Introduction to Logic: Propositional Logic & Truth Tables

Overview

• Introduction by Mark Doors B.
• Course covers basics of categorical propositions and predicate logic.
• Focus on propositional logic, specifically truth tables.

Key Topics

1. Truth Tables for Propositions

   • Definition and purpose of truth tables.
   • Building and analyzing truth tables for compound propositions.
   • Classifying and comparing propositional arguments.

2. Compound Statements

   • Every compound statement is bivalent (true or false).
   • Example: A and B therefore C.
   • Calculate truth value of entire proposition by main operator.
   • Address unknown truth values in arguments.

3. Truth Tables

   • Calculate all possible truth values for propositions.
   • Classify statements as tautological (always true), contradictory, or contingent (some true, some false).
   • Compare separate compound statements for consistency or contradiction.

Building Truth Tables

• Use the equation: L = 2^n (L = number of lines, n = number of propositional variables).
• Example: For A and B implication C, with 3 unique variables, table needs 8 lines (2^3 = 8).
• Process:
  • Write first variable values (half true, half false).
  • Divide by two for subsequent variables.
  • Ensure each variable follows the same pattern.

Truth Table Example

• Example proposition: C and not D therefore E.
• Identify unique variables and determine number of lines.
• Fill in truth table by comparing lines under main operators.
• Analyze which conditions make the statement true or false.

Classifying Statements

1. Tautology: All lines under main operator are true.
2. Contradiction: All lines under main operator are false.
3. Contingent: Mix of true and false values.

Comparing Statements

1. Logically Equivalent: Identical truth values for each line.
2. Contradictory: Opposite truth values on each line.
3. Consistent: At least one line where both statements are true.
4. Inconsistent: No line where both statements are true.

Examples and Exercises

• Practice creating truth tables and classifying statements using examples from the textbook.
• Compare compound statements to determine logical relationships (equivalence, contradiction, consistency).

Notes

• Use graph paper for clarity in manual calculations.
• Highlight or use different colors to mark main operators and key comparisons.
• Ensure consistency in labeling and variable order.

Upcoming Topics

• Truth tables for arguments.
• Efficiency shortcuts for analyzing complex propositions with many variables.

This summary captures the main points from the lecture on propositional logic and truth tables, focusing on building and analyzing truth tables, classifying statements, and comparing them for logical relationships.