Inference Rules

Logical inference rules provide the foundation for constructing correct arguments. These are patterns that ensure if the starting statements are true, the conclusion will also be true.

Modus Ponens

If p is true, and p implies q, then q is true.

Example: If it is raining (p), then the road is wet (q). It is raining → therefore the road is wet.

Modus Tollens

If p → q is true, and q is false, then p cannot be true.

Example: If it is raining (p), then the road is wet (q). The road is not wet → therefore it is not raining.

Hypothetical Syllogism

If p implies q, and q implies r, then p implies r.

Example: If you study (p), then you understand the material (q). If you understand the material, then you will succeed on the exam (r). Therefore: if you study, you will succeed on the exam.

Disjunctive Syllogism

If p or q is true, and p is false, then q is true.

Example: Either Anna is at home (p), or at the store (q). If she is not at home, then she must be at the store.

Double Negation Rule

Double negation returns to the original statement: if it is not true that not p, then p is true.

Summary

Inference rules ensure correct reasoning. The most well-known are: modus ponens, modus tollens, hypothetical and disjunctive syllogism, as well as the double negation rule.

Practice Exercise