The reason you teach algebra is because algebra teaches symbolic manipulation
The tool you should be using is symbolic and philosophical logic
How do you propose to teach someone symbolic logic without teaching them what symbols are, and how to manipulate them?
The reason is not algebra's application to daily life. The reason you teach algebra is because algebra teaches symbolic manipulation. Learning math teaches you not just how to add two numbers. Addition is almost unnecessary in daily life (we do have calculators). Learning algebra is critical because it teaches us to think in terms of abstractions, of models. We do not teach mathematics to teach you how to add, we teach mathematics to teach you how to solve, to teach you how to think.
What hath Bob wrought?