Inference: Disjunctive and Hypothetical Syllogism
Learn about disjunctive and hypothetical syllogism.
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Disjunctive syllogism
If we know that $\left(q_1 \lor q_2\right)$ is true and further we know that $\neg q_1$ is true, then we can conclude that $q_2$ is true. We can write it as the following tautology:
$\left(q_1 \lor q_2\right)\land \neg q_1\Rightarrow q_2.$
Examples
To further comprehend the application of this rule, let’s look at a few examples.
Consider the following propositions:
 $W_{C}$: William is a cricket player.
 $W_{F}$: William is a football player.
Now, assume that the following statements are facts:

$W_{C}\lor W_{F}$: William is a cricket player or a football player.

$\neg W_{C}$: William is not a cricket player.
Then we can conclude that $W_{F}$ is true.
For the next example, consider the following propositions:
 $W_{P}$: Wilma is going to Paris for a vacation.
 $W_{L}$: Wilma is going to London for a vacation.
Assume that the following propositions are true.

$W_{P}\lor W_{L}$: Wilma is going to Paris or London for a vacation.

$\neg W_{P}$: Wilma is not going to Paris for a vacation.
Then, we can conclude that the following proposition is true:
 $W_{L}$: Wilma is going to London to spend her vacation.
Hypothetical syllogism
If we know that $q_1\Rightarrow q_2$ is true and $q_2 \Rightarrow q_3$ is also true, then we can conclude that $q_1 \Rightarrow q_3$ is true. We can verify it by the following truth table:
$\left(q_1,q_2,q_3\right)$  $q_1\Rightarrow q_2$  $q_2\Rightarrow q_3$  $q_1\Rightarrow q_3$ 

1 – (T,T,T)  T  T  T 
2 – (T,T,F)  T  F  F 
3 – (T,F,T)  F  T  T 
4 – (T,F,F)  F  T  F 
5 – (F,T,T)  T  T  T 
6 – (F,T,F)  T  F  T 
7 – (F,F,T)  T  T  T 
8 – (F,F,F)  T  T  T 
We can observe that $\left(q_1\Rightarrow q_2\right)$ and $\left(q_2\Rightarrow q_3\right)$ both are true in rows number one, five, seven, and eight (shown in bold); and in all these four cases $\left(q_1\Rightarrow q_3\right)$ is also true. We can write it as the following tautology:
$\left(\left(q_1\Rightarrow q_2\right)\land \left(q_2\Rightarrow q_3\right)\right)\Rightarrow \left(q_1\Rightarrow q_3\right).$
Examples
Let’s look at a few examples to comprehend further and apply hypothetical syllogism.
Consider the following propositions:
 $L_{B}$: Lina wants to buy a pizza.
 $L_{P}$: Lina has to pay for pizza.
 $L_{M}$: Lina needs money.
Now assume that the following propositions are true:

$L_{B}\Rightarrow L_{P}$: If Lina wants to buy a pizza, (then) she has to pay for it.

$L_{P}\Rightarrow L_{M}$: If Lina has to pay for pizza, then she needs money.
Then by applying hypothetical syllogism, we can conclude that the following proposition is true:
 $L_{B}\Rightarrow L_{M}$: If Lina wants to buy a pizza, then she needs money.
For the next example, consider the following propositions.
 $S_{F}$: Sam wants to fly to Florida.
 $S_{T}$: Sam needs to buy an air ticket to Florida.
 $S_{M}$: Sam needs money.
Now assume that the following propositions are true:

$S_{F} \Rightarrow S_{T}$: If Sam wants to fly to Florida, (then) he needs to buy an air ticket to Florida.

$S_{T} \Rightarrow S_{M}$: If Sam needs to buy an air ticket to Florida, (then) he needs money.
Then, by applying hypothetical syllogism, we can conclude that the following proposition is true.
 $S_{F}\Rightarrow S_{M}$: If Sam wants to fly to Florida, then he needs money.
Quiz
Test your understanding of the disjunctive and hypothetical syllogism.
(Select all that apply.) Consider the following propositions:
$V$: Oliver is eating vanillaflavored ice cream.
$C$: Oliver is eating chocolateflavored ice cream.
According to disjunctive syllogism, two of the following statements must be true to conclude that Oliver is eating chocolateflavored ice cream. What are those two statements?
Oliver is eating vanillaflavored ice cream and not chocolateflavored ice cream.
Oliver is not eating vanillaflavored ice cream.
Oliver is eating vanillaflavored ice cream or chocolateflavored ice cream.
Oliver is neither eating vanillaflavored ice cream nor chocolateflavored ice cream.