CSCE 235 Logic - In Depth Notes

Introduction to Logic

  • Logic is the study of the relationships between objects.
  • Provides the foundation for mathematical reasoning and automated reasoning.

Overview of Propositional Logic (PL)

  • Propositional Logic (PL), also known as Propositional Calculus or Sentential Logic.
  • Definition of a Proposition: A statement that can either be true or false, but not both.
  • Represented by letters such as p, q, r, s, etc.

Key Components of Propositional Logic

  • Outline:

    • Defining Propositional Logic (Propositions, Connectives, Precedence of Operators, Truth Tables).
    • Usefulness of Logic (Bitwise operations, Logic in Theoretical Computer Science, Logic in Programming).
    • Logical Equivalences (Terminology, Truth tables, Equivalence rules).

  • Truth Value:

    • Denoted as T (true) or F (false).
    • Truth values signify whether propositions are true or false.

Propositions Examples

  • Examples of Propositions:

    • "Today is Monday" (M)
    • "The grass is wet" (W)
    • "It is raining" (R)

  • Non-Propositions:

    • "C++ is the best language" (Opinion).
    • "When is the pretest?" (Interrogative).
    • "Do your homework" (Imperative).

Logical Connectives

  • Connectives are used to form compound propositions:

    • Negation (¬): Indicates the opposite of a proposition.
    • AND (∧): True only if both propositions are true.
    • OR (∨): True if at least one proposition is true.
    • Exclusive OR (⊕): True if exactly one proposition is true.
    • Implication (→): True unless a true proposition implies a false one.
    • Biconditional (↔): True when both propositions share the same truth value.

  • Precedence of Logical Operators:

    1. Negation (¬)
    2. Conjunction (∧)
    3. Disjunction (∨)
    4. Implication (→)
    5. Biconditional (↔)

Truth Tables

  • Truth Tables demonstrate the relationships between the truth values of propositions:
    • Example:
    • p | q | p ∧ q | p ∨ q | p ⊕ q | p → q | p ↔ q
    • 0 | 0 | 0 | 0 | 0 | 1 | 1
    • 0 | 1 | 0 | 1 | 1 | 1 | 0
    • 1 | 0 | 0 | 1 | 1 | 0 | 0
    • 1 | 1 | 1 | 1 | 0 | 1 | 1

Useful Applications of Logic

  • Logic is used in:
    • Digital logic design for hardware.
    • Enabling precise communication compared to natural language.
    • Specification and verification in software engineering.

Bitwise Operations

  • Computers use binary digits (bits) for information.
  • Bitwise logical operations (AND, OR, XOR) can be applied on bit strings of equal length.

Logic in Theoretical Computer Science (TCS)

  • SAT (Satisfiability Problem): Determines if a propositional logic sentence is satisfiable.
    • A sentence in PL is satisfiable if there exists a truth assignment that makes it true.

Logical Equivalences and their Importance

  • Logical equivalences allow for substitution of statements that share truth values.

  • Key concepts include Tautology (always true), Contradiction (always false), and Contingency (neither).

  • Definitions:

    • Tautology: p ∨ ¬p
    • Contradiction: p ∧ ¬p

Examples of Logical Equivalences

  • To prove (p → q) is equivalent to (¬p ∨ q), construct truth tables and find identical columns.

Conclusion

  • Logic provides a framework essential for reasoning, computation, programming, and theoretical foundations in computer science. Understanding its principles enhances clarity and correctness in mathematical and logical argumentation.