Inference: Constructive and Destructive Dilemma

Constructive dilemma

If we know that $\left(q_1\Rightarrow q_2\right)\land\left(q_3\Rightarrow q_4\right)$ is true, and $\left(q_1 \lor q_3\right)$ is also true, then we can conclude that $\left(q_2\lor q_4\right)$ is true. We can write it as the following tautology:

$$\left(\left(q_1\Rightarrow q_2\right)\land\left(q_3\Rightarrow q_4\right)\land \left(q_1 \lor q_3\right)\right)\Rightarrow \left(q_2\lor q_4\right).$$

To understand why this is a tautology, we observe that, if $\left(q_1 \lor q_3\right)$ is true, there are three possibilities; let’s look at them one by one.

$\bold {q_1}$ is true: In this case, $q_2$ has to be true otherwise, $\left(q_1\Rightarrow q_2\right)$ will become false. Hence, $\left(q_2\lor q_4\right)$ is true.

$\bold {q_3}$ is true: In this case, $q_4$ has to be true otherwise, $\left(q_3\Rightarrow q_4\right)$ will become false. Hence, $\left(q_2\lor q_4\right)$ is true.

$\bold {q_1}$ and $\bold {q_3}$ both are true: In this case, $q_2$ has to be true otherwise, $\left(q_1\Rightarrow q_2\right)$ will become false; and $q_4$ has to be true otherwise, $\left(q_3\Rightarrow q_4\right)$ will become false. Hence, $\left(q_2\lor q_4\right)$ is true.

Examples

Let’s look at a few examples to understand and apply the rule of constructive dilemma.

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Consider the following propositions:

• $F_{S}$: Harry wants to fly to Sydney.
• $A_{T}$: Harry needs an air ticket to Sydney.
• $T_{S}$: Harry wants to take a train to Sydney.
• $T_{T}$: Harry needs a train ticket to Sydney.

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Now assume that the following propositions are true:

• $F_{S}\Rightarrow A_{T}$: If Harry wants to fly to Sydney, (then) he needs an air ticket to Sydney.

• $T_{S}\Rightarrow T_{T}$: If Harry wants to take a train to Sydney, (then) he needs a train ticket to Sydney.

• $F_{S}\lor T_{S}$: Harry wants to fly or take a train to Sydney.