# Deductive Logic/Inference Rules

## Inference Rules

editA 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

edit(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

edit(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

edit

p

q

p•q

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

### Addition

edit

p

p∨q

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

### Simplification

edit

p•q

p

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

### Elimination

edit

p∨q

¬p

q

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

### Transitivity

edit

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

edit

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

edit

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.