Elementary Logic

• Logic is a branch of science that studies correct forms of reasoning. It plays a key role in philosophy, mathematics, and computer science.
• Ancient Roots: Logic dates back over 2000 years.
• Greek philosophers, such as Aristotle (384–322 BCE), wrote about deduction and logical reasoning.
• Chinese thinkers also explored logical paradoxes during the same era.
• Aristotle is considered the father of formal logic. His work formed the foundation for centuries of logical study.
• Gottfried Wilhelm Leibniz (1646–1716) aimed to formalize logical arguments using a universal scientific language. While he did not succeed, his ideas inspired others.
• George Boole (1815–1864): Introduced Boolean Algebra through his works like The Mathematical Analysis of Logic and An Investigation of the Laws of Thought.
• Gottlob Frege (1848–1925): Created a symbolic logic system, linking logic to mathematics.
• Bertrand Russell and Alfred North Whitehead: Published Principia Mathematica, providing a modern account of logic and its mathematical foundation.
• Logic helps determine whether an argument is valid by using principles and methods to appraise reasoning.
• Logic simplifies verbal arguments into symbolic logic for analysis.
• Logic helps evaluate reasoning by identifying valid and invalid arguments.
• Logic roots go back to Aristotle, while mathematicians like Boole modernized it with symbolic and Boolean logic.
• A proposition is a statement that is either true or false, but not both. Propositions must express a complete thought.
• Types of Propositions:

  • Simple Proposition
  • Compound Proposition
• Compound Proposition: Combines two or more simple propositions using logical connectives like and, or, etc.
• Statement Variables: Lowercase letters (p, q, r, etc.) represent arbitrary propositions whose truth value is unspecified.
• Statement Constants: Uppercase letters (e.g., P, Q) represent specific statements with known truth values.
• Symbols in Propositions:

  • Statement Variables
  • Statement Constants
• Conjunction (AND, ∧):

  • Combines two propositions with and.
  • True only when both propositions are true.
• Disjunction (OR, ∨):

  • Combines two propositions with or.
  • True if at least one proposition is true.
• Negation (NOT, ∼):

  • Reverses the truth value of a proposition.
• Material Implication (IF-THEN, →):

  • A conditional statement, “If p, then q.”
  • False only when p is true, and q is false.
• Material Equivalence (IF AND ONLY IF, ↔):

  • A biconditional statement: “p if and only if q.”
  • True when p and q have the same truth value.
• A truth table is a way to determine the truth value of a compound proposition by listing all possible values of its variables. It helps analyze logical expressions systematically.
• In logic, a conditional statement (p→q \to q→q) can be rewritten in different ways:

  • Converse (q→p)
  • Inverse (∼p→∼q)
  • Contrapositive (∼q→∼p)
• The contrapositive is always logically equivalent to the original statement, while the converse and inverse are equivalent to each other but not necessarily to the original.
• Tautology

  • A proposition that is always true, regardless of the truth values of its components.
  • Since the final column is all T’s, this is a tautology.
• Contradiction

  • A proposition that is always false, no matter what.
  • Since the final column is all F’s, this is a contradiction.
• Contingency

  • A proposition that is sometimes true and sometimes false, depending on the values of its variables.
  • Since the final column has both T’s and F’s, this is a contingency.
• Simple Proposition: Conveys a single thought without any connecting words.
• Parentheses & Order of Operations

  • Grouping symbols ( , { , [ ) are used to avoid confusion.
  • Example:
  • English: "Both P and Q, or R."
  • Incorrect: P∧Q∨R
  • Correct: (P∧Q)∨R
• The contrapositive (∼q→∼p ) is always logically equivalent to the original conditional (p→q).
• The converse (q→p) and inverse (∼p→∼q) are logically equivalent to each other, but not necessarily to the original statement.
• The conditional (p→q) is false only when p is true and q is false.