RULES of INFERENCE - DISCRETE MATHEMATICS

Introduction

This tutorial covers the rules of inference in discrete mathematics, which are essential for constructing valid logical arguments. Understanding these rules is crucial for students studying logic, mathematics, and computer science. We will explore several key rules, including modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplification, and conjunction.

Step 1: Understand Modus Ponens

Modus Ponens is a fundamental rule of inference that can be summarized as follows:

• If you have a statement "If P, then Q" (P → Q)
• And you know that P is true
• You can conclude that Q is true.

Example

• Premise 1: If it rains, then the ground is wet (P → Q).
• Premise 2: It is raining (P).
• Conclusion: Therefore, the ground is wet (Q).

Practical Tip: Always ensure that the first premise is a conditional statement before applying this rule.

Step 2: Understand Modus Tollens

Modus Tollens is another important rule of inference:

• If you have "If P, then Q" (P → Q)
• And you know that Q is false
• You can conclude that P is false.

Example

• Premise 1: If it rains, then the ground is wet (P → Q).
• Premise 2: The ground is not wet (¬Q).
• Conclusion: Therefore, it is not raining (¬P).

Common Pitfall: Ensure that the second premise explicitly negates the conclusion of your conditional statement.

Step 3: Explore Hypothetical Syllogism

Hypothetical Syllogism allows you to chain together conditional statements:

• If you have "If P, then Q" (P → Q)
• And "If Q, then R" (Q → R)
• You can conclude "If P, then R" (P → R).

Example

• Premise 1: If it rains, then the ground is wet (P → Q).
• Premise 2: If the ground is wet, then the plants are happy (Q → R).
• Conclusion: If it rains, then the plants are happy (P → R).

Real-World Application: This rule is useful in programming logic and decision-making processes.

Step 4: Understand Disjunctive Syllogism

Disjunctive Syllogism is used when dealing with "or" statements:

• If you have "P or Q" (P ∨ Q)
• And you know that P is false
• You can conclude that Q is true.

Example

• Premise 1: Either it rains or it is sunny (P ∨ Q).
• Premise 2: It is not raining (¬P).
• Conclusion: Therefore, it is sunny (Q).

Practical Tip: This rule is particularly useful in scenarios where multiple possibilities exist.

Step 5: Learn Addition

The addition rule allows you to introduce a new disjunct:

• If you know that P is true
• You can conclude "P or Q" (P ∨ Q).

Example

• Premise: It is raining (P).
• Conclusion: It is raining or it is sunny (P ∨ Q).

Common Pitfall: Ensure that the new disjunct (Q) is relevant to maintain logical integrity.

Step 6: Understand Simplification

Simplification lets you derive a single proposition from a conjunction:

• If you have "P and Q" (P ∧ Q)
• You can conclude P or Q individually.

Example

• Premise: It is raining and the ground is wet (P ∧ Q).
• Conclusion: It is raining (P) or the ground is wet (Q).

Practical Tip: Use simplification cautiously, as it does not imply that both conditions are necessary for the conclusion.

Step 7: Explore Conjunction

Conjunction allows you to combine two true statements:

• If you know P is true and Q is true
• You can conclude "P and Q" (P ∧ Q).

Example

• Premise 1: It is raining (P).
• Premise 2: The ground is wet (Q).
• Conclusion: It is raining and the ground is wet (P ∧ Q).

Real-World Application: This is commonly used in compiling results from multiple conditions in programming.

Conclusion

Understanding the rules of inference is vital for logical reasoning in mathematics and computer science. The key rules discussed—modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplification, and conjunction—provide a foundation for constructing valid arguments. Familiarizing yourself with these rules will enhance your logical thinking skills and improve your problem-solving capabilities. Consider practicing with real-world examples to solidify your understanding and application of these concepts.