Hypothetical Syllogism/Formulation 4

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Theorem

$\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$


Proof 1

Let us use the following abbreviations

\(\displaystyle \phi\) \(\text{ for }\) \(\displaystyle p \implies q\)
\(\displaystyle \psi\) \(\text{ for }\) \(\displaystyle q \implies r\)
\(\displaystyle \chi\) \(\text{ for }\) \(\displaystyle p \implies r\)


By the tableau method of natural deduction:

$\left({p \implies q}\right) \implies \left({\left({q \implies r}\right) \implies \left({p \implies r}\right)}\right)$
Line Pool Formula Rule Depends upon Notes
1 $\left({\phi \land \psi}\right) \implies \chi$ Theorem Introduction (None) Hypothetical Syllogism: Formulation 3
2 $\phi \implies \left({\psi \implies \chi}\right)$ Sequent Introduction 1 Rule of Exportation

Expanding the abbreviations leads us back to:

$\vdash \left({p \implies q}\right) \implies \left({\left({q \implies r}\right) \implies \left({p \implies r}\right)}\right)$

$\blacksquare$


Proof 2

This proof is derived in the context of the following proof system: instance 1 of a Hilbert proof system.

By the tableau method:

$\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Assumption (None)
2 2 $p \implies q$ Assumption (None)
3 1, 2 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 2
4 4 $q \implies r$ Assumption (None)
5 1, 2, 4 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 3, 4
6 2, 4 $p \implies r$ Deduction Rule 5
7 2 $\paren {q \implies r} \implies \paren {p \implies r}$ Deduction Rule 6
8 $\paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$ Deduction Rule 7

$\blacksquare$


Sources