

To prove that conclusion P is true:

1. Assume P is false (Assume ¬P)
2. Apply the premises and ¬P together
3. Derive a contradiction with the premises (A∧¬A)
4. ∴ ¬P is false → P is true

Example of Proof by Contradiction

Premises: ① A → B ② B → ¬A (If B, then not A) ③ A is true.

Candidate Conclusion: “A and B cannot be true at the same time” — Let’s prove it

By contradiction: Assume both A and B are true (Assume ¬Conclusion)

From ③, A=T → from ①, B=T (OK)

B=T → from ②, ¬A=T (A=F)

However, from ③, A was T. A=T and A=F → Contradiction!

Therefore, the conclusion “A and B cannot be true at the same time” is proven to be true. ✓

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6. Tips for Rapid Reasoning Without Truth Tables

You don’t have time to draw full truth tables in the exam room. Practical strategies for rapid reasoning:

Strategy 1: Start from what is true (Given)

If a specific proposition is given as true among the premises, use it as a starting point and apply Modus Ponens consecutively.

Strategy 2: Go backwards from what is false (Given)

If a specific proposition is given as false, apply the contrapositive (Modus Tollens) consecutively.

Strategy 3: Draw the chain

For complex problems, connect conditionals into a chain and visualize them:

• Original chain: A → B → C → D
• Contrapositive direction: ¬D → ¬C → ¬B → ¬A — negate the end and the negation travels back up the chain

Strategy 4: Immediately decompose ∧/∨

When you see (A∨B)→C, immediately decompose it into “A→C and B→C”. When you see A→(B∧C), immediately decompose it into “A→B and A→C”.

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7. Comprehensive Summary of Frequently Tested Practical Patterns

Situation	Correct Inference
A→B, A=T	B=T (Modus Ponens)
A→B, B=F	A=F (Modus Tollens / Contrapositive)
A→B, B=T	Truth value of A is unknown (Affirming the Consequent = Error)
A→B, A=F	Truth value of B is unknown (Denying the Antecedent = Error)
A→B→C, A=T	C=T (Chain)
A→B→C, C=F	A=F (Reverse chain)
A∨B, ¬A	B=T (Disjunctive Syllogism)

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Core Summary

1. Hypothetical Syllogism: A→B, B→C → A→C (Chain connection)
2. Modus Ponens: A→B, A=T → B=T
3. Modus Tollens: A→B, B=F → A=F (Contrapositive)
4. Disjunctive Syllogism: A∨B, ¬A → B=T
5. Proof by Contradiction: Assume ¬P and derive a contradiction → P=T
6. Forbidden Inferences: Affirming the consequent (error), Denying the antecedent (error)

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In the next episode, Ch5: Complete Guide to Necessary and Sufficient Conditions, we cover how to convert expressions like “In A→B, B is a necessary condition for A, and A is a sufficient condition for B” into conditionals, and address the most commonly confused misconceptions about necessary and sufficient conditions in actual exams.