Principle of Dilemma/Formulation 1/Forward Implication

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Theorem

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


Proof 1

By the tableau method of natural deduction:

$\left({p \implies q}\right) \land \left({\neg p \implies q}\right) \vdash q$
Line Pool Formula Rule Depends upon Notes
1 1 $\left({p \implies q}\right) \land \left({\neg p \implies q}\right)$ Premise (None)
2 1 $p \implies q$ Rule of Simplification: $\land \mathcal E_1$ 1
3 1 $\neg p \implies q$ Rule of Simplification: $\land \mathcal E_2$ 1
4 1 $p \lor \neg p \implies q \lor q$ Sequent Introduction 2, 3 Constructive Dilemma
5 $p \lor \neg p$ Law of Excluded Middle (None)
6 1 $q \lor q$ Modus Ponendo Ponens: $\implies \mathcal E$ 4, 5
7 1 $q$ Sequent Introduction 6 Rule of Idempotence: Disjunction

$\blacksquare$


Proof 2

By the tableau method of natural deduction:

$\left({p \implies q}\right) \land \left({\neg p \implies q}\right) \vdash q$
Line Pool Formula Rule Depends upon Notes
1 1 $\left({p \implies q}\right) \land \left({\neg p \implies q}\right)$ Premise (None)
2 1 $p \implies q$ Rule of Simplification: $\land \mathcal E_1$ 1
3 1 $\neg p \implies q$ Rule of Simplification: $\land \mathcal E_2$ 1
4 $p \lor \neg p$ Law of Excluded Middle (None)
5 5 $p$ Assumption (None)
6 1, 5 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 5
7 7 $\neg p$ Assumption (None)
8 1, 7 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 3, 7
9 1 $q$ Proof by Cases: $\text{PBC}$ 4, 5 – 6, 7 – 8 Assumptions 5 and 7 have been discharged

$\blacksquare$


Proof 3

From the Constructive Dilemma we have:

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

from which, changing the names of letters strategically:

$p \implies q, \neg p \implies q \vdash p \lor \neg p \implies q \lor q$

From Law of Excluded Middle, we have:

$\vdash p \lor \neg p$

From the Rule of Idempotence we have:

$q \lor q \vdash q$

and the result follows by Hypothetical Syllogism.

$\blacksquare$

Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.


Sources