HH

Lecture 3 - Propositional Logic Laws

Tautology and Contradiction

  • A tautology is a compound proposition that is always true.
  • A contradiction is a compound proposition that is always false.
  • Truth table basics:
    • If a proposition evaluates to true for all possible truth assignments, it is a tautology.
    • If a proposition evaluates to false for all possible truth assignments, it is a contradiction.
  • Example truth table (from transcript):
    • Columns: p, ¬p, p ∨ ¬p, p ∧ ¬p
    • Rows:
    • p = T, ¬p = F → p ∨ ¬p = T, p ∧ ¬p = F
    • p = F, ¬p = T → p ∨ ¬p = T, p ∧ ¬p = F
    • Conclusion: p o
      eg p is not the focus here; rather, p \u2228
      eg p is always true, and p \u2227
      eg p is always false.
  • Summary relationships:
    • p \lor \neg p \equiv \top (Law of excluded middle in this context)
    • p \land \neg p \equiv \bot

Truth-table-based example: (p → q) ∨ p

  • We can determine tautology/contingency by evaluating the expression under all combinations of p and q.
  • Expression: (p \rightarrow q) \lor p
  • Using the truth table:
    • p = T, q = T: p → q = T; (p → q) ∨ p = T
    • p = T, q = F: p → q = F; (p → q) ∨ p = T
    • p = F, q = T: p → q = T; (p → q) ∨ p = T
    • p = F, q = F: p → q = T; (p → q) ∨ p = T
  • Since the value is always true, the expression is a tautology.

Logical Equivalences

  • Two compound propositions p and q are logically equivalent if they always have the same truth value.
  • Denoted as: p \equiv q or sometimes p \Leftrightarrow q.
  • Significance:
    • Enables replacing a subformula with an equivalent one to simplify expressions or prove properties (tautology/contradiction/equivalence).
  • Example from transcript:
    • Prove that \boldsymbol{\diameter} p \lor q (interpreted as a certain expression in transcript) is equivalent to p \to q by comparing truth tables across all assignments.
    • Truth table shows: for all assignments, the two expressions have identical truth values, so they are logically equivalent.

Example: Prove the equivalence Øp Ú q ≡ p → q

  • Given truth table (p, q, ¬p, ¬p ∨ q, p → q):
    • p q | ¬p | ¬p ∨ q | p → q
    • T T | F | T | T
    • T F | F | F | F
    • F T | T | T | T
    • F F | T | T | T
  • Observation: The column for ¬p ∨ q matches p → q in every row.
  • Conclusion: Øp Ú q is logically equivalent to p → q (as per transcript's notation and truth table).
  • Note: In the transcript, "Øp Ú q" is presented in a way that aligns with ¬p ∨ q; the key takeaway is that the two expressions share identical truth values across all possible p, q.

Laws of Propositional Logic

  • Purpose: Provide a toolkit to simplify propositions, prove logical equivalence, tautology, or contradiction.
  • Common laws (Expression 1 ↔ Expression 2):
    • Idempotent laws:
    • p \lor p \equiv p
    • p \land p \equiv p
    • Associative laws:
    • (p \lor q) \lor r \equiv p \lor (q \lor r)
    • (p \land q) \land r \equiv p \land (q \land r)
    • Commutative laws:
    • p \lor q \equiv q \lor p
    • p \land q \equiv q \land p
    • Distributive laws:
    • p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)
    • p \land (q \lor r) \equiv (p \land q) \lor (p \land r)
    • Identity laws:
    • p \lor \bot \equiv p
    • p \land \top \equiv p
    • Domination laws:
    • p \lor \top \equiv \top
    • p \land \bot \equiv \bot
    • Double negation law:
    • \neg\neg p \equiv p
    • Complement laws (the principle of excluded middle and contradiction):
    • p \lor \neg p \equiv \top
    • p \land \neg p \equiv \bot
    • De Morgan’s laws:
    • \neg(p \lor q) \equiv \neg p \land \neg q
    • \neg(p \land q) \equiv \neg p \lor \neg q
    • Absorption laws:
    • p \lor (p \land q) \equiv p
    • p \land (p \lor q) \equiv p
    • Conditional identities:
    • p \rightarrow q \equiv \neg p \lor q
    • Biconditional: (optional from transcript discussion)
    • p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p)

