The Law of Syllogism
Chain two conditionals together: if and , then . Logic lets you skip the middle step.
Introduction
The Law of Syllogism is like a logical chain: it lets you link conditionals end-to-end. If A implies B, and B implies C, then A implies C — directly.
Past Knowledge
Law of Detachment (2.2.1). Conditional statements (2.1.3).
Today's Goal
Chain two or more conditionals into a single new conditional using the Law of Syllogism.
Future Success
Multi-step proofs (2.3) chain many syllogism + detachment steps together.
Key Concepts
Law of Syllogism
If both of these are true:
Then you can conclude:
The Chain Must Connect
The conclusion of the first conditional must match the hypothesis of the second. If they don't connect, the Law of Syllogism does not apply.
Worked Examples
Everyday Chain
(1) If I study, then I pass. (2) If I pass, then I graduate. What can you conclude?
The conclusion of (1) — “I pass” — is the hypothesis of (2). The chain connects.
Conclusion: If I study, then I graduate.
Geometric Chain
(1) If two angles form a linear pair, then they are supplementary. (2) If two angles are supplementary, then their measures sum to 180°.
“supplementary” is the link between the two statements.
Conclusion: If two angles form a linear pair, then their measures sum to 180°.
No Valid Chain
(1) If , then . (2) If , then is positive. Can you chain them?
The conclusion of (1) is “”, which does not match the hypothesis of (2) “”.
Answer: No valid conclusion — the chain doesn't connect.
Common Pitfalls
Forcing a Non-Matching Chain
The conclusion of the first statement must exactly match the hypothesis of the second. Don't force connections that don't exist.
Real-Life Applications
Food Chains in Ecology
A food chain is a syllogism: if grass feeds rabbits, and rabbits feed foxes, then the energy from grass reaches foxes. Scientists trace energy transfer by chaining cause-and-effect steps — the same logic as the Law of Syllogism.
Practice Quiz
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