# inverting ji'i

Mentioned on existential requantification:

See the Book, chapter 18.

*su'o da* means *naku no da* and *no da* means *naku su'o da*; if you don't have at least one then you have zero, and if you don't have zero then you have at least one. We can say that *su'o* inverts to *no* and vice versa. Knowing how to invert quantifiers is handy if you want to baffle your critics by precision-dropping *naku* into the middle of your bridi.

Other numbers used as existential quantifiers can be inverted too. *mu* inverts to *vei ga me'i mu gi za'u mu ve'o*, or "(less than 5 or greater than 5)", meaning "not five". Maybe there's a better way to say this--anybody know one? *su'e mu* (at most five) inverts as *za'u mu* (more than five). You get the idea.

But I don't see any way to invert *ji'i mu* that's convincingly symmetrical. Suppose you claim it's *vei ga me'i ji'i mu gi za'u ji'i mu ve'o*--would you say that "not around five" means "either more than around five or less than around five"? And vice versa? Maybe, but you'd rather have a number that means *not that close to five*, and I don't see one.

Other vague numbers have the same problem; for example, *so'i* and *no'o* (depending on what it means in context).

*mi'e jezrax*

OK, let's say that ji'imu defines a probability function shaped like a bell curve centered at 5. Call that function **j(n)**. Now, the inversion of **j**, call it **i(n)**, is nothing other than **i(n) = 1 - j(n)**. Perhaps you knew this and were asking for something else? --la xod

Are you sure it's so simple? Say j(9)=0.05 and j(10)=0.03, then using your formula, we get i(9)=0.95 and i(10)=0.97, and the probability of 9 or 10 is 1.92, which seems a bit high. The sum for all n of j(n) has to be 1. The sum of i(n) should also be 1, but you formula gives infinity. --xorxes

*If we normalize the inversion, i(n) is infinitesimal for all n except "near" 5, where it dips even lower. --la xod*

I think xod's solution is sort of OK if we consider **j(n)** to be a fuzzy truth-value function rather than a probability distribution. The problem is that *ji'imu* is vaguer than **j** ; the function contains strictly more information than the Lojban quantifier. In a given context people could come up with many fuzzy truth-functions which they might all agree are reasonable interpretations of *ji'imu*--or might disagree about. In any case, inventing an elaborate way to invert *ji'imu* certainly won't happen in usual practice; people who are being precise will not move *naku* freely about the bridi when there's such a tricky point, and people who are not being precise won't care! *mi'e jezrax*