> This is what they're doing!! They're calling pre-orders "weak total orders". No. Giving one thing two names is not an incompatibility. Giving two things on

> This is what they're doing!! They're calling pre-orders "weak total orders". No. Giving one thing two names is not an incompatibility. Giving two things one name is an incompatibility. > "weak total order" already means "total order" Does it? I've never heard it called that. I think anyone on hearing the term "weak total order" would reasonably infer it refers to some notion than a total order. > "Weak" and "total" are both adjectives, and they're interchangeable. No. This is completely wrong. Terminology is very frequently not compositional in such a simple way. I mean, this works perfectly fine with words that convey additional conditions. It does not work with words that convey a removal of conditions, which is what the word "weak" does! Like, in the phrase "weak order" (meaning total preorder), the word "weak" is not imposing the conditions of reflexivity, transitivity, etc. It is starting from a baseline of "order" meaning "total order", and then weakening this to a preorder. The word "weak" only weakens! Anyone, on seeing the phrase "weak total order", assuming they know the general usage of the word "weak", can reasonably infer that it refers to something weaker than a total order. (Likely a preorder.) Yes, this makes "weak total order" something of a ridiculous phrase, given existing terminology. But again, remember that "weak order" is starting from a baseline where "order" means "total order"; in a sense, "weak order" is really something of an abbreviation for "weak total order". (...of course, all of this highlights the odd way that terminology for orders themselves work. After all, "partial" is also a weakener. I haven't studied the history, but I'd bet you that "order" originally meant "total order" (it's still often used that way), then "partial order" came later as a weakener, then the meaning of "order" shifted as the study of partial orders became more common, then "total order" was coined as a retronym.) > If you want to make a comparison, what they're doing would be like using "a nonnegative positive solution" to mean "a nonnegative solution" (rather than "a positive nonnegative solution"). Or using "an fractional real number" to mean "a fractional complex number" rather than a "real fractional number" (i.e. real fraction). Which would be completely nuts. All of these are examples where you're using two terms that both impose conditions, rather than removing them. The word "weak" simply does not work that way.