## 14.16. Interval connectives and forethought non-logical connection

In addition to the non-logical connectives of selma'o JOI explained in Section 14.14 and Section 14.15, 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:

Example 14.138.

 mi ca sanli I [present] stand-on-surface
 la drezdn. bi'i la frankfurt. that-named Dresden [interval] that-named Frankfurt.
 I am standing between Dresden and Frankfurt.

In Example 14.138, 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.)

Example 14.139.

 mi cadzu ca la pacac. I walk simultaneous-with First-hour
 bi'o la recac. [ordered-interval] Second-hour.
 I walk from one o'clock to two o'clock.

In Example 14.139, 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 14.139 is equivalent to:

Example 14.140.

 mi cadzu ca la recac. I walk simultaneous-with Second-hour
 se bi'o la pacac. [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:

Example 14.141.

 le jbama pu daspo la .uacintyn. The bomb [past] destroys Washington
 mi'i lo minli be li muno [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 14.138, 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:

Example 14.142.

 mi ca sanli la drezdn. ga'o I [present] stand that-named Dresden [inclusive]
 bi'i ga'o la frankfurt. [interval] [inclusive] that-named Frankfurt.
 I am standing between Dresden and Frankfurt, inclusive of both.

Example 14.143.

 mi ca sanli la drezdn. ga'o I [present] stand that-named Dresden [inclusive]
 bi'i ke'i la frankfurt. [interval] [exclusive] that-named Frankfurt.
 I am standing between Dresden (inclusive) and Frankfurt (exclusive).

Example 14.144.

 mi ca sanli la drezdn. ke'i I [present] stand that-named Dresden [exclusive]
 bi'i ga'o la frankfurt. [interval] [inclusive] that-named Frankfurt.
 I am standing between Dresden (exclusive) and Frankfurt (inclusive).

Example 14.145.

 mi ca sanli la drezdn. ke'i I [present] stand that-named Dresden [exclusive]
 bi'i ke'i la frankfurt. [interval] [exclusive] that-named 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):

Example 14.146.

 do dicra .e'a mi ca la daucac. You disturb (allowed) me at that-named 10
 bi'onai la gaicac. not-from-...-to that-named 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 14.111 in forethought becomes:

Example 14.147.

 joigi la djan. gi la .alis. bevri le pipno [Together] that-named John and that-named 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 14.143 can be expressed in forethought as:

Example 14.148.

 mi ca sanli ke'i bi'i I [present] stand [exclusive] between
 ga'o gi la drezdn. gi la frankfurt. [inclusive] and that-named Dresden and that-named 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.