### 15. More about non-logical connectives

The final three JOI cmavo, “jo'e”, “ku'a”, and “pi'u”, are probably only useful when talking explicitly about sets. They represent three standard set operators usually called “union”, “intersection”, and “cross product” (also known as “Cartesian product”). The union of two sets is a set containing all the members that are in either set; the intersection of two sets is a set containing all the members that are in both sets. The cross product of two sets is the set of all possible ordered pairs, where each ordered pair contains a single element from the first set followed by a single element from the second. This may seem very abstract; hopefully, the following examples will help:

```15.1)  lo'i ricfu ku jo'e lo'i dotco cu barda
The-set-of rich-things union the-set-of German-things is large.

15.2)  lo'i ricfu ku ku'a lo'i dotco cu cmalu
The set-of rich-things intersection the-set-of German-things is small.
```
There is a parallelism between logic and set theory that makes Example 15.1 and Example 15.2 equivalent respectively to:
```15.3)  lo'i ricfu ja dotco cu barda
The-set-of rich-or-German-things is large.
```
and
```15.4)  lo'i ricfu je dotco cu cmalu
The-set-of rich-and-German-things is small.
```

The following example uses “se remei”, which is a set (not a mass) of two elements:

```15.5)  la djeimyz. ce[bo] la djordj. pi'u la meris. cebo la martas.
cu prami se remei
James and-set George cross-product Mary and-set Martha
are-lover type-of-pairs.
```
means that each of the pairs James/Mary, George/Mary, James/Martha, and George/Martha love each other. Therefore it is similar in meaning to Example 14.13; however, that example speaks only of the men loving the women, not vice versa.

Joiks may be combined with “bo” or with “ke” in the same way as eks and jeks; this allows grouping of non-logical connections between sumti and tanru units, in complete parallelism with logical connections:

```15.6)  mi joibo do ce la djan. joibo la djein.
cu gunma se remei
(I massed-with you) and (John massed-with Jane)
are-a-mass type-of-two-set
```
asserts that there is a set of two items each of which is a mass.

Non-logical connection is permitted at the joint of a termset; this is useful for associating more than one sumti or tagged sumti with each side of the non-logical connection. The place structure of “casnu” is:

```      casnu: the mass x1 discusses/talks about x2
```
so the x1 place must be occupied by a mass (for reasons not explained here); however, different components of the mass may discuss in different languages. To associate each participant with his or her language, we can say:
```15.7)  mi ce'e bau la lojban.
pe'e joi do ce'e bau la gliban. nu'u casnu
( I [plus] in-language Lojban
massed-with you [plus] in-language English ) discuss.
```

Like all non-logical connectives, the usage shown in Example 15.7 cannot be mechanically converted into a non-logical connective placed at another location in the bridi. The forethought equivalent of Example 15.7 is:

```15.8)  nu'i joigi mi bau la lojban gi do bau la gliban. nu'u casnu
```

Non-logical forethought termsets are also useful when the things to be non-logically connected are sumti preceded with tense or modal (BAI) tags:

```15.9)  la djan. fa'u la frank. cusku nu'i bau la lojban.
nu'u fa'u bai tu'a la djordj. [nu'u]
John respectively-with Frank express [start termset] in-language Lojban
[joint] respectively-with under-compulsion-by George.
John and Frank speak in Lojban and under George’s compulsion, respectively.
```
Example 15.9 associates speaking in Lojban with John, and speaking under George’s compulsion with Frank. We do not know what language Frank uses, or whether John speaks under anyone’s compulsion.

Joiks may be prefixed with “.i” to produce ijoiks, which serve to non-logically connect sentences. The ijoik “.ice'o” indicates that the event of the second bridi follows that of the first bridi in some way other than a time relationship (which is handled with a tense):

```15.10) mi ba gasnu la'edi'e
.i tu'e kanji lo ni cteki
.ice'o lumci le karce
.ice'o dzukansa le gerku tu'u
I [future] do the-referent-of-the-following:
( Compute the quantity of taxes.
And-then wash the car.
And-then walkingly-accompany the dog. )
List of things to do:
Figure taxes.
Wash car.
Walk dog.
```
Example 15.10 represents a list of things to be done in priority order. The order is important, hence the need for a sequence connective, but does not necessarily represent a time order (the dog may end up getting walked first). Note the use of “tu'e” and “tu'u” as general brackets around the whole list. This is related to, but distinct from, their use in Section 8, because there is no logical connective between the introductory phrase “mi ba gasnu la'edi'e” and the rest. The brackets effectively show how large an utterance the word “di'e”, which means “the following utterance”, refers to.

Similarly, “.ijoi” is used to connect sentences that represent the components of a joint event such as a joint cause: the Lojban equivalent of “Fran hit her head and fell out of the boat, so that she drowned” would join the events “Fran hit her head” and “Fran fell out of the boat” with “.ijoi”.

The following “nai”, if present, does not negate either of the things to be connected, but instead specifies that some other connection (logical or non-logical) is applicable: it is a scalar negation:

```15.11) mi jo'unai do cu remei
I in-common-with [not!] you are-a-twosome
```
The result of “mi jo'u do” would be two individuals, not a mass, therefore “jo'u” is not applicable; “joi” would be the correct connective.

There is no joik question cmavo as such; however, joiks and ijoiks may be uttered in isolation in response to a logical connective question, as in the following exchange:

```15.12) do djica tu'a
loi ckafi ji loi tcati