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Lemma: Modus Tollens

Modus tollens (lat. "the way that denies by denying") is a valid logical argument consisting of two premises and a conclusion, which altogether are constructed from two propositions $p$ and $q$ as follows:

• The first premise (called the major premise) is the implication $p\Rightarrow q$.
• The second premise (called the minor premise) is the negated proposition $\neg q$.
• The negated proposition $\neg p$ is the conclusion of the modus tollens.

Formally, modus tollens is the following logical argument: $$\begin{array}{rll} p\Rightarrow q&\text{major premise}&\text{e.g. If it is raining, then the roads are wet.}\\ \neg q&\text{minor premise}&\text{e.g. The roads are not wet.}\\ \hline \neg p&\text{conclusion}&\text{e.g. It is not raining.}\\ \end{array} $$

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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016