Principle of Commutation/Formulation 2

Theorem

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

This can of course be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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

Proof

Proof of Forward Implication

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \implies \paren {q \implies r}$	Assumption	(None)
2		1	$q \implies \paren {p \implies r}$	Sequent Introduction	1	Principle of Commutation: Forward Implication: Formulation 1
3			$\paren {p \implies \paren {q \implies r} } \implies \paren {q \implies \paren {p \implies r} }$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged

$\blacksquare$

Proof of Reverse Implication

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$q \implies \paren {p \implies r}$	Assumption	(None)
2		1	$p \implies \paren {q \implies r}$	Sequent Introduction	1	Principle of Commutation: Reverse Implication: Formulation 1
3			$\paren {q \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged

$\blacksquare$

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1			$\paren {p \implies \paren {q \implies r} } \implies \paren {q \implies \paren {p \implies r} }$	Theorem Introduction	(None)	Principle of Commutation: Forward Implication: Formulation 2
2			$\paren {q \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$	Theorem Introduction	(None)	Principle of Commutation: Reverse Implication: Formulation 2
3			$\paren {p \implies \paren {q \implies r} } \iff \paren {q \implies \paren {p \implies r} }$	Biconditional Introduction: $\iff \II$	1, 2

$\blacksquare$

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text T 107$