Formal Logic for Beginners

Introduction

This tutorial provides a foundational understanding of formal logic, addressing key concepts and operators essential for beginners. It draws on insights from Professor Hansen's video response to Logic for Kids, clarifying distinctions between material and formal logic, and introducing logical operators and their applications.

Step 1: Understand the Basics of Logic

• Two Aspects of Reality:
  • Material Aspect: Perceived through senses (physical objects).
  • Formal Aspect: Related to ideas and relationships between objects.
• Types of Logic:
  • Material logic deals with the material world (e.g., abduction, adduction, induction).
  • Formal logic focuses on deductions and logical necessity.

Step 2: Grasp Logical Values

• Two Logical Values:
  • True and False, represented by symbols (up arrow for true, down arrow for false).
• Logical Operators:
  • Not (¬): Negates a statement.
    • Syntax: ¬True = False and ¬False = True.
  • Or (∨): Requires at least one true component.
    • Combinations:
      • True ∨ True = True
      • True ∨ False = True
      • False ∨ True = True
      • False ∨ False = False
  • And (∧): Requires both components to be true.
    • Combinations:
      • True ∧ True = True
      • True ∧ False = False
      • False ∧ True = False
      • False ∧ False = False

Step 3: Explore Implication and Equivalence

• Implication (→): Represents conditional relationships.
  • Combinations:
    • True → True = True
    • False → True = True
    • False → False = True
    • True → False = False
• Equivalence (↔): Indicates that two statements are logically equivalent.
  • Only true if both statements have the same logical value.

Step 4: Apply the Axioms of Logic

• Axioms of Commutation: The order of propositions does not affect the truth.
  • Example: P ∨ Q = Q ∨ P and P ∧ Q = Q ∧ P.
• Axioms of Association: Grouping of propositions does not affect truth.
  • Example: (P ∨ Q) ∨ R = P ∨ (Q ∨ R).
• Distributive Law: Allows for distributing operators over propositions.
  • Example: P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R).

Step 5: Practice Logical Proofs

• Use the rules and axioms to develop logical proofs.
• Example Proofs:
  • Tautology: Prove that P ∨ ¬P is always true.
  • Disjunctive Addition: If P is true, then P ∨ Q is true.

Step 6: Understand the Laws of Thought

• Law of Identity: A proposition is always equivalent to itself (P ↔ P).
• Law of Non-Contradiction: A statement cannot be both true and false at the same time.
• Law of Excluded Middle: A statement is either true or not true.

Conclusion

This tutorial outlines the key concepts of formal logic, including logical values, operators, and foundational axioms. Understanding these principles is essential for engaging in logical reasoning and constructing valid arguments. For further study, consider practicing logical proofs and exploring more complex logical systems.