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.