Hypothetical Syllogism

Chain reasoning: If A then B, If B then C, therefore If A then C.

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Chain reasoning: If A then B, If B then C, therefore If A then C.

Hypothetical Syllogism (also called Chain Reasoning or Hypothetical Chain) allows us to link conditional statements together. If we know that A implies B, and B implies C, we can conclude that A implies C. This rule is essential for building chains of reasoning and is widely used in mathematical proofs and everyday logic.

P → Q, Q → R ⊢ P → R

If it rains, the streets get wet. If the streets get wet, traffic slows down. Therefore, if it rains, traffic slows down.
If you exercise regularly, you build stamina. If you build stamina, you can run longer. Therefore, if you exercise regularly, you can run longer.
If a shape is a square, it is a rectangle. If a shape is a rectangle, it is a quadrilateral. Therefore, if a shape is a square, it is a quadrilateral.

How to recognize it

Look for two 'If...then...' statements (conditionals)
The consequent of the first should match the antecedent of the second
The conclusion links the first antecedent to the last consequent

Common mistakes

Reversing the chain: 'If A then B, If C then B' does NOT give 'If A then C'
The middle terms must match exactly for the chain to be valid

Step-by-step walkthrough

If it rains, the streets get wet. If the streets get wet, traffic slows. What can we conclude?

Identify first conditional: 'If rain → streets wet'
Identify second conditional: 'If streets wet → traffic slows'
The consequent of first (Q) matches antecedent of second (Q)
Combine: If it rains, traffic slows

Answer: If it rains, traffic slows

Practice guidance

Logical Deduction