16.12. Logical Connectives and DeMorgan's Law

DeMorgan's Law states that when a logical connective between terms falls within a negation, then expanding the negation requires a change in the connective. Thus (where p and q stand for terms or sentences) not (p or q) is identical to not p and not q, and not (p and q) is identical to not p or not q. The corresponding changes for the other two basic Lojban connectives are: not (p equivalent to q) is identical to not p exclusive-or not q, and not (p whether-or-not q) is identical to both not p whether-or-not q and not p whether-or-not not q. In any Lojban sentence having one of the basic connectives, you can substitute in either direction from these identities. (These basic connectives are explained in Chapter 14.)

The effects of DeMorgan's Law on the logical connectives made by modifying the basic connectives with nai, na and se can be derived directly from these rules; modify the basic connective for DeMorgan's Law by substituting from the above identities, and then, apply each nai, na and se modifier of the original connectives. Cancel any double negatives that result.

When do we apply DeMorgan's Law? Whenever we wish to distribute a negation over a logical connective; and, for internal naku negation, whenever a logical connective moves in to, or out of, the scope of a negation – when it crosses a negation boundary.

Let us apply DeMorgan's Law to some sample sentences. These sentences make use of forethought logical connectives, which are explained in Section 14.5. It suffices to know that ga and gi, used before each of a pair of sumti or bridi, mean either and or respectively, and that ge and gi used similarly mean both and and. Furthermore, ga, ge, and gi can all be suffixed with nai to negate the bridi or sumti that follows.

We have defined na and naku zo'u as, respectively, internal and external bridi negation. These forms being identical, the negation boundary always remains at the left end of the prenex. Thus, exporting or importing negation between external and internal bridi negation forms never requires DeMorgan's Law to be applied. Example 16.94 and Example 16.95 are exactly equivalent:

Example 16.94. 

la djan. na klama ga
that-named John [false] goes-to either
la paris. gi la rom.
that-named Paris or that-named Rome.

Example 16.95. 

naku zo'u la djan. klama
It-is-false that: that-named John goes-to
ga la paris. gi la rom.
either that-named Paris or that-named Rome.

It is not an acceptable logical manipulation to move a negator from the bridi level to one or more sumti. However, Example 16.94 and related examples are not sumti negations, but rather expand to form two logically connected sentences. In such a situation, DeMorgan's Law must be applied. For instance, Example 16.95 expands to:

Example 16.96. 

ge la djan. la paris. na klama
[It-is-true-that] both that-named John, to-that-named Paris, [false] goes,
gi la djan. la rom. na klama
and that-named John, to-that-named Rome, [false] goes.

The ga and gi, meaning either-or, have become ge and gi, meaning both-and, as a consequence of moving the negators into the individual bridi.

Here is another example of DeMorgan's Law in action, involving bridi-tail logical connection (explained in Section 14.9):

Example 16.97. 

la djein. le zarci na ge dzukla gi bajrykla
that-named Jane to-the market [false] both walks and runs.

Example 16.98. 

la djein. le zarci ganai dzukla ginai bajrykla
that-named Jane to-the market either-([false] walks) or-([false] runs.
that-named Jane to-the market if walks then-([false] runs).

(Placing le zarci before the selbri makes sure that it is properly associated with both parts of the logical connection. Otherwise, it is easy to erroneously leave it off one of the two sentences.)

It is wise, before freely doing transformations such as the one from Example 16.97 to Example 16.98, that you become familiar with expanding logical connectives to separate sentences, transforming the sentences, and then recondensing. Thus, you would prove the transformation correct by the following steps. By moving its na to the beginning of the prenex as a naku, Example 16.97 becomes:

Example 16.99. 

naku zo'u la djein. le zarci
It-is-false-that : that-named Jane to-the market
ge dzukla gi bajrykla
(both walks and runs).

And by dividing the bridi with logically connected selbri into two bridi,

Example 16.100. 

naku zo'u ge la djein. le zarci cu dzukla
It-is-false that: both (that-named Jane to-the market walks)
gi la djein. le zarci cu bajrykla
and (that-named Jane to-the market runs).

is the result.

At this expanded level, we apply DeMorgan's Law to distribute the negation in the prenex across both sentences, to get

Example 16.101. 

ga la djein. le zarci na dzukla
Either that-named Jane to-the market [false] walks,
gi la djein. le zarci na bajrykla
or that-named Jane to-the market [false] runs.

which is the same as

Example 16.102. 

ganai la djein. le zarci cu dzukla
If that-named Jane to-the market walks,
ginai la djein. le zarci cu bajrykla
then-([false] that-named Jane to-the market runs).

If Jane walks to the market, then she doesn't run.

which then condenses down to Example 16.98.

DeMorgan's Law must also be applied to internal naku negations:

Example 16.103. 

ga la paris. gi la rom.
(Either that-named Paris or that-named Rome)
naku se klama la djan.
is-not gone-to-by that-named John.

Example 16.104. 

la djan. naku klama ge
that-named John doesn't go-to both
la paris. gi la rom.
that-named Paris and that-named Rome.

That Example 16.103 and Example 16.104 mean the same should become evident by studying the English. It is a good exercise to work through the Lojban and prove that they are the same.