In addition to the non-logical connectives of selma'o JOI explained in Sections 14 and 15, there are three other connectives which can appear in joiks: “bi'i”, “bi'o”, and “mi'i”, all of selma'o BIhI. The first two cmavo are used to specify intervals: abstract objects defined by two endpoints. The cmavo “bi'i” is correct if the endpoints are independent of order, whereas “bi'o” or “sebi'o” are used when order matters.
An example of “bi'i” in sumti connection:
16.1) mi ca sanli la drezdn. bi'i la frankfurt.
I [present] stand-on-surface Dresden [interval] Frankfurt.
I am standing between Dresden and Frankfurt.
In Example 16.1, it is all the same whether I am standing between Dresden and Frankfurt or between Frankfurt and Dresden, so “bi'i” is the appropriate interval connective. The sumti “la drezdn. bi'i la frankfurt.” falls into the x2 place of “sanli”, which is the surface I stand on; the interval specifies that surface by its limits. (Obviously, I am not standing on the whole of the interval; the x2 place of “sanli” specifies a surface which is typically larger in extent than just the size of the stander’s feet.)
16.2) mi cadzu ca la pacac. bi'o la recac.
I walk simultaneous-with First-hour [ordered-interval] Second-hour.
I walk from one o’clock to two o’clock.
In Example 16.2, on the other hand, it is essential that “la pacac.” comes before “la recac.”; otherwise we have an 11-hour (or 23-hour) interval rather than a one-hour interval. In this use of an interval, the whole interval is probably intended, or at least most of it.
Example 16.2 is equivalent to:
16.3) mi cadzu ca la recac. sebi'o la pacac.
I walk simultaneous-with Second-hour [reverse] [ordered] First-hour.
English cannot readily express “sebi'o”, but its meaning can be understood by reversing the two sumti.
The third cmavo of selma'o BIhI, namely “mi'i”, expresses an interval seen from a different viewpoint: not a pair of endpoints, but a center point and a distance. For example:
16.4) le jbama pu daspo la .uacintyn. mi'i lo minli be li muno
The bomb [past] destroys Washington [center] what-is measured-in-miles by 50.
The bomb destroyed Washington and fifty miles around.
Here we have an interval whose center is Washington and whose distance, or radius, is fifty miles.
In Example 16.1, is it possible that I am standing in Dresden (or Frankfurt) itself? Yes. The connectives of selma'o BIhI are ambiguous about whether the endpoints themselves are included in or excluded from the interval. Two auxiliary cmavo “ga'o” and “ke'i” (of cmavo GAhO) are used to indicate the status of the endpoints: “ga'o” means that the endpoint is included, “ke'i” that it is excluded:
16.5) mi ca sanli la drezdn. ga'o bi'i ga'o la frankfurt.
I [present] stand Dresden [inclusive] [interval] [inclusive] Frankfurt.
I am standing between Dresden and Frankfurt, inclusive of both.
16.6) mi ca sanli la drezdn. ga'o bi'i ke'i la frankfurt.
I [present] stand Dresden [inclusive] [interval] [exclusive] Frankfurt.
I am standing between Dresden (inclusive) and Frankfurt (exclusive).
16.7) mi ca sanli la drezdn. ke'i bi'i ga'o la frankfurt.
I [present] stand Dresden [exclusive] [interval] [inclusive] Frankfurt.
I am standing between Dresden (exclusive) and Frankfurt (inclusive).
16.8) mi ca sanli la drezdn. ke'i bi'i ke'i la frankfurt.
I [present] stand Dresden [exclusive] [interval] [exclusive] Frankfurt.
I am standing between Dresden and Frankfurt, exclusive of both.
As these examples should make clear, the GAhO cmavo that applies to a given endpoint is the one that stands physically adjacent to it: the left-hand endpoint is referred to by the first GAhO, and the right-hand endpoint by the second GAhO. It is ungrammatical to have just one GAhO.
(Etymologically, “ga'o” is derived from “ganlo”, which means “closed”, and “ke'i” from “kalri”, which means “open”. In mathematics, inclusive intervals are referred to as closed intervals, and exclusive intervals as open ones.)
BIhI joiks are grammatical anywhere that other joiks are, including in tanru connection and (as ijoiks) between sentences. No meanings have been found for these uses.
Negated intervals, marked with a “-nai” following the BIhI cmavo, indicate an interval that includes everything but what is between the endpoints (with respect to some understood scale):
16.9) do dicra .e'a mi ca la daucac. bi'onai la gaicac.
You disturb (allowed) me at 10 not-from ... to 12
You can contact me except from 10 to 12.
The complete syntax of joiks is:
[se] JOI [nai]
[se] BIhI [nai]
GAhO [se] BIhI [nai] GAhO
Notice that the colloquial English translations of “bi'i” and “bi'o” have forethought form: “between ... and” for “bi'i”, and “from ... to” for “bi'o”. In Lojban too, non-logical connectives can be expressed in forethought. Rather than using a separate selma'o, the forethought logical connectives are constructed from the afterthought ones by suffixing “gi”. Such a compound cmavo is not unnaturally called a “joigik”; the syntax of joigiks is any of:
[se] JOI [nai] GI
[se] BIhI [nai] GI
GAhO [se] BIhI [nai] GAhO GI
Joigiks may be used to non-logically connect bridi, sumti, and bridi-tails; and also in termsets.
Example 14.3 in forethought becomes:
16.10) joigi la djan. gi la .alis. bevri le pipno
[Together] John and Alice carry the piano.
The first “gi” is part of the joigik; the second “gi” is the regular gik that separates the two things being connected in all forethought forms.
Example 16.6 can be expressed in forethought as:
16.11) mi ca sanli ke'i bi'i ga'o gi la drezdn. gi la frankfurt.
I [present] stand [exclusive] between [inclusive] Dresden and Frankfurt.
I am standing between Dresden (exclusive) and Frankfurt (inclusive).
In forethought, unfortunately, the GAhOs become physically separated from the endpoints, but the same rule applies: the first GAhO refers to the first endpoint.