Inference: Constructive and Destructive Dilemma
Learn about the constructive and destructive dilemma.
We'll cover the following
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.
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.
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.
Get handson with 1200+ tech skills courses.