A Universal Truth Table

A Universal Truth Table

Introduction

In digital logic, understanding how different operations relate to each other can sometimes lead to surprising insights. This page explores the relationship between NAND and conditional statements through a series of thought experiments involving logical expressions.

Key Concepts

• NAND Gate: A basic digital logic gate that outputs false only when all its inputs are true.
• Conditional Statement: A statement that expresses a condition; typically denoted as A → B (if A, then B).

The Thought Experiment

1. If you examine the truth table for NAND:

   * A | B | A NAND B
   * 0 | 0 | 1
   * 0 | 1 | 1
   * 1 | 0 | 1
   * 1 | 1 | 0

2. Next, consider the truth table for a conditional statement (A → B):

   * A | B | A → B
   * 0 | 0 | 1
   * 0 | 1 | 1
   * 1 | 0 | 0
   * 1 | 1 | 1

3. Rewrite A → B as A → NOT B:

   * A | B | A → NOT B
   * 0 | 0 | 1
   * 0 | 1 | 1
   * 1 | 0 | 0
   * 1 | 1 | 0

Conclusion

Upon this exploration, it appears that the logic statement of "if A then NOT B" reflects the same truth table as NAND. This suggests that this conditional expression captures the essence of the NAND operation in terms of logical productivity.

Practical Application

Let:

• A = "cheese"
• B = "rainbow"

Then define C as follows:

C = ifnot(ifnot(ifnot(ifnot(ifnot(ifnot(ifnot(ifnot(A, B), ifnot(A, B)), ifnot(A, B)), ifnot(A, B)), ifnot(A,B)), ifnot(A,B), ifnot(A, B)), ifnot(A,B))

Reflection

This leads one to question the complexity and usability of logical operators, highlighting that certain expressions may become unnecessarily convoluted, reminiscent of programming languages like Brainfuck or Lisp.