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# Sau La Lodtua (From the Logic-Worker = Logician)

## The Logic of "Respectively"

by Randall Holmes

The Keugru recently approved the introduction of a grammatical form for designating ordered lists of objects: the list (A,B,C...) will be expressed in Loglan as

• lou A, B, C, ... luo
This parallels the recent introduction of lau A, B, C, ... lua for the finite set (or unordered list) {A,B,C...}.

Ordered lists were originally considered for introduction because of the problems posed by the English idiom respectively, as in

• (1) Tom, Dick, and Harry love Mary, Alice, and Susan, respectively.
The logical problem posed by respectively is still unsolved, in spite of the introduction of ordered lists; ordered lists are only part of a solution the final form of which is not yet clear. In this column, I will discuss the logical issues raised by this problem.

It might seem that sentence (1) could be translated into Loglan as

• (1a) La Tam, e la Dik, e la Heris, cluva la Meris, e la Alis, e la Suzn.
But sentence (1a) says too much. It can be expanded into the conjunction of nine sentences asserting that each of the three men loves each of the three women, while it is the effect of respectively in English that sentence (1) is equivalent to the conjunction of only three sentences, expressed in Loglan thus:
• (1b) La Tam, cluva la Meris, ice la Dik, cluva la Alis, ice la Heris, cluva la Suzn.
It is quite possible that Harry has entertained a lifelong dislike for Mary, for example, in which case (1a) but not (1b) would be false.

I am now going to play one of the oft-repeated refrains in my logical analysis of Loglan; I beg my readers' patience while I remind them that an argument like

• La Tam, e la Dik, e la Heris
built with logical connectives is a non-designating argument; there is no object referred to by this construction, especially not any kind of list or set. The grammatical form of a sentence involving this kind of non-designating argument is deceptive; the argument instructs us to expand the sentence containing it into a conjunction of three sentences, one about Tom, one about Dick, and one about Harry.

A sentence like

• (2) La Tam, e la Dik, e la Heris, tera.
which asserts Tom, Dick, and Harry are a threesome, is false: for it expands into the sentence
• (2a) La Tam, tera, ice la Dik, tera, ice la Heris, tera.
which makes the unlikely assertion that each of these gentlemen is a set with three elements. Of course, the proper way to say what we really wanted to say with (2) is
• (2b) Lau la Tam, la Dik, la Heris, lua tera.
I can't resist one further diversion before I return to respectively, lest anyone still entertain doubts about the non-designating status of logically-connected arguments. We all know that the order of implicitly-quantified arguments, as in
• (3a) Raba cluva be
• (3b) Be nu cluva raba
makes a difference in meaning. The English sentences Everyone loves someone and Someone is loved by everyone exhibit the same distinction. The same effect can be achieved with logically connected arguments (in fact, arguments connected with and and or can be thought of as expressing implicit universal and existential quantification (respectively) over finite sets).
• (4a) La Tam, e la Dik, cluva la Meris, a la Alis.
• (4b) La Meris, a la Alis, nu cluva la Tam, e la Dik.
The first sentence asserts that each of the gentlemen loves one or both of the ladies (and if exactly one, not necessarily the same one!); the second asserts that at least one of the ladies is loved by both of the gentlemen. This behavior is of course not exhibited by arguments which designate genuine referents: A cluva B is precisely equivalent to B nu cluva A when and B are designating arguments.

It is because of the non-designating character of arguments constructed with logical connectives that the ordered and unordered list-constructions have been introduced. It should be noted here that arguments formed with ze (mixed arguments) are normal designating arguments: la Tam, ze la Heris designates a composite object made up of Tom and Harry.

Now we come to ordered lists and respectively . One way of understanding sentence (1) is that each element of the ordered list (Tom, Dick, Harry) loves the corresponding element of the list (Mary, Alice, Susan). So some believe that a correct translation of (1) is

• (1d) Lou la Tam, la Dik, la Heris, luo cluva lou la Meris, la Alis, la Suzn, luo.
This is logically wrong. For the arguments in this sentence designate definite objects (two ordered lists), and it is not correct to assert that one ordered list loves another. We know very little about the emotional lives of mathematical objects!

To illustrate why this will not do, assign the predicate predu the following temporary definition: X predu Y means X is the first term of ordered list Y. Now consider the sentence

• (5) Lou A B luo predu lou lou A B luo C luo.
This can be translated into mathematical English as
• (5a) (A,B) is the first term of the ordered list ((A,B),C).
Only a mathematician would want to say this, but then mathematicians should be encouraged to speak logical languages!

If the logical transformation being proposed by advocates of (1d) were allowed, this sentence would be equivalent to

• (5b) A predu (A,B), ice B predu C.
which asserts in mathematical English
• (5c) A is the first term of (A,B) and B is the first term of C.
The first conjunct is true (accidentally) but we certainly did not intend to assert the second in the original sentence (5)!

