## 16.9. Negation boundaries

This section, as well as Section 16.10 through Section 16.12 , are in effect a continuation of Chapter 15 , introducing features of Lojban negation that require an understanding of prenexes and variables. In the examples below, there is a Y and the like must be understood as there is at least one Y, possibly more.

As explained in Section 15.2 , the negation of a bridi is usually accomplished by inserting na at the beginning of the selbri:

Example 16.57.

 mi na klama le zarci I [false] go-to the store.
 It is false that I go to the store. I don't go to the store.

The other form of bridi negation is expressed by using the compound cmavo naku in the prenex, which is identified and compounded by the lexer before looking at the sentence grammar. In Lojban grammar, naku is then treated like a sumti. In a prenex, naku means precisely the same thing as the logician's it is not the case that in a similar English context. (Outside of a prenex, naku is also grammatically treated as a single entity – the equivalent of a sumti – but does not have this exact meaning; we'll discuss these other situations in Section 16.11.)

To represent a bridi negation using a prenex, remove the na from before the selbri and place naku at the left end of the prenex. This form is called external bridi negation , as opposed to internal bridi negation using na. The prenex version of Example 16.57 is

Example 16.58.

 naku zo'u mi klama le zarci It-is-not-the-case-that : I go-to the store.
 It is false that: I go to the store.

However, naku can appear at other points in the prenex as well. Compare

Example 16.59.

 naku de zo'u de zutse It-is-not-the-case-that: for-some-Y : Y sits. It-is-false-that: for-at-least-one-Y : Y sits.
 It is false that something sits. Nothing sits.

with

Example 16.60.

 su'ode naku zo'u de zutse For-at-least-one-Y, it-is-false-that : Y sits.
 There is something that doesn't sit.

The relative position of negation and quantification terms within a prenex has a drastic effect on meaning. Starting without a negation, we can have:

Example 16.61.

 roda su'ode zo'u da prami de For-every-X, there-is-a-Y, such-that X loves Y.
 Everybody loves at least one thing (each, not necessarily the same thing).

or:

Example 16.62.

 su'ode roda zo'u da prami de There-is-a-Y, such-that-for-each-X : X loves Y.
 There is at least one particular thing that is loved by everybody.

The simplest form of bridi negation to interpret is one where the negation term is at the beginning of the prenex:

Example 16.63.

 naku roda su'ode zo'u da prami de It-is-false-that: for-every-X, there-is-a-Y, such-that: X loves Y.
 It is false that: everybody loves at least one thing. (At least) someone doesn't love anything.

the negation of Example 16.61 , and

Example 16.64.

 naku su'ode roda zo'u da prami de It-is-false-that: there-is-a-Y such-that for-each-X : X loves Y.
 It is false that: there is at least one thing that is loved by everybody. There isn't any one thing that everybody loves.

the negation of Example 16.62.

The rules of formal logic require that, to move a negation boundary within a prenex, you must invert any quantifier that the negation boundary passes across. Inverting a quantifier means that any ro (all) is changed to su'o (at least one) and vice versa. Thus, Example 16.63 and Example 16.64 can be restated as, respectively:

Example 16.65.

 su'oda naku su'ode zo'u da prami de For-some-X, it-is-false-that: there-is-a-Y such-that: X loves Y.
 There is somebody who doesn't love anything.

and:

Example 16.66.

 rode naku roda zo'u da prami de For-every-Y, it-is-false-that: for-every-X : X loves Y.
 For each thing, it is not true that everybody loves it.

Another movement of the negation boundary produces:

Example 16.67.

 su'oda rode naku zo'u da prami de There-is-an-X such-that-for-every-Y, it-is-false-that : X loves Y.
 There is someone who, for each thing, doesn't love that thing.

and

Example 16.68.

 rode su'oda naku zo'u da prami de For-every-Y, there-is-an-X, such-that-it-is-false-that : X loves Y.
 For each thing there is someone who doesn't love it.

Investigation will show that, indeed, each transformation preserves the meanings of Example 16.63 and Example 16.64.

The quantifier no (meaning zero of) also involves a negation boundary. To transform a bridi containing a variable quantified with no , we must first expand it. Consider

Example 16.69.

 noda rode zo'u da prami de There-is-no-X, for-every-Y, such-that X loves Y.
 Nobody loves everything.

which is negated by:

Example 16.70.

 naku noda rode zo'u da prami de It-is-false-that: there-is-no-X-that, for-every-Y : X loves Y.
 It is false that there is nobody who loves everything.

We can simplify Example 16.70 by transforming the prenex. To move the negation phrase within the prenex, we must first expand the no quantifier. Thus for no x means the same thing as it is false that for some x , and the corresponding Lojban noda can be replaced by naku su'oda. Making this substitution, we get:

Example 16.71.

 naku naku su'oda It-is-false-that it-is-false-that there-is-some-X-such-that
 …rode zo'u da prami de for-every-X : X loves Y
 It is false that it is false that: for an X, for every Y: X loves Y.

Adjacent pairs of negation boundaries in the prenex can be dropped, so this means the same as:

Example 16.72.

 su'oda rode zo'u da prami de There-is-an-X-such-that, for-every-Y : X loves Y.
 At least one person loves everything.

which is clearly the desired contradiction of Example 16.69.

The interactions between quantifiers and negation mean that you cannot eliminate double negatives that are not adjacent. You must first move the negation phrases so that they are adjacent, inverting any quantifiers they cross, and then the double negative can be eliminated.