Why does ISA includes instruction for logical operation?

I'm a junior student in Electronic Engineering.

Recently, I learned about Gödel's incompleteness theorem. One of the concepts related to this theorem is Gödel numbering, which shows that every logical operation can be converted into an arithmetic operation.

This leads me to think that an Instruction Set Architecture (ISA) might not need to include specific logical operation instructions, as these operations could theoretically be performed using arithmetic operations instead.

I assume that the reason for including direct logical operation instructions is that they are much faster than performing logical operations through arithmetic operations, but I'm not entirely sure.

• The reason for including anything in an instruction set is performance. mov is Turing-complete. Commented Jun 5 at 15:40
• I do not even understand a digital circuit which performs [an] arithmetic operation without being a subset of a logical operation - how can a circuit be part of an operation? How does this answer Why include both if one can be expressed in terms of the other?? Commented Jun 23 at 5:55