There are two possible solutions to this problem that I see. One uses ordered lists and one does not. One solution, foreshadowed in the approach of (1d), is to introduce a new series of logical connectives, constructing non-designating arguments with different distribution rules than those of the conventional logical connectives. Suppose we introduce a new CVV-form little word xxx, and the connectives are xxxa, xxxe, etc. We could then have sentence

• (1e) La Tam, xxxe la Dik, xxxe la Heris, cluva la Meris, xxxe laAlis, xxxe la Suzn.
Arguments linked by xxxe would distribute according to a different rule than the usual connectives: only those sentences would be included in a conjunction (or disjunction, for example, if xxxa were used) that had arguments taken from corresponding positions in arguments formed with xxxe (and so all such arguments in a sentence would need to have the same number of connected arguments for the sentence to make sense). (1e) would expand to the form given above as (1b).

One virtue of this approach is that it would be easy to mix arguments linked with the usual connectives with the novel arguments formed with the new connectives:

• (5a) La Tam, xxxe la Dik, xxxe la Heris, fundi la Meris, xxxe la Alis, xxxe la Suzn, la Me'redeq, e la Qiydoras.
asserts that Tom likes Mary more than Meredith, Tom (also) likes Mary more than Theodora, Dick likes Alice more than Meredith, Dick (also) likes Alice more than Theodora, and so forth; the restriction that arguments in corresponding positions be used to form conjuncts only applies to the arguments formed with xxxe, while the third argument is distributed in the usual aggressive way.

A disadvantage of this form is that it requires a brand-new logical construction, one not using the ordered list construction which we have managed to agree on. Another disadvantage, relative to the alternative we will now present, is that the marker xxx- is used a lot; the sentence above has a lot of extra syllables in it!

The alternative which I favor is to introduce an operation on predicates which has the effect of allowing the distribution which we rejected in sentence (1d). Let the little word needed be yyy; then A yyy preda B C... will mean corresponding elements of the lists A, B, C... (preda may have any number of arguments) stand in the preda relation to one another. Then we have sentence

• (1f) Lou la Tam, la Dik, la Heris, luo yyy cluva lou la Meris, la Alis, la Suzn, luo.
with the meaning intended by (1).

A refinement of the meaning of yyy will enable the mixing of conjuncts distributing respectively with conjuncts distributing aggressively: allow unordered lists to be arguments of yyy preda as well, with the convention that elements of an unordered list are taken to correspond to all elements of each of the other lists (ordered or unordered). So the effect of (5a) is duplicated by

• (5b) Lou la Tam, la Dik, la Heris, luo yyy fundi lou la Meris, la Alis, la Suzn, luo lau la Me'redeq, la Qiydoras, lua.
Another pair of examples are needed to indicate the effects of the appearance of single arguments (not formed by any logical connective) in sentences under each of these proposals.
• (6a) La Tam, xxxe la Dik, fundi la Meris, xxxe la Alis, la Me'redeq.
This asserts (under the first proposal) that Tom loves Mary more than Meredith and that Dick loves Alice more than Meredith.

One might think that the corresponding form under the second proposal would be

• (6b) Lou la Tam, la Dik, luo yyy fundi lou la Meris, la Alis, luo la Me'redeq.
but this is not correct. The arguments of yyy preda need to be ordered or unordered lists. The correct form would be
• (6c) Lou la Tam, la Dik luo yyy fundi lou la Meris, la Alis, luo lau la Me'redeq, lua.
in which Meredith appears by herself in a set. It might appear that allowing sentences like (6b) with the obviously intended meaning would be harmless, but problems would arise with sentences using mathematical predicates of lists or sets, analogous to the problems with the (1d) proposal illustrated in (5).

The second proposal requires one further refinement: if the sentences constructed to translate respectively are to be linked with a logical connective other than ice , the little word yyy needs to be marked in some way with the logical connective. Perhaps I should have used yyye through the examples above, but it seems logical that and would be the default connective. An example is

• (7) Lou la Tam, la Dik, luo yyya cluva lou la Meris, la Alis, luo.
which asserts that Tom loves Mary, or Dick loves Alice; Tom or Dick loves Mary or Alice, respectively, if this can be said in English.

The main reason that I favor the second proposal over the first is that I don't see any reason to introduce new constructions of non-designating arguments into the language if they can be avoided. The grammatical structure of a logical language should follow its semantic structure as much as possible.

Whatever phonemic values are given to yyy, the new word if adopted would belong to the NU Lexeme, as would its variants yyya, etc.

--Hue Rendl Holmz

I have a question for our Lodtua: would (2c) La Tam, ze la Dik, ze la Heris, tera, which is (2) made with ze instead of e, also be false? Or do you think that cardinal predicates like tera can be truly asserted of single, team-like, "composite" objects formed with ze if these have been composed of three distinct members, chunks, or parts? It would be a convenience for the speakers of this logical language if we could safely interpret these non-mathematical, composite objects (teams, etc.) in this essentially mathematical way. For we could then say La Tam, ze la Dik. ze la Heris, yyy cluva la Meris, ze la Alis, ze la Suzn = Tom, Dick, and Harry respectively love Mary, Alice, and Susan with very nearly the same economy as the E-word respectively provides.--JCB