Chapter 16: “Who Did You Pass On The Road? Nobody”: Lojban And Logic
10. bridi negation and logical connectives
A complete discussion of logical connectives appears in Chapter 14. What is said here is intentionally quite incomplete and makes several oversimplifications.
A logical connective is a cmavo or compound cmavo. In this chapter, we will make use of the logical connectives “and” and “or” (where “or” really means “and/or”, “either or both”). The following simplified recipes explain how to make some logical connectives:
- To logically connect two Lojban sumti with “and”, put them both in the bridi and separate them with the cmavo “.e”.
- To logically connect two Lojban bridi with “and”, replace the regular separator cmavo “.i” with the compound cmavo “.ije”.
- To logically connect two Lojban sumti with “or”, put them both in the bridi and separate them with the cmavo “.a”.
- To logically connect two Lojban bridi with “or”, replace the regular separator cmavo “.i” with the compound cmavo “.ija”.
More complex logical connectives also exist; in particular, one may place “na” before “.e” or “.a”, or between “.i” and “je” or “ja”; likewise, one may place “nai” at the end of a connective. Both “na” and “nai” have negative effects on the sumti or bridi being connected. Specifically, “na” negates the first or left-hand sumti or bridi, and “nai” negates the second or right-hand one.
Whenever a logical connective occurs in a sentence, that sentence can be expanded into two sentences by repeating the common terms and joining the sentences by a logical connective beginning with “.i”. Thus the following sentence:
✥10.1 mi .e do klama ti I and you come here.
can be expanded to:
✥10.2 mi klama ti .ije do klama ti I come here, and, you come here.
The same type of expansion can be performed for any logical connective, with any valid combination of “na” or “nai” attached. No change in meaning occurs under such a transformation.
Clearly, if we know what negation means in the expanded sentence forms, then we know what it means in all of the other forms. But what does negation mean between sentences?
The mystery is easily solved. A negation in a logical expression is identical to the corresponding bridi negation, with the negator placed at the beginning of the prenex. Thus:
✥10.3 mi .enai do prami roda I, and not you, love everything.
✥10.4 mi prami roda .ijenai do prami roda I love everything, and-not, you love everything.
and then into prenex form as:
✥10.5 roda zo'u mi prami da .ije naku zo'u do prami da For each thing: I love it, and it is false that you love (the same) it.
By the rules of predicate logic, the “ro” quantifier on “da” has scope over both sentences. That is, once you've picked a value for “da” for the first sentence, it stays the same for both sentences. (The “da” continues with the same fixed value until a new paragraph or a new prenex resets the meaning.)
Thus the following example has the indicated translation:
✥10.6 su'oda zo'u mi prami da .ije naku zo'u do prami da For at least one thing: I love that thing. And it is false that: you love that (same) thing. There is something that I love that you don't.
If you remember only two rules for prenex manipulation of negations, you won't go wrong:
- Within a prenex, whenever you move “naku” past a bound variable (da, de, di, etc.), you must invert the quantifier.
- A “na” before the selbri is always transformed into a “naku” at the left-hand end of the prenex, and vice versa.