Let us consider the English sentence

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).when converted to the external negation form produces:

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).

Lojban provides a negation form which more closely emulates natural language negation. This involves putting “naku” before the selbri, instead of a “na”. “naku” is clearly a contradictory negation, given its parallel with prenex bridi negation. Using “naku”, Example 11.1 can be expressed as:

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.

Although it is not technically a sumti, “naku” can be used in most of the places where a sumti may appear. We’ll see what this means in a moment.

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.or rather, Example 11.8 means something completely different from Example 11.4. Conversion with “se” under “naku” negation is not symmetric; not all sumti are treated identically, and some sumti are not invariant under conversion. Thus, internal negation with “naku” is considered an advanced technique, used to achieve stylistic compatibility with natural languages.

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.but in Example 11.9, the bound variables “da” and “de” have been hidden.

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.from which we can restore the quantified variables to the sentence, giving:

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.or more briefly

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

As noted in Section 5, a sentence with two different quantified variables, such as Example 11.13, cannot always be converted with “se” without first exporting the quantified variables. When the variables have been exported, the sentence proper can be converted, but the quantifier order in the prenex must remain unchanged:

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.

While you can’t freely convert with “se” when you have two quantified variables in a sentence, you can still freely move sumti to either side of the selbri, as long as the order isn’t changed. If you use “na” negation in such a sentence, nothing special need be done. If you use “naku” negation, then quantified variables that cross the negation boundary must be inverted.

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:

11.15) 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.