Hypothetical Syllogism/Formulation 1

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Theorem

$p \implies q, q \implies r \vdash p \implies r$


Proof 1

By the tableau method of natural deduction:

$p \implies q, q \implies r \vdash p \implies r$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $q \implies r$ Premise (None)
3 3 $p$ Assumption (None)
4 1, 3 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3
5 1, 2, 3 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 4
6 1, 2 $p \implies r$ Rule of Implication: $\implies \mathcal I$ 3 – 5 Assumption 3 has been discharged

$\blacksquare$


Proof 2

We apply the Method of Truth Tables to the propositions in turn.

As can be seen for all boolean interpretations by inspection, where the truth values under the main connective on the left hand side is $T$, that under the one on the right hand side is also $T$:


$\begin{array}{|ccccccc||ccc|} \hline (p & \implies & q) & \land & (q & \implies & r) & p & \implies & r \\ \hline F & T & F & T & F & T & F & F & T & F \\ F & T & F & T & F & T & T & F & T & T \\ F & T & T & F & T & F & F & F & T & F \\ F & T & T & T & T & T & T & F & T & T \\ T & F & F & F & F & T & F & T & F & F \\ T & F & F & F & F & T & T & T & T & T \\ T & T & T & F & T & F & F & T & F & F \\ T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$

Hence the result.

$\blacksquare$


Proof 3

This proof uses $\mathscr H_2$, Instance 2 of the Hilbert proof systems.

Recall the sequent form of the Hypothetical Syllogism:

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

Applying the Rule of Detachment $RST3$ twice, we obtain:

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

and subsequently:

$\vdash p \implies r$

as desired.

$\blacksquare$


Sources