Deductive Logic/Inference Rules

Inference Rules

A number of valid argument schemas are useful for drawing conclusions from the premises. These are called inference rules. Inference rules preserve truth—if the premises are true, the conclusions must also be true. These rules of inference can be written in the following standard form:

Premise 1 Premise 2 … Premise #n Conclusion

For example, the inference rule known as modus ponens is written as:

p → q p q

And can be read as: if p then q, and p imply (therefore) q.

Several valid rules of inference will be introduced, including the name, form, and example for each.

Modus Ponens

(affirming the antecedent)

p → q p q

If it is Saturday, then I will wash my car. It is Saturday, therefore I will wash my car.

A common fallacy of modus ponens is affirming the consequent. This has the general form of asserting q and then concluding p from the premise p→q. In the above example, knowing that I will wash my car it is invalid to conclude that it is Saturday, because I may wash my car on other days as well.

Modus Tollens

(denying the consequent)

p → q ¬ q ¬ p

If it is Saturday, then I will wash my car. I am not washing my car, therefore it is not Saturday.

A common fallacy of modes tollens is denying the antecedent. This has the general form of denying p and then concluding ¬ q. In the above example, knowing that it is not Saturday it is invalid to conclude that I am not washing my car, because I may wash my car on other days as well.

Conjunction

p q p • q

Today is Saturday. It is raining. Therefore it is Saturday and it is raining.

Addition

p p ∨ q

It is Saturday. Therefore it is Saturday or pigs fly.

Simplification

p • q p

It is Saturday and it is raining. Therefore, it is Saturday.

Elimination

p ∨ q ¬ p q

Pigs can fly, or it is Saturday. Pigs cannot fly. Therefore it is Saturday.

Transitivity

p → q q → r p → r

If today is Saturday, then I will wash my car. If I wash my car, then my car will shine. Therefore if today is Saturday then my car will shine.

Constructive Dilemma

p → q r → s p ∨ r q ∨ s

If it is Saturday, then I will wash my car. If it rains, I will use an umbrella. Either it is Saturday, or it is raining. Therefore, either I am washing my car or I will use an umbrella.

Destructive Dilemma

p → q r → s ¬q ∨ ¬ s ¬p ∨ ¬ r

If it is Saturday, then I will wash my car. If it rains, I will use an umbrella. I am not washing my car or I am not using an umbrella. Therefore, it is not Saturday, or it is not raining.