Introduction to Logic: Indirect Truth Tables

Overview

• Presenter: Mark Doors B
• Subject: Basics of categorical propositions, predicate logic, and propositional logic.
• Goal: Discuss how to build indirect truth tables in propositional logic.

Truth Tables

• Building truth tables for arguments can be time-consuming with many variables.
• Number of lines in a truth table: (2^n) where (n) is the number of unique variables.
  • Example: 10 variables require over 1000 lines.

Propositional Logic and Complexity

• Ordinary language arguments often have many propositions.
• Human capacity to analyze multiple propositions is impressive but becomes challenging with many variables.
• Higher number of propositions increases likelihood of invalid arguments.

Indirect Truth Table Method

• Introduced as a quicker method than traditional truth tables.
• Steps for Indirect Truth Tables:
  1. Assume Invalidity: Assume the argument is invalid.
  2. Enter Assumption: Enter the assumption into the truth table.
     • Premises must be true; conclusion must be false.
  3. Work Backwards: Check for contradictions by working backwards through the argument.
     • Enter values consistently across equations.
  4. Contradiction Check: Finding a contradiction means the argument is valid. No contradiction means it's invalid.

Example Problems

• Problem 1:
  • Argument: (\neg A \rightarrow (B \lor C), \neg B, \therefore C, C \rightarrow A)
  • Analyze using indirect method to see if assumptions hold without contradictions.
• Problem 2:
  • Argument: (A \rightarrow B, C \rightarrow B, \therefore A \rightarrow C)
  • Analyze using indirect method to explore consistency and identify potential contradictions.

Consistency Check with Indirect Tables

• Used to test if a set of statements are consistent.
• No conclusion involved, just testing consistency among statements.
• Assume all statements are true, and check if any logical contradictions arise.
• Steps: Similar to indirect method for arguments.

Validity and Consistency

• Contradiction discovered: Argument is valid.
• No contradiction: Argument is invalid.
• Consistent statements show no contradictions.

Final Thoughts

• Indirect truth tables provide a faster alternative to direct truth tables.
• Practice increases familiarity and efficiency.
• Upcoming lectures will focus on natural deduction and proving validity.

---

Additional Notes

• Consider practical applications such as analyzing multiple witness statements in detective work.
• Exercises involve deriving truth values and determining argument validity through indirect methods.