Chapter 16: “Who Did You Pass On The Road? Nobody”: Lojban And Logic
8. The problem of “any”
Consider the English sentence
✥8.1 Anyone who goes to the store, walks across the field.
Using the facilities already discussed, a plausible translation might be
✥8.2 ro da poi klama le zarci cu cadzu le foldi All X such-that-it goes-to the store walks-on the field. Everyone who goes to the store walks across the field.
But there is a subtle difference between ✥8.1 and ✥8.2. ✥8.2 tells us that, in fact, there are people who go to the store, and that they walk across the field. A sumti of the type “ro da poi klama” requires that there are things which “klama”: Lojban universal claims always imply the corresponding existential claims as well. ✥8.1, on the other hand, does not require that there are any people who go to the store: it simply states, conditionally, that if there is anyone who goes to the store, he or she walks across the field as well. This conditional form mirrors the true Lojban translation of ✥8.1:
✥8.3 ro da zo'u ganai da klama le zarci gi cadzu le foldi For-every X: if X is-a-goer-to the store then X is-a-walker-on the field.
Although ✥8.3 is a universal claim as well, its universality only implies that there are objects of some sort or another in the universe of discourse. Because the claim is conditional, nothing is implied about the existence of goers-to-the-store or of walkers-on-the-field, merely that any entity which is one is also the other.
There is another use of “any” in English that is not universal but existential. Consider
✥8.4 I need any box that is bigger than this one.
✥8.4 does not at all mean that I need every box bigger than this one, for indeed I do not; I require only one box. But the naive translation
✥8.5 mi nitcu da poi tanxe gi'e bramau ti I need some-X which is-a-box and is-bigger-than this-one
does not work either, because it asserts that there really is such a box, as the prenex paraphrase demonstrates:
✥8.6 da poi tanxe gi'e bramau ti zo'u mi nitcu da There-is-an-X which is-a-box and is-bigger-than this : I need X.
What to do? Well, the x2 place of “nitcu” can be filled with an event as well as an object, and in fact ✥8.5 can also be paraphrased as:
✥8.7 mi nitcu lo nu mi ponse lo tanxe poi bramau ti I need an event-of I possess some box(es) which-are bigger-than this-one.
Rewritten using variables, ✥8.7 becomes
✥8.8 mi nitcu lo nu da zo'u da se ponse mi gi'e tanxe gi'e bramau ti I need an event-of there-being an-X such-that X is-possessed-by me and is-a-box and is-bigger-than this-thing.
So we see that a prenex can be attached to a bridi that is within a sentence. By default, a variable always behaves as if it is bound in the prenex which (notionally) is attached to the smallest enclosing bridi, and its scope does not extend beyond that bridi. However, the variable may be placed in an outer prenex explicitly:
✥8.9 da poi tanxe gi'e bramau ti zo'u mi nitcu le nu mi ponse da There-is-an-X which is-a-box and is-bigger-than this-one such-that I need the event-of my possessing X.
But what are the implications of ✥8.7 and ✥8.9? The main difference is that in ✥8.9, the “da” is said to exist in the real world of the outer bridi; but in ✥8.7, the existence is only within the inner bridi, which is a mere event that need not necessarily come to pass. So ✥8.9 means
✥8.10 There's a box, bigger than this one, that I need
which is what ✥8.6 says, whereas ✥8.7 turns out to be an effective translation of our original ✥8.1. So uses of “any” that aren't universal end up being reflected by variables bound in the prenex of a subordinate bridi.