11. Using ``naku'' outside a prenex

11.1)    Some children do not go to school.

We cannot express this directly with ``na''; the apparently obvious translation

11.2)    su'oda poi verba
        na klama su'ode poi ckule
    At-least-one X which-are child(ren)
        [false] go-to at-least-one Y which-are school(s).

11.3)    naku zo'u su'oda poi verba
        cu klama su'ode poi ckule
    It is false that some which are children
        go-to some which are schools.
    All children don't go to some school
        (not just some children).

11.4)    su'oda poi verba naku klama
        su'ode poi ckule
    Some which-are children don't go-to
        some which-are schools.
    Some children don't go to a school.

When you use ``naku'' within a bridi, you are explicitly creating a negation boundary. As explained in Section 9, when a prenex negation boundary expressed by ``naku'' moves past a quantifier, the quantifier has to be inverted. The same is true for ``naku'' in the bridi proper. We can move ``naku'' to any place in the sentence where a sumti can go, inverting any quantifiers that the negation boundary crosses. Thus, the following are equivalent to Example 11.4 (no good English translations exist):

11.5)    su'oda poi verba cu klama rode poi ckule naku
    For some children, for every school,
        they don't go to it.

11.6) su'oda poi verba cu klama naku su'ode poi ckule
    Some children don't go to (some) school(s).

11.7) naku roda poi verba cu klama su'ode poi ckule
    It is false that all children go to some school(s).

In Example 11.5, we moved the negation boundary rightward across the quantifier of ``de'', forcing us to invert it. In Example 11.7 we moved the negation boundary across the quantifier of ``da'', forcing us to invert it instead. Example 11.6 merely switched the selbri and the negation boundary, with no effect on the quantifiers.

The same rules apply if you rearrange the sentence so that the quantifier crosses an otherwise fixed negation. You can't just convert the selbri of Example 11.4 and rearrange the sumti to produce

11.8)    su'ode poi ckule ku'o naku se klama roda poi verba
    Some schools aren't gone-to-by every child.

It isn't always easy to see which quantifiers have to be inverted in a sentence. Example 11.4 is identical in meaning to:

11.9)    su'o verba naku klama su'o ckule
    Some children don't go-to some school.

It is trivial to export an internal bridi negation expressed with ``na'' to the prenex, as we saw in Section 9; you just move it to the left end of the prenex. In comparison, it is non-trivial to export a ``naku'' to the prenex because of the quantifiers. The rules for exporting ``naku'' require that you export all of the quantified variables (implicit or explicit) along with ``naku'', and you must export them from left to right, in the same order that they appear in the sentence. Thus Example 11.4 goes into prenex form as:

11.10)  su'oda poi verba ku'o naku
        su'ode poi ckule zo'u
            da klama de
    For some X which is a child, it is not the case that
        there is a Y which is a school such that:
            X goes to Y.

We can now move the ``naku'' to the left end of the prenex, getting a contradictory negation that can be expressed with ``na'':

11.11)  naku roda poi verba
        su'ode poi ckule zo'u
            da klama de
    It is not the case that for all X's which are children,
        there is a Y which is a school such that:
            X goes to Y.

11.12)  naku zo'u roda poi verba cu klama su'ode poi ckule
    It is not the case that all children go to some school.

11.13)  ro verba cu na klama su'o ckule
    All children [false] go-to some school(s).

11.14)  roda poi verba
        su'ode poi ckule zo'u
            de na se klama da
    It is not the case that for all X's which are children,
        there is a Y which is a school such that:
            Y is gone to by X.

Clearly, if all of Lojban negation was built on ``naku'' negation instead of ``na'' negation, logical manipulation in Lojban would be as difficult as in natural languages. In Section 12, for example, we'll discuss DeMorgan's Law, which must be used whenever a sumti with a logical connection is moved across a negation boundary.

Since ``naku'' has the grammar of a sumti, it can be placed almost anywhere a sumti can go, including ``be'' and ``bei'' clauses; it isn't clear what these mean, and we recommend avoiding such constructs.

You can put multiple ``naku''s in a sentence, each forming a separate negation boundary. Two adjacent ``naku''s in a bridi are a double negative and cancel out:

11.16)  mi naku naku le zarci cu klama

Other expressions using two ``naku''s may or may not cancel out. If there is no quantified variable between them, then the ``naku''s cancel.

Negation with internal ``naku'' is clumsy and non-intuitive for logical manipulations, but then, so are the natural language features it is emulating.