Rule of Inference

Question

B. Identify the Rule of Inference that establishes the ff. arguments’ validity.

1.     If the promises of elected official are fulfilled, then development in his jurisdiction is observed.

Development in elected official’s jurisdiction is not observed.

Therefore, the promises of elected official are not fulfilled.

2.     Hardworking student gets passing grades.

If hardworking student gets passing grades, then he will pursue his education.

Therefore, hardworking student will pursue his education.

3.  JAMES PLAYS VIOLIN OR HE PLAYS PIANO. JAMES DOES NOT PLAY VIOLIN.

     THEREFORE, HE PLAYS PIANO.

4.  10 > 3 OR 5 ≤ 2.  

     10 ≤ 3. therefore, 5 ≤ 2.

5. IF REY CAN DRIVE A CAR, THEN HE CAN DRIVE A MOTORCYCLE.

 REY CANNOT DRIVE A MOTORCYCLE.

THEREFORE, REY CANNOT DRIVE A CAR.

Answer 1

The first step to solve questions of this type is to translate them into a propositional language. In other words, replace the propositions with propositional letters (a.k.a variables).

For question 1 we have:

1. If the promises of elected official are fulfilled, then development in his jurisdiction is observed.

2. Development in elected official’s jurisdiction is not observed.

3. Therefore, the promises of elected official are not fulfilled.

Replacing propositions with variables:

1. P -> Q

2. ~Q

3. ~P

in which P represents "If the promises of elected official are fulfilled" and Q represents "development in his jurisdiction is observed".

Now we will use the propositional logic representation to determine the rule of inference used in this argument. If you have a table of rules with their definitions, you will simply compare this logical form to the definition. Some courses/texts use different rules of inference (because often one is derivable from another). Therefore, it will depend on which rules you are using. In this case, the rule used is very common and is usually called Modus Tollens, Modus Tollendo Tollens, or Denying the Consequent.

In propositional form:

2.

1. P
2. P -> Q
3. Q

This is Modus Ponens.

3.

1. P v Q
2. ~P
3. Q

This is Disjunctive Syllogism.

4.

1. P or Q
2. ~P
3. Q

This is Disjunctive Syllogism again.

5.

1. P -> Q
2. ~Q
3. ~P

Modus Tollens, again.