Lecture on Truth Tables and Evaluating Statements

Introduction

• Building on last class's truth table operator definitions
• Focus on constructing truth tables and using them to evaluate statements
• Future focus: Using truth tables to evaluate arguments

Function of a Truth Table

• Covers all possible combinations of truth values in a statement
  • Simple statement (e.g., "Matthew likes ice cream"): two possibilities (true or false)
  • Compound statements add complexity with more truth combinations

Combinations Example

• "Matthew likes ice cream and pizza"
  • True + True
  • True + False
  • False + True
  • False + False
• Adding a third component (e.g., Cheetos) increases complexity further
  • Total combinations: 8 (2^3)

Formula for Combinations

• General formula: L = 2^N
  • L = number of rows
  • N = number of different letters

Defining Rows and Columns

• Row: Horizontal (left to right)
• Column: Vertical (up and down)

Example Scenarios

• Two letters: P and Q
  • Rows: 4 (2^2)
  • P provides 2 rows
  • Q doubles to 4 rows
• Three letters: P, Q, and R
  • Rows: 8 (2^3)
  • Example: P horseshoe Q, triple bar R

Common Sense Approach

• First letter: 2 rows
• Each additional letter doubles the number of rows

Creating the Key

• Read letters left to right: P, Q, R
• Double rows with each new letter: 2, 4, 8
• Example key setup: P, Q, R (8 rows)
• Procedure: True/False pattern
  • Divide column in half each time moving right (4, 2, 1)

Using the Key

• Plug the key into the expression to evaluate
• Solve the most inner parentheses first, then work outward
• Example evaluation:
  • Left side: P horseshoe R
  • Right side: Q dot R
  • Main operator: Wedge

Solving the Statement

• Step-by-step solving of each operator
  • P horseshoe R
  • Q dot R
  • Combine with main operator (wedge)

Interpretation of Results

• Main operator column determines the statement's logical status:
  • All true: Logically true or tautology
  • All false: Logically false or self-contradictory
  • Mixed true/false: Contingent
• Example result: Contingent (both true and false present)

Practice Scenario

• Example statement: "(P horseshoe Q) dot P horseshoe Q"
  • Determine number of rows
  • Set up key with truth values
  • Evaluate inner expressions and main operator
  • Result: Tautology (all true)

Next Steps

• Continue practicing statement evaluation
• Next class: Evaluating arguments using truth tables

Conclusion

• Focus on mastering keys and basic evaluation techniques today
• Prepare for argument evaluation in the next class