Rules of Inference

Logical rules of inference describe particular ways in which certain combinations of propositions can be used to derive the truth of additional propositions. Such rules of inference are useful for the construction of logically valid arguments, since it is by application of these rules of inferences that premises can be combined to logically entail particular conclusions. There are a large number of such rules, so only a few of the most important are outlined here. Misapplication of one or more rules of inference results in a formal logical fallacy, which are the subject of the next section.

Modus ponens

P1. If A then B

P2. A

C. Therefore B

Example: If you get a grade of B+ or higher, you will be accepted into the course. You got a grade of A-, which is higher than a B+, so you will be accepted into the course.

Modus tollens

P1. If A then B

P2. Not B

C. Therefore not A

Example: If everything was going according to plan then he would be here by now. But he isn’t here yet, so something must not have gone according to plan.

Hypothetical Syllogism

P1. If A then B

P2. If B then C

C. Therefore if A then C

Example: If it is cloudy it will probably rain, and if it will probably rain that it is a good idea to bring an umbrella. So if it is cloudy then it’s a good idea to bring an umbrella.

Disjunctive syllogism

P1. A or B

P2. Not B

C. Therefore A

Example: Either she’s at home or she has gone out. We know she hasn’t gone out, so she must be at home.

Further Reading

Rules of inference: a simple introduction from

Rules of inference: a discussion of some of the most common rules of inference used in philosophical logic

Rules of inference and logical proofs: a more advanced discussion incorporating methods of constructing proofs