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Symbolic Logic Overview

Sep 30, 2025

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Overview

This lecture introduces symbolic (propositional) logic, its benefits, the shift from categorical logic, key logical operators, and how to symbolize and evaluate statements using truth tables.

Why Study Symbolic Logic?

  • Symbolic logic helps understand the foundations of computers and mathematics.
  • It trains the brain in logical thinking and ingrains logical habits.
  • Symbolization allows quick, accurate assessment of argument validity.
  • Modern logic enables handling entire propositions as variables, not just categories or terms.

Categorical vs. Propositional Logic

  • Categorical logic uses terms and categories (e.g., "All S are P").
  • Propositional logic uses symbols to represent whole statements (e.g., D = "All dinosaurs are funny").
  • Only statements making clear truth claims can be symbolized; commands or questions cannot.

Symbolizing Statements

  • Assign simple letters (P, Q, S, etc.) to positive statements.
  • Use "not" (ÂŹ, ~, or a tilde) to negate statements rather than making negatives primary.
  • Compound statements use operators to combine propositions (e.g., S ∧ P, C ∨ S).

Logical Operators and Symbols

  • Negation (ÂŹ/ ~ / tilde): "Not" or "It is not the case that..."
  • Conjunction (¡ or ∧): "And"
  • Disjunction (∨): "Or" (inclusive)
  • Conditional (⊃ or →): "If...then..." (material implication)
  • Biconditional (≡ or ↔): "If and only if"

Main Operator and Parentheses

  • The main operator governs the largest part of the statement and determines its overall truth value.
  • Parentheses clarify which operators apply to which parts of a compound statement.

Truth Tables Basics

  • Negation reverses the truth value (if P is true, ÂŹP is false, and vice versa).
  • Conjunction is true only if both components are true.
  • Disjunction is true if at least one component is true.
  • Conditional is false only if the antecedent is true and the consequent is false; otherwise true.
  • Biconditional is true when both components share the same truth value.

Key Terms & Definitions

  • Proposition — A statement that is either true or false.
  • Negation — Logical operation turning a statement into its opposite truth value.
  • Conjunction — Connects two propositions with "and"; true only if both are true.
  • Disjunction — Connects with "or"; true if at least one is true.
  • Conditional (Implication) — "If...then..." statement; false only if the first is true and the second false.
  • Biconditional (Equivalence) — "If and only if"; true when both propositions share the same truth value.
  • Main Operator — The operator that covers the entire compound statement.

Action Items / Next Steps

  • Practice translating English statements into symbolic form using the five logical operators.
  • Review and memorize the truth tables for each logical operator.
  • Prepare for the next video on well-formed formulas and more practice with compound statements.

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