Chapter 16: “Who Did You Pass On The Road? Nobody”: Lojban And Logic
9. Negation boundaries
This section, as well as c16-§10 through c16-§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 Chapter 15, the negation of a bridi is usually accomplished by inserting “na” at the beginning of the selbri:
✥9.1 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 c16-§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 ✥9.1 is
✥9.2 naku zo'u la djan. klama It is not the case that: John comes. It is false that: John comes.
However, “naku” can appear at other points in the prenex as well. Compare
✥9.3 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.
✥9.4 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:
✥9.5 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).
✥9.6 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:
✥9.7 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 ✥9.5, and
✥9.8 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 ✥9.6.
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, ✥9.7 and ✥9.8 can be restated as, respectively:
✥9.9 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.
✥9.10 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:
✥9.11 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 ✥9.12 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 ✥9.7 and ✥9.8.
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
✥9.13 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:
✥9.14 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 ✥9.14 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 for some x”, and the corresponding Lojban “noda” can be replaced by “naku su'oda”. Making this substitution, we get:
✥9.15 naku naku su'oda rode zo'u da prami de It is false that it is false that: for an X, for every Y: X loves Y.
Adjacent double negation boundaries in the prenex can be dropped, so this means the same as:
✥9.16 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 ✥9.13.
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.