Rules of Inference (Discrete Math)

Introduction

This tutorial will guide you through the fundamental rules of inference in discrete mathematics. Understanding these rules is essential for constructing valid arguments and engaging with propositional and predicate logic. Whether you're studying for a course or just interested in logical reasoning, this guide will provide a clear and actionable framework for applying these rules effectively.

Step 1: Understanding Propositional Logic

• Propositional logic deals with statements or propositions that can be either true or false.
• Key components include:
  • Premises: Statements assumed to be true.
  • Conclusion: The statement derived from the premises.
• Familiarize yourself with logical operators such as AND, OR, and NOT, which are used to connect propositions.

Step 2: Learning the Rules of Inference

Rules of inference allow you to derive conclusions from premises. Here are the major rules:

Modus Ponens

• Structure: If P, then Q. P is true, therefore Q is true.
• Example:
  • If it rains (P), then the ground will be wet (Q).
  • It rains (P).
  • Conclusion: The ground is wet (Q).

Modus Tollens

• Structure: If P, then Q. Not Q, therefore not P.
• Example:
  • If it rains (P), then the ground will be wet (Q).
  • The ground is not wet (Not Q).
  • Conclusion: It does not rain (Not P).

Hypothetical Syllogism

• Structure: If P, then Q. If Q, then R. Therefore, if P, then R.
• Example:
  • If it rains (P), then the ground is wet (Q).
  • If the ground is wet (Q), then the flowers will bloom (R).
  • Conclusion: If it rains (P), then the flowers will bloom (R).

Disjunctive Syllogism

• Structure: P or Q. Not P, therefore Q (or vice versa).
• Example:
  • It is either raining (P) or sunny (Q).
  • It is not raining (Not P).
  • Conclusion: It is sunny (Q).

Step 3: Additional Rules of Inference

Familiarize yourself with these additional rules:

Addition Rule

• If P is true, then P or Q is true.

Conjunction Rule

• If P is true and Q is true, then P and Q is true.

Simplification Rule

• If P and Q is true, then P is true (and Q is also true).

Resolution Rule

• If P or Q is true, and Not P is true, then Q is true.

Conclusion

By mastering these rules of inference, you can enhance your logical reasoning skills and construct valid arguments in both propositional and predicate logic. Practice applying these rules with various statements to solidify your understanding. For further study, consider exploring more complex logical structures or engaging with exercises related to discrete mathematics. Refer to the lecture notes and additional resources provided in the video description for deeper insights.