1. (Arabic) Introduction to propositional logic

Introduction

This tutorial provides a comprehensive introduction to propositional logic, a fundamental area of mathematical logic that deals with propositions and their logical relationships. Understanding propositional logic is essential for various fields, including computer science, mathematics, and philosophy, as it forms the basis for logical reasoning and critical thinking.

Step 1: Understanding Propositions

• A proposition is a declarative statement that can be classified as either true or false, but not both.
• Examples of propositions:
  • "The sky is blue." (This can be true or false depending on the context.)
  • "2 + 2 equals 4." (This is a universally true proposition.)

Practical Tip

• Distinguish between propositions and non-propositional statements such as questions or commands. For example, "What time is it?" is not a proposition.

Step 2: Logical Connectives

• Learn about the basic logical connectives that combine propositions:
  • Conjunction (AND): Symbolized as ∧
    • Example: P ∧ Q is true only if both P and Q are true.
  • Disjunction (OR): Symbolized as ∨
    • Example: P ∨ Q is true if at least one of P or Q is true.
  • Negation (NOT): Symbolized as ¬
    • Example: ¬P is true if P is false.
  • Implication (IF...THEN): Symbolized as →
    • Example: P → Q is false only if P is true and Q is false.
  • Biconditional (IF AND ONLY IF): Symbolized as ↔
    • Example: P ↔ Q is true if both P and Q are either true or false together.

Common Pitfall

• Misunderstanding the implication operator. Remember that P → Q is not simply saying that if P is true, Q must also be true; it’s about the relationship between the truth values of P and Q.

Step 3: Truth Tables

• Construct truth tables to understand the truth values of propositions and their connectives.
• To create a truth table:
  1. List all possible combinations of truth values for the involved propositions.
  2. Apply the logical connectives to derive the resulting truth values.

Example Truth Table for P ∧ Q

| P | Q | P ∧ Q | |-------|-------|-------| | True | True | True | | True | False | False | | False | True | False | | False | False | False |

Step 4: Logical Equivalence

• Understand that two propositions are logically equivalent if they have the same truth values in all possible scenarios.
• Common equivalences include:
  • De Morgan's Laws:
    • ¬(P ∧ Q) is equivalent to ¬P ∨ ¬Q
    • ¬(P ∨ Q) is equivalent to ¬P ∧ ¬Q

Practical Application

• Use logical equivalences to simplify logical expressions in proofs or programming logic.

Conclusion

In this tutorial, we covered the basics of propositional logic, including the definition of propositions, logical connectives, truth tables, and logical equivalence. Mastering these concepts is crucial for further study in logic and its applications. Next steps could include practicing more complex logical expressions and exploring predicate logic for a deeper understanding of logical reasoning.