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.

Then by applying the rule of constructive dilemma, we can conclude that the following proposition is true:

• $A_{T}\lor T_{T}$: Harry needs an air ticket or a train ticket to Sydney.

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For the next example, consider the following propositions:

• $R_{P}$: Johnson wants to reduce the probability of getting the flu.
• $S_{D}$: Johnson should keep social distance.
• $I_{I}$: Johnson wants to improve his immunity against the flu.
• $G_{V}$: Johnson should get the flu vaccine.

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Now, assume that the following propositions are true.

• $R_{P} \Rightarrow S_{D}$: If Johnson wants to reduce the probability of getting the flu, (then) he should keep social distance.

• $I_{I} \Rightarrow G_{V}$: If Johnson wants to improve his immunity against the flu, (then) he should get the vaccination.

• $R_{P} \lor I_{I}$: Johnson wants to reduce the probability of getting the flu or improve his immunity against the virus.

Then, by applying the rule of constructive dilemma, we can conclude that the following proposition is true.

• $S_{D} \lor G_{V}$: Johnson should keep social distance, or he should get the flu vaccination.

Destructive dilemma

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

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

To understand why this is a tautology, we observe that if $\left(\neg q_2 \lor \neg q_4\right)$ is true, there are three possibilities; let’s look at them one by one.

$\bold {\neg q_2}$ is true: We know that $\left(q_1\Rightarrow q_2\right)$ is true. So, its contrapositive $(\neg q_2\Rightarrow \neg q_1)$ is also true. Therefore, if $\neg q_2$ is true, then $\neg q_1$ has to be true. Consequently, $\left(\neg q_1\lor \neg q_3\right)$ is true.

$\bold {\neg q_4}$ is true: We assume that $\left(q_3\Rightarrow q_4\right)$ is true. Additionally, its contrapositive $\left(\neg q_4\Rightarrow \neg q_3\right)$ is also true. Therefore, if $\neg q_4$ is true, then $\neg q_3$ has to be true. So, $\left(\neg q_1\lor \neg q_3\right)$ is true.

$\bold {\neg q_2}$ and $\bold {\neg q_4}$ both are true: We assume that $\left(q_1\Rightarrow q_2\right)$ and $\left(q_3\Rightarrow q_4\right)$ are true. Therefore, their contrapositives $\left(\neg q_2\Rightarrow \neg q_1\right)$ and $\left(\neg q_4\Rightarrow \neg q_3\right)$ are also true. Consequently, if $\neg q_2$ is true, then $\neg q_1$ has to be true and if $\neg q_4$ is true, then $\neg q_3$ has to be true. So, $\left(\neg q_1\lor \neg q_3\right)$ is true.

Examples

Let’s look at a few examples to see how we can use the rule of destructive dilemma.

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For the first example, once again, consider the propositions $R_{P}, S_{D}, I_{I},$ and $G_{V}$.

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Now, assume that the following propositions are true:

• $R_{P} \Rightarrow S_{D}$: If Johnson wants to reduce the probability of getting the flu virus, (then) he should keep social distance.
• $I_{I} \Rightarrow G_{V}$: If Johnson wants to improve his immunity against the flu virus, (then) he should get the flu vaccination.
• $\neg S_{D} \lor \neg G_{V}$: Johnson is not keeping social distance, or he is not getting the flu vaccination.

Then, by applying the rule of destructive dilemma, we can conclude that the following proposition is true:

• $\neg R_{P} \lor \neg I_{I}$: Johnson does not want to reduce the probability of getting the flu, or he does not want to improve his immunity against the flu.

• $S_{H} \Rightarrow S_{M}$: if Sarim was home at 9:00 a.m., (then) he went to the market with his father.

• $\neg S_{Z} \lor \neg S_{M}$: Sarim did not go to the zoo or (he did not go to) the market with his father.

Then, by applying the rule of destructive dilemma, we can conclude that the following proposition is true.

• $\neg S_{T} \lor \neg S_{H}$: Sarim did not go on the school trip or was not home at 9:00 a.m.

Quiz