Do you think "twice as expensive" is meaningless? Because by that same logic, if I start with "once as expensive" that's not a number and multiplying by two gives me a meaningless concept.

Once as expensive is a number, it's 1 x the cost. Twice as expensive is 2x the cost. Once as cheap might arguably be 1x the cost, but twice as cheap is undefined.

> Once as expensive is a number, it's 1 x the cost.

Nobody says that. If someone said "once as expensive" I'd interpret "once" as talking about time.

"Once" can be a counter but not a multiple.

It seems like we're taking at cross purposes. I'm not trying to convince you that anyone says "once as X". I'm just saying "twice as cheap" is nonsensical. Because it's a multiple of something undefined.