Hypothetical Syllogism
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Theorem
The (rule of the) hypothetical syllogism is a valid deduction sequent in propositional logic:
- If we can conclude that $p$ implies $q$, and if we can also conclude that $q$ implies $r$, then we may infer that $p$ implies $r$.
Formulation 1
\(\ds p\) | \(\implies\) | \(\ds q\) | ||||||||||||
\(\ds q\) | \(\implies\) | \(\ds r\) | ||||||||||||
\(\ds \vdash \ \ \) | \(\ds p\) | \(\implies\) | \(\ds r\) |
Formulation 2
\(\ds p\) | \(\implies\) | \(\ds q\) | ||||||||||||
\(\ds q\) | \(\implies\) | \(\ds r\) | ||||||||||||
\(\ds p\) | \(\) | \(\ds \) | ||||||||||||
\(\ds \vdash \ \ \) | \(\ds r\) | \(\) | \(\ds \) |
Formulation 3
- $\vdash \paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$
Formulation 4
- $\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$
Formulation 5
- $\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Also known as
It is referred to by some authors as the principle of syllogism
It is also known as the transitivity law.
Its abbreviation in a tableau proof is $\textrm{HS}$.
Examples
Ancient Chinese Proverb
- If there is light in the soul,
- then there will be beauty in the person.
- If there is beauty in the person,
- then there will be harmony in the house.
- If there is harmony in the house,
- then there will be order in the nation.
- If there is order in the nation,
- then there will be peace in the world.
The conclusion is:
- If there is light in the soul, then there will be peace in the world.
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.12$: Laws of sentential calculus
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.9$: Derivation by Substitution
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hypothetical syllogism