Symbolic Logic

Fundamentals of Symbolic Logic

• Proposition: A statement that can be classified as either true (T) or false (F). Commands, questions, and algebraic expressions with unknown variables are not propositions.
• Logical Operations:
  • Conjunction (p ∧ q): True only if both p and q are true.
  • Disjunction (p ∨ q): An inclusive "or"; true if p is true, q is true, or both are true. False only if both are false.
  • Negation (~p): The opposite truth value of p.
  • Conditional (p → q): "If p, then q." False only when p is true and q is false. p is the hypothesis; q is the conclusion.
  • Biconditional (p ↔ q): "p if and only if q." True only when p and q have the same truth value.
• Conditional Variations:
  • Converse (q → p): Switches the hypothesis and conclusion.
  • Inverse (~p → ~q): Negates both the hypothesis and conclusion.
  • Contrapositive (~q → ~p): Switches and negates both. It is logically equivalent to the original conditional statement.
• Truth Classifications:
  • Tautology: A statement that is always true regardless of the circumstances.
  • Contradiction: A statement that is always false.
  • Contingency: A statement whose truth value depends on the situation.