| "A propositional function [roughly, property or relation] is an incomplete
object whose completion is a proposition" (Frege, loose trat). So, every {ka} insofar as it creates a propositional function, property or relation, contains at least one hole and that is marked by {ce'u}. Did it contain no holes, it would be a complete object and, asuming the original type was right, a proposition. I*think* everyone agrees thus far. So, the disagreement is about whether the {ce'u} must always be written in and, if not, where the implicit one is. 1) Every {ce'u} must be explicit. . Any slot not filled by an overt marker is filled by {zo'e} or some such thing. At least one slot must be filled by {ce'u} (unless we collapse the distinction between {ka} and {du'u}, in which case, {du'u} are the {ce'u}-less {ka} -- or conversely). An easy rule and ambiguity-proof, but possibly verbose. 2) Not so -- some {ce'u} may be implicit, so long as there is a rule for identifying the place(s). The rule may now be somewhat more complex, but the results will be less verbose (generally). A) The implicit {ce'u} is always the first (x1) place. This runs into immediate conflict with the possibility (indeed, reality) of {ka} phrases in which the first place is filled with a content sumti. This is not ungrammatical, so it needs an interpretation. i) Assuming there are no explicit {ce'u} in the phrase, this is treated like other {ce'u}-less {ka} -- reduced to {du'u}. With an explicit {ce'u} elsewhere, it is taken as the propositional function defined by the explicit places, with no implicit ones (it was on a permission, after all). ii) The {ce'u} is always in the first place, even if something else is also there. The something else is a) an exemplary argument to which the function applies to produce a true proposition or a new propositional function (depending on whether there are explicit {ce'u}), but the {ka} phrase refers to this an all other such functions. This does not seem to really give the first-place phrase any role that would justify it being there, unless it is to suggest a range of values for {ce'u}, and that would better be done using explicit predicates. b) as in a), but now the {ka} phrase indicates just the function with the sumti in first place. This reduces to i). c) like a) in all respects except that the sumti in first place markes a special relationship between its referent and the function, which is different from (but may include) the application relationship a) assumes. {leka do xunre} may or may not mean that {do xunre} is true, but it indicates a special relationship between you and leka ce'u xunre. The nature of this relationship is not specified anywhere that I can find, and so the whole does not seem different from {ledo ka ce'u xunre} (to be on the safe side), which is also unexplained, but in the same way. B) The implicit {ce'u} is the first unfilled place in the bridi as written (if none then {du'u}). This comes, in a way that A) does not, into conflict with other Lojbanic habits, in particular, not filling uninteresting places, dropping {zo'e}. Using it correctly requires noticing that the place (assuming it is not the first, as it most often will be) is important, since it gets a {ce'u} and then dropping that {ce'u}. It thus is harder to use than A when it does not have the same effect as A and so harder than A altogether, and more likely to errors in what is said. C) A) and B) save at most a couple of syllables, so could B be generalized to, say, all the unfilled spaces up to the first explicit {zo'e}. This is actually simpler than B) since we only have to decide that something is unimportant and put in a {zo'e}, not decide it is important and then leave out a {ce'u} we were going to put in. It also gives rather natural results, e.g. {le ka prami} is the love relationship, not either the property of being loved or of being a lover. It could be extended (but I doubt it is worth it) by returning to {ce'u} after an explicit one after a {zo'e}. The last example raises a general issue: each occurrence of {ce'u} is new, independent of others in the context (like {ma} and unlike {ke'a}). How, then, do we force two occurrences to be the same, as we can do with the lambda operator from which {ce'u} derives. How, for example, do we talk about self-love, leka prami with the two implicit {ce'u} identified. Notice, we can't do this with any identity predicate, since that just introduces two more {ce'u}, unconnected with the earlier ones. For this and general reasons, I suggest that {ce'u}, like KOhA generally, be taken as having implicit subscripts (starting with 0) assigned in left to right order. So, self -love is le ka ce'uxino prami ce'uxino, which might be shortened somehow (to, for example, {le ka prami ce'uxino}) but probably shouldn't be. |