16.10. 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:

Example 16.73.

mi	.e	do	klama	ti
I	and	you	come-to	this-here

I and you come here.

can be expanded to:

Example 16.74.

mi	klama	ti	.ije	do	klama	ti
I	come-to	this-here	and	you	come-to	this-here

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:

Example 16.75.

mi	.enai	do	prami	roda
I	and-not	you	love	everything

I, and not you, love everything.

expands to:

Example 16.76.

mi	prami	roda	.ijenai	do	prami	roda
I	love	everything,	and-not,	you	love	everything.

and then into prenex form as:

Example 16.77.

roda	zo'u	mi	prami	da	.ije
For-each-thing	:	I	love	it,	and

naku	do	prami	da
it-is-false-that	you	love	(the-same)-it.

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:

Example 16.78.

su'oda	zo'u	mi	prami	da
For-at-least-one-thing	:	I	love	that-thing.

.ije	naku	zo'u	do	prami	da
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.