(A page from the Loglan web site.)

(From Lognet 92/3. Used with the permission of The Loglan Institute, Inc.)

(from the Logic-Worker = Logician)


by M. Randall Holmes

I won't give another installment of the "introduction to logic" this time, because some real discussion of logical issues is going on!

The issue which I will try to tackle in this column can be vaguely summarized as the problem of "sets". There are a number of different kinds of ways of referring (or appearing to refer) to several objects at once (thus "putting things together".) In English, these are handled in an extremely sloppy fashion (as will be seen in examples below); in Loglan, they must be handled in a clear and unambiguous fashion. This, of course, is easier said than done.

We start with a typically horrible example:

proposed as meaning BIE is the infix predicate of set membership: "X bie Y" means "X is an element of set Y". The speaker seems to want to say "X is an element of the set of men I have in mind". But the speaker has not succeeded in saying this: what he has said is "X is a member of the man I have in mind (or is a member of each of the men I have in mind)". An even worse example of the same phenomenon is does this mean "X is an element of the set I have in mind (or of each of the sets I have in mind)" (the correct interpretation) or "X is one of the sets I have in mind" (the interpretation parallel to the incorrect interpretation proposed for the sentence above)?

The problem here is a confusion between plural reference (reference to more than one individual) and reference to a set of individuals. It is true that LE MRENU may have plural reference: it may refer to Tom, Dick and Harry, for example. In this case, the sentence DA BIE LE MRENU will mean DA BIE LA TAM, ICE DA BIE LA DIK, ICE DA BIE LA HERI (X is a member of Tom, X is a member of Dick, and X is a member of Harry). Since we do not think of the three gentlemen as being sets, we do not know what to make of this sentence. What the speaker wanted to say was "X is a member of {Tom, Dick, Harry}", and the problem is that LE MRENU cannot be made to refer to the set {Tom, Dick, Harry}, but only to each of its elements individually.

There is a way to say what the speaker originally wanted to say: we can use the ME operator of predification to build the predicate ME LE MRENU, which means "is one of the men I have in mind"; in our present context, this means "is either Tom, Dick, or Harry". We can then use the LEA operator of set formation to build LEA ME LE MRENU, which means (hey, presto!) "the set of those men I have in mind (the ones designated by LE MRENU)", or, in the present context, {Tom, Dick, Harry}. We can then say

which really does mean The same mistake can arise in other ways. Suppose that we wanted to say that the men are a threesome (true in our working context!). We attempt But what we have said (assuming reference to Tom, Dick and Harry as before) is "La Tam ga tera, ice la Dik ga tera, ice la Heris ga tera", which means, in English, "Tom is a three-element set, Dick is a three-element set, and Harry is a three-element set". This is a very odd assertion! The error is easy to correct: says what we want to say.

I would like to say at this point that the key is in the ME operator: an argument LE PREDA always refers to each intended "preda" individually, while a reference to the set of "predas" to which LE PREDA refers individually must be made via the predicate ME LE PREDA. But this is not quite true. There are pre-ME constructions using numerals which do operate on the set of "predas".

Consider the argument TE LE MRENU. This is a non-designating argument meaning "three of the men I have in mind"; it clearly uses the predicate "is one of the men I have in mind (that I designate currently with LE MRENU)\'d3. Here, the use of ME one might expect can be made explicit! TE ME LE MRENU is precisely synonymous to TE LE MRENU!

This introduction of an implicit ME can even be grammatically useful. Consider TO LE TE MRENU ("two of the three men"). Suppose we want to attach the modifier JI BLANU ("which is/are blue") to this. We could mean two different things: we might want to identify the three men as blue, as we do in TO LE TE MRENU JI BLANU, or we might want to identify the two taken out of the three as blue. This form of the argument does not allow us to do this! But introducing ME allows us to do this as well: in TO ME LE TE MRENU GU GU JI BLANU, the two men are identified as blue, not the three. The two GU's are a little unpleasant, it is true. Without the ME, this does not parse -- the grammatical structure of the pre-ME way of attaching the TO to the argument did not allow for this.

The other kind of attachment of numerals to arguments, found in LE TO MRENU (the two men), cannot be eliminated so easily with a circumlocution involving ME. The TO here does give us information about the set of men being individually referred to (there are two of them) although LE TO MRENU can only refer to each of these two men individually.

does not mean but the surprising while says what we originally intended.

There is some bad news and some good news. The bad news is that these mistakes are easy to make, because English does not train us to distinguish between plural reference and reference to sets. The good news is that Loglan does provide the tools for correct solution of the problem, via use of the ME operation to gain the ability to refer to the set of things whose individual elements are referred to by an argument with plural reference.

