Inference: Disjunctive and Hypothetical Syllogism

Learn about disjunctive and hypothetical syllogism.

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Disjunctive syllogism

If we know that (q1∨q2)\left(q_1 \lor q_2\right) is true and further we know that ¬q1\neg q_1 is true, then we can conclude that q2q_2 is true. We can write it as the following tautology:

(q1∨q2)∧¬q1⇒q2.\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:

  • WCW_{C}: William is a cricket player.
  • WFW_{F}: William is a football player.

Now, assume that the following statements are facts:

  • WC∨WFW_{C}\lor W_{F}: William is a cricket player or a football player.

  • ¬WC\neg W_{C}: William is not a cricket player.

Then we can conclude that WFW_{F} is true.

William

For the next example, consider the following propositions:

  • WPW_{P}: Wilma is going to Paris for a vacation.
  • WLW_{L}: Wilma is going to London for a vacation.

Assume that the following propositions are true.

  • WP∨WLW_{P}\lor W_{L}: Wilma is going to Paris or London for a vacation.

  • ¬WP\neg W_{P}: Wilma is not going to Paris for a vacation.

Then, we can conclude that the following proposition is true:

  • WLW_{L}: Wilma is going to London to spend her vacation.
Wilma

Hypothetical syllogism

If we know that q1⇒q2q_1\Rightarrow q_2 is true and q2⇒q3q_2 \Rightarrow q_3 is also true, then we can conclude that q1⇒q3q_1 \Rightarrow q_3 is true. We can verify it by the following truth table:

(q1,q2,q3)\left(q_1,q_2,q_3\right) q1⇒q2q_1\Rightarrow q_2 q2⇒q3q_2\Rightarrow q_3 q1⇒q3q_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 (q1⇒q2)\left(q_1\Rightarrow q_2\right) and (q2⇒q3)\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 (q1⇒q3)\left(q_1\Rightarrow q_3\right) is also true. We can write it as the following tautology:

((q1⇒q2)∧(q2⇒q3))⇒(q1⇒q3).\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:

  • LBL_{B}: Lina wants to buy a pizza.
  • LPL_{P}: Lina has to pay for pizza.
  • LML_{M}: Lina needs money.

Now assume that the following propositions are true:

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

  • LP⇒LML_{P}\Rightarrow L_{M}: If Lina has to pay for pizza, then she needs money.

Lina

Then by applying hypothetical syllogism, we can conclude that the following proposition is true:

  • LB⇒LML_{B}\Rightarrow L_{M}: If Lina wants to buy a pizza, then she needs money.

For the next example, consider the following propositions.

  • SFS_{F}: Sam wants to fly to Florida.
  • STS_{T}: Sam needs to buy an air ticket to Florida.
  • SMS_{M}: Sam needs money.

Now assume that the following propositions are true:

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

  • ST⇒SMS_{T} \Rightarrow S_{M}: If Sam needs to buy an air ticket to Florida, (then) he needs money.

Sam wants to fly to Florida

Then, by applying hypothetical syllogism, we can conclude that the following proposition is true.

  • SF⇒SMS_{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.

1

(Select all that apply.) Consider the following propositions:

VV: Oliver is eating vanilla-flavored ice cream.

CC: Oliver is eating chocolate-flavored ice cream.

According to disjunctive syllogism, two of the following statements must be true to conclude that Oliver is eating chocolate-flavored ice cream. What are those two statements?

A)

Oliver is eating vanilla-flavored ice cream and not chocolate-flavored ice cream.

B)

Oliver is not eating vanilla-flavored ice cream.

C)

Oliver is eating vanilla-flavored ice cream or chocolate-flavored ice cream.

D)

Oliver is neither eating vanilla-flavored ice cream nor chocolate-flavored ice cream.

Question 1 of 20 attempted