scope of na
The cmavo na is always used to negate a bridi, i.e. it is a bridi operator. Other bridi operators include tenses, quantifiers and logical connectives. In general, the order in which a bridi operator is applied to a bridi matters: two operators applied in a given order give a different meaning than when applied in the opposite order. The general rule for Lojban is that an operator has scope over operators that appear later in the sentence, and is within the scope of operators that appeared before it in the sentence. There is however one glaring exception to this rule: the operator na when it appears directly in front of the selbri is traditionally taken to have maximal scope, scoping even over those operators that appeared before it in the sentence. The motivation for this exception has never been very clear. I will argue here that this exception either leads to contradictions or, in order to avoid the contradictions, it requires complicating the rules so much that the exception should be scrapped.
2. Some simple examples of bridi operators
2.1a) mi na dunda ti do
The operator na is used to negate the bridi mi dunda ti do, i.e. to indicate that the situation described by the bridi mi dunda ti do does not occur.
2.2a) mi ta'e dunda ti do
The operator ta'e is used to indicate that the situation described by the bridi mi dunda ti do occurs habitually.
2.3a) mi ciroi dunda ti do
The operator ciroi is used to indicate that the situation described by the bridi mi dunda ti do occurs exactly three times.
2.4a) mi dunda mu da do
The operator mu is used to indicate that the situation described by the bridi mi dunda da do occurs for exactly five different values of the variable da.
2.5a) mi dunda ge ti gi ta do
The operator ge ... gi ... is used to indicate that both of the situations described by the bridis mi dunda ti do and mi dunda ta do occur.
2.6a) mi ti do ga dunda gi lebna
The operator ga ... gi ... is used to indicate that at least one of the situations described by the bridis mi dunda ti do and mi lebna ti do occurs.
The last two examples are binary operators, the operators take two bridis as arguments and return one. The other operators take one bridi as argument and return another bridi.
In all of these examples, there is a single bridi operator, so it doesn't matter where it is palced in the bridi. These are all logically equivalent to their a) version:
2.1b) naku mi dunda ti do
2.2b) mi dunda ti ta'eku do
2.3b) mi dunda ti do ciroiku
2.4b) fi do fe mu da fa mi dunda
2.5b) ge ti gi ta se dunda mi do
2.6b) mi ga dunda gi lebna vau ti do
3. Combinations of operators on the same bridi
When two operators act on the same bridi, the order in which they are applied matters. For example, these two say different things:
3.1a) ciroiku mi dunda mu da do
3.1b) mi dunda mu da do ciroiku
The first one says that it happened exactly three times that I gave you exactly five things. The second one says that there are exactly five things that I gave you exactly three times.
3.2a) mi ti do ta'e gonai dunda gi lebna
3.2b) mi ti do gonai ta'e dunda gi ta'e lebna
The first one says that habitually it happens that I either give this to you or else I take it from you.
The second one says that either I habitually give this to you, or else habitually I take it from you.
3.3a) naku mi dunda ge ti gi ta do
3.3b) mi dunda ge ti gi ta do naku
The first one says that it is not the case that I gave you both this and that.
The second one says that for both this and that, I didn't give them to you.
3.4a) mi su'oroiku naku dunda ti do
3.4b) mi naku su'oroiku dunda ti do
The first one says that at least once I didn't give this to you.
The second one says that I didn't even once give this to you.
3.5a) mi su'oroi na dunda ti do
3.5b) mi na su'oroi dunda ti do
According to the traditional view, these both say the same as (3.4b), even though in (3.5a) the negation appears after the tense, it is supposed to have scope over it because it appears as a selbri tag.
4. The problem
One problem of the scope exception for the operator na is the unintuitive result shown in examples (3.5a) and (3.5b) compared with (3.4a) and (3.4b). But there is also a more serious problem. Let's consider the following example:
4.1) mi su'o da do ge na dunda gi na lebna
The obvious reading for this is there is at least one thing such that both I did not give it to you and I did not take it from you. This means the scope of the operators is, su'o scopes over ge ... gi ... which scopes over both instances of na.
How are we to apply the rule here that na, which appears as selbri tag, has scope over everything else?
Also, ge na dunda gi na lebna is logically equivalent to na ga dunda gi lebna. What if we wrote:
4.2) mi su'o da do na ga dunda gi lebna
Is (4.2) equivalent to (4.1), or are we to take na in (4.2) as having scope over su'o?
5. A possible solution
A possible solution is to say that in (4.1) the scope of na is blocked by being in a logically connected bridi, and so in such cases it does not have scope over every other operator, and that in (4.2) it does have scope over everything and so (4.2) and (4.1) are not equivalent and that the application of De Morgan's laws have to take this into account. Perhaps even more exceptions to the exception will have to be introduced as we consider more complex cases.
6. A better solution
Eliminate the exception for na. Make na behave like every other bridi operator, with scope over following operators and within the scope of preceding operators, and then no conflicts arise.