Now we discuss briefly how to define sets explicitly. To explicitly refer to the set {Tom, Dick, Harry}, one might try first "Tom, Dick, and Harry". There are two ways to say this in Loglan:

and Neither is satisfactory. The first we dismiss quickly; anything asserted of "Tom, Dick, and Harry" in this sense is actually asserted of each of the three gentlemen. The second is the mass individual made up of Tom, Dick, and Harry. The latter might seem at first to be a candidate for {Tom, Dick, Harry}; the next paragraph refutes this.

Sets are not mass individuals. The best example of this I know is the following: consider the mass individual LA INGLYND, ZE LA FRANS, ZE LA ITALIAS. This is the same as the mass individual LA INGLYND, ZE LA FRANS, ZE LA ITALIAS, ZE LA ALTO-ADIJ (Alto-Adige is a province of Italy), but the sets {England, France, Italy}and {England, France, Italy, Alto-Adige}are different; the first has three elements and the second has four elements. Masses can be used in place of sets where the elements of the sets are disjoint (in the sense that they do not share any parts) and it is understood what kinds of parts are in use (one needs to be able to distinguish between the set whose elements are France and Italy and the set whose elements are the departments of France and the provinces of Italy). Such "partitioned masses" cannot do all the work of sets, though; consider the set of all administrative divisions of Europe, large and small!

The confusion of sets with masses goes with another confusion. The relation of membership in a set is frequently confused with the relation of parts to a whole. An easy way to see that membership is not analogous to a part-whole relationship is to observe that if x is a member of y and y is a member of z, it is not true necessarily that x is a member of z; membership is not transitive. An example: let x be 2, y be {2,3}and z be the set of all two-element sets of numbers. 2 is not a two-element set of numbers, though it is an element of a two-element set of numbers. The relation among sets which is analogous to a part-whole relation is the relation which a subset has to the set in which it is included. This relation is transitive; a subset of a subset of a set is still a subset of the original set. It is better to think of a subset like {1,2}as a part of {1,2,3}than to think of 1 or 2 as parts thereof.

A correct Loglan name for {Tom, Dick, Harry}is LEA MELA TAM, CA MELA DIK, CA MELA HERIS; the set of all objects which have the property of either being the Tom we have in mind, or the Dick we have in mind, or the Harry we have in mind. Since English talks about the set whose elements are Tom, Dick, AND Harry, we might be surprised by the use of CA ("or") here; but if we used CE instead, we would be defining the set of all objects equal to all three of them at the same time, which is empty! The use of "and" in the English is more related to Loglan ZE than to Loglan E. A dedicated Loglan construction for naming finite sets has been proposed but not yet officially adopted (so far as la Lodtua knows at this time!).

We briefly show how to describe sets as they are described in mathematics: consider Russell's paradoxical class "The set of all sets which are not elements of themselves". This cannot exist (is it a member of itself or not?) but it exemplifies the mathematical method of describing sets, by giving a property which holds of their elements. In Loglan, this is LEA ME TAI JIO NO TAI BIE TAI. "NO TAI BIE TAI" is a sentence asserting the defining property of elements T of the Russell class; T will not be an element of T. "TAI JIO NO TAI BIE TAI" is a designation which refers to each such T; LEA ME TAI JIO NO TAI BIE TAI is the collection of all T's referred to by this designation. In general, the set of all T such that P, where P is a sentence which refers to T, will be LEA ME TAI JIO P. This usage seems compact enough to be used in mathematics as it stands.

As if keeping track of plurals and sets were not bad enough, there is another kind of reference to structures made up of several objects which Loglan does not support now, and for which we will need a construction. This is reference to finite ordered lists of elements. Consider the English sentence "I'm betting on Greased Lightning, Mashed Turnips, and Tired Commissar to win, place, and show in the fourth race." Here, the order of the three horses is very important; this could be made clearer by inserting the word "respectively" into the sentence. The horrible temptation for the Loglan speaker whose native language is English is to try to refer to lists of objects with lists of Loglan arguments linked by E. Loglan has a predicate NU TERI meaning, "is a sequence of at least three elements". The naive speaker might try to say

but this means not a believable English sentence, but rather the shocking Loglan does not now have any economical way of talking about finite lists, and it needs one. The pending proposal for naming finite sets mentioned above will also provide finite sequences. A parallel construction on predicates will also be needed; observe that in the eventual Loglan translation of the sentence on horse-racing, the predicates "win", "place" and "show" are linked to the successive arguments of the list in a fashion which cannot be handled by simple logical conjunction. It appears that building finite sequences and corresponding predicate structures is yet another of the many functions of the underrated English word "and", along with logical conjunction, construction of mass individuals, and building finite sets.

It is certain that I will have more to say about this in future columns.

Copyright 1992 by The Loglan Institute, Inc. All rights reserved.

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