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Lemma: Hypothetical Syllogism

The hypothetical syllogism is a valid logical argument consisting of two premises and a conclusion, which altogether are constructed from three propositions $p,q$ and $r$ as follows:

• The first premise (called the major premise) is the implication $p\vee q$.
• The second premise (called the minor premise) is the implication $q\Rightarrow r$.
• The implication $p\Rightarrow r$ is the conclusion of the hypothecal syllogism.

Formally, the hypothecal syllogism is the following logical argument: $$\begin{array}{rll} p\Rightarrow q&\text{major premise}&\text{e.g. If it is raining then the roads are wet.}\\ q\Rightarrow r&\text{minor premise}&\text{e.g. If the roads are wet then they are slippery.}\\ \hline p\Rightarrow r&\text{conclusion}&\text{e.g. If it is raining, then the roads are slippery.}\\ \end{array} $$

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References

Bibliography

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