Chapter 16: “Who Did You Pass On The Road? Nobody”: Lojban And Logic
5. Dropping the prenex
It isn't really necessary for every Lojban bridi involving variables to have a prenex on the front. In fact, none of the examples we've seen so far required prenexes at all! The rule for dropping the prenex is simple: if the variables appear in the same order within the bridi as they did in the prenex, then the prenex is superfluous. However, any “ro” or “poi” appearing in the prenex must be transferred to the first occurrence of the variable in the main part of the bridi. Thus, ✥2.3 becomes just:
✥5.1 da viska mi There-is-an-X-which sees me. Something sees me.
and ✥4.4 becomes:
✥5.2 ro da poi gerku cu vasxu For-every X which is-a-dog, it-breathes. Every dog breathes.
You might well suppose, then, that the purpose of the prenex is to allow the variables in it to appear in a different order than the bridi order, and that would be correct. Consider
✥5.3 ro da poi prenu ku'o de poi gerku ku'o zo'u de batci da For-every X which is-a-person, there-is-a-Y which is-a-dog: Y bites X.
The prenex of ✥5.3 is like that of ✥3.4 (but with relative clauses): it notes that the following bridi is true of every person with respect to some dog, not necessarily the same dog for each. But in the main bridi part, the “de” appears before the “da”. Therefore, the true translation is
✥5.4 Every person is bitten by some dog (or other).
If we tried to omit the prenex and move the “ro” and the relative clauses into the main bridi, we would get:
✥5.5 de poi gerku cu batci ro da poi prenu There-is-a-Y which is-a-dog which-bites every Y which is-a-person Some dog bites everyone.
which has the structure of ✥3.5: it says that there is a dog (call him Fido) who bites, has bitten, or will bite every person that has ever existed! We can safely rule out Fido's existence, and say that ✥5.5 is false, while agreeing to ✥5.3.
Even so, ✥5.3 is most probably false, since some people never experience dogbite. Examples like 5.3 and 4.4 (might there be some dogs which never have breathed, because they died as embryos?) indicate the danger in Lojban of universal claims even when restricted. In English we are prone to say that “Everyone says” or that “Everybody does” or that “Everything is” when in fact there are obvious counterexamples which we are ignoring for the sake of making a rhetorical point. Such statements are plain falsehoods in Lojban, unless saved by a context (such as tense) which implicitly restricts them.
How can we express ✥5.3 in Lojban without a prenex? Since it is the order in which variables appear that matters, we can say:
✥5.6 ro da poi prenu cu se batci de poi gerku Every-X which is-a-person is-bitten-by some-Y which is-a-dog.
using the conversion operator “se” (explained in Chapter 5) to change the selbri “batci” (“bites”) into “se batci” (“is bitten by”). The translation given in ✥5.4 uses the corresponding strategy in English, since English does not have prenexes (except in strained “logician's English”). This implies that a sentence with both a universal and an existential variable can't be freely converted with “se”; one must be careful to preserve the order of the variables.
If a variable occurs more than once, then any “ro” or “poi” decorations are moved only to the first occurrence of the variable when the prenex is dropped. For example,
✥5.7 di poi prenu zo'u ti xarci di di There-is-a-Z which is-a-person : this-thing is-a-weapon for-use-against-Z by-Z This is a weapon for someone to use against himself/herself.
(in which “di” is used rather than “da” just for variety) loses its prenex as follows:
✥5.8 ti xarci di poi prenu ku'o di This-thing is-a-weapon-for-use-against some-Z which is-a-person by-Z.
As the examples in this section show, dropping the prenex makes for terseness of expression often even greater than that of English (Lojban is meant to be an unambiguous language, not necessarily a terse or verbose one), provided the rules are observed.