What are modus tollens and the chain argument (hypothetical syllogism), and why are they valid?

Modus tollens: if A then B, not-B, therefore not-A. Chain: if A then B, if B then C, therefore if A then C. Both are valid conditional forms.

Two valid argument forms built on conditionals ("if A, then B"):

• Modus tollens ("denying the consequent"): (P1) If A, then B. (P2) Not-B. (K) Therefore not-A. — If B is required by A but B is absent, then A can't hold either.
• Chain argument / transitivity: (P1) If A, then B. (P2) If B, then C. (K) Therefore if A, then C. — The conditionals link up transitively.

Both are valid: in each, the conclusion follows necessarily from the premises by the logical structure alone.

Tip: Modus tollens is the engine of falsification: find that the predicted consequence (B) is false, and you've refuted the antecedent (A).