4: Lojbanic Relations

Ordered sequences: {ce'o}, {nomoi}, {pamoi},

{ce'o} builds ordered sequences, which generalize ordered pairs by having more than two elements. We will need some metasyntactic help to address these sequences in a uniform way; we want to be able to address each part of a sequence numerically. In the following rules, the ellipsis ... metasyntactically represents additional variables linked by {ce'o}, forming the "rest" or "tail" of the sequence. This tail could well be empty.

Note for the future: Experimental {ra'ei} could represent additional elements, but is in entirely the wrong selma'o.

ro da zo'u:

da nomoi da ce'o ... (nomoi)

ro de zo'u:

de pamoi da ce'o de ce'o ... (pamoi)

ro di zo'u:

di remoi da ce'o de ce'o di ce'o ... (remoi)

...

Ternary relations: {bridi},

This entire section rests on whether {bridi} is semantic or syntactic; that is, whether it operates on live sets and values, or Lojban ASTs.

At last, we can define {bridi}. First, let's relate it to other selbri. {bridi} internalizes the rule of {ckaji} and {ckini}, which instantiate relations. When we connect terbri to selbri, we are asking the selbri whether the given terbri are present in the underlying relation. This is precisely the characteristic function! We use {du'u} to track whether a bridi is satisfying the characteristic of its selbri.

ro da poi ke'a broda ro de poi broda ke'a zo'u:

                           da broda de
================================================================== (bridi-intel-2)
pa du'u da broda de kei bridi pa ka ce'u broda ce'u kei da ce'o de

Just like with {ka}, we only get one {du'u} bound at a time. A {du'u} is definitely inhabited or not inhabited. On the top of the rule, we have one result for every pair in the selbri; on the bottom of the rule, we have one result for every pair in the Cartesian product (the full relation), and also a {du'u} indicating whether or not the pair is also in the selbri.

Reactions have been mixed, to say the least, to this approach. la gleki opines that terbri do not work this way. However, CLL insists that selbri and bridi and {bridi} work this way. The main contention is what parts of the bridi are syntactic and which are semantic, since Lojban does support text literals.

For posterity: selbri are syntactic representations of semantic equivalence classes. This gives us isomorphism-invariance; any particular {pa du'u} really does only have one class of equivalent bridi underneath it. This has consequences for discursive logic, in that {lo du'u mlatu} should be taken as a vague pattern which the listener must disambiguate from context, and not as a literal reference to an equivalence class of logical sentences.