Let us consider the English sentence

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

when converted to the external negation form produces:

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 16.79 can be expressed as:

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 16.9 , 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 16.82 (no good English translations exist):

In Example 16.83 , we moved the negation boundary rightward across the quantifier of de , forcing us to invert it. In Example 16.85 we moved the negation boundary across the quantifier of da , forcing us to invert it instead. Example 16.84 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 16.82 and rearrange the sumti to produce

or rather, Example 16.86 means something completely different from Example 16.82. 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 16.82 is identical in meaning to:

but in Example 16.87 , 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 16.9 ; 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 16.82 goes into prenex form as:

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

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

or more briefly

As noted in Section 16.5 , a sentence with two different quantified variables, such as Example 16.91 , 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:

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 16.12 , 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 compounds in a sentence, each forming a separate negation boundary. Two adjacent naku compounds in a bridi are a double negative and cancel out:

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

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