How to use the laws

  • Purpose: Use the laws to simplify a compound proposition, and to prove logical equivalence, tautology, or contradiction.
  • Rules of thumb:
    • Remove a biconditional (↔) by applying the conditional identity: p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p) and/or transform implications.
    • Remove implications (→) by using the identity: p \rightarrow q \equiv \neg p \lor q .
    • Apply De Morgan’s laws to move a negation inside: \neg(p \lor q) \equiv \neg p \land \neg q and \neg(p \land q) \equiv \neg p \lor \neg q .
    • Apply other laws (idempotent, associative, commutative, distributive, etc.) as needed to reach the desired form (e.g., a tautology, contradiction, or a simpler equivalent expression).

Example: Prove p ∨ (p → q) is a tautology

  • Target: Prove p \lor (p \rightarrow q) is a tautology.
  • Steps:
    • Use conditional identity: p \rightarrow q \equiv \neg p \lor q
    • Then p \lor (\neg p \lor q)
    • By associativity: (p \lor \neg p) \lor q
    • By complement law: \top \lor q
    • By domination law: \top
  • Conclusion: p \lor (p \rightarrow q) is a tautology.

Example: Prove ¬p → (q → r) ≡ q → (p ∨ r)

  • Target: Show \neg p \rightarrow (q \rightarrow r) \equiv q \rightarrow (p \lor r)
  • Steps (as in transcript):
    • Start with left: \neg p \rightarrow (q \rightarrow r)
    • Apply conditional identity: \neg p \rightarrow (q \rightarrow r) \equiv p \lor (q \rightarrow r)
    • Replace inner implication: p \lor (\neg q \lor r)
    • Reassess with associativity/commutativity to form (\neg q \lor p) \lor r or equivalently \neg q \lor (p \lor r)
    • Recognize that q \rightarrow (p \lor r) \equiv \neg q \lor (p \lor r)
  • Therefore: \neg p \rightarrow (q \rightarrow r) \equiv q \rightarrow (p \lor r)
  • Note: This derivation aligns with the transcript’s final equivalence and uses standard transformation steps.

Connections to foundational principles

  • These topics underpin boolean algebra, digital logic design, and automated theorem proving.
  • Logical equivalence, tautology, and contradiction form the basis for simplifying expressions and for proving properties about logical systems.
  • The laws provide a finite and systematic toolkit to transform and reason about propositional formulas.

Practical implications and context

  • In computer science, these concepts underpin circuit design, compiler optimizations, and verification; proving a circuit is a tautology corresponds to showing it always outputs true.
  • In mathematics, logical equivalences enable proving theorems via rewrite rules.
  • In philosophy, these ideas relate to formal epistemology and the study of valid arguments.

Quick reference of key formulas (LaTeX)

  • Tautology/Contradiction:
    • p \lor \neg p \equiv \top
    • p \land \neg p \equiv \bot
  • Implication:
    • p \rightarrow q \equiv \neg p \lor q
  • De Morgan:
    • \neg(p \lor q) \equiv \neg p \land \neg q
    • \neg(p \land q) \equiv \neg p \lor \neg q
  • Identity/Domination:
    • p \lor \bot \equiv p
    • p \land \top \equiv p
    • p \lor \top \equiv \top
    • p \land \bot \equiv \bot
  • Absorption:
    • p \lor (p \land q) \equiv p
    • p \land (p \lor q) \equiv p
  • Equivalences:
    • p \equiv q (logical equivalence)
    • p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p)
  • Example results:
    • p \lor (p \rightarrow q) \equiv \top
    • \neg p \rightarrow (q \rightarrow r) \equiv q \rightarrow (p \lor r)