Rules of Inference

Modus Ponens

Summary

”P implies Q. P is true. Therefore, Q must also be true.”

The form of a modus ponens argument is a mixed hypothetical syllogism, with two premises and a conclusion:

1. If P, then Q.
2. P.
3. Therefore, Q.

Modus Tollens

Summary

”P implies Q and Q is false, then P must also be false.”

The form of a modus tollens argument is a mixed hypothetical syllogism, with two premises and a conclusion:

1. If P, then Q.
2. Not Q.
3. Therefore, not P.

Hypothetical Syllogism

Summary

”P implies Q and Q implies R, then P implies R.”

1. If P, then Q.
2. If Q, then R.
3. Therefore, If P, then R.

Disjunctive Syllogism

Summary

”P or Q is true, and P is false, then Q must be true.”

1. If P and Q.
2. Not P.
3. Therefore, Q.

or

1. If P and Q.
2. Not Q.
3. Therefore, P.

Simplification

Summary

”From a conjunction, infer either of the conjuncts.”

1. P and Q.
2. Therefore, P.

or

1. P and Q.
2. Therefore, Q.

Addition

Summary

”From a proposition, infer the disjunction with another proposition.”

1. P.
2. Therefore, P or Q.

or

1. Q.
2. Therefore, P or Q.

Conjunction

Summary

”From two propositions, infer their conjunction.”

1. P.
2. Q.
3. Therefore, P or Q.