Chapter 14: If Wishes Were Horses: The Lojban Connective System

7. More than two propositions

So far we have seen logical connectives used to connect exactly two sentences. How about connecting three or more? Is this possible in Lojban? The answer is yes, subject to some warnings and some restrictions.

Of the four primitive truth functions A, E, O, and U, all but O have the same truth values no matter how their component sentences are associated in pairs. Therefore,

✥7.1  mi dotco .ije mi ricfu .ije mi nanmu
I am-German.  And I am-rich.  And I am-a-man.

means that all three component sentences are true. Likewise,

✥7.2  mi dotco .ija mi ricfu .ija mi nanmu
I am-German.  Or I am-rich.  Or I am-a-man.

means that one or more of the component sentences is true.

O, however, is different. Working out the truth table for

✥7.3  mi dotco .ijo mi ricfu
    .ijo mi nanmu
I am-German.  If-and-only-if I am-rich.
    If-and-only-if I am-a-man.

shows that ✥7.3 does not mean that either I am all three of these things or none of them; instead, an accurate translation would be:

Of the three properties — German-ness, wealth, and manhood -- I possess either exactly one or else all three.
Because of the counterintuitiveness of this outcome, it is safest to avoid O with more than two sentences. Likewise, the connectives which involve negation also have unexpected truth values when used with more than two sentences.

In fact, no combination of logical connectives can produce the “all or none” interpretation intended (but not achieved) by ✥7.3 without repeating one of the bridi. See ✥8.11.

There is an additional difficulty with the use of more than two sentences. What is the meaning of:

✥7.4  mi nelci la djan. .ije mi nelci la martas.
    .ija mi nelci la meris.
I like John.  And I like Martha.
    Or I like Mary.

Does this mean:

✥7.5  I like John, and I like
    either Martha or Mary or both.

Or is the correct translation:

✥7.6  Either I like John and I like Martha,
    or I like Mary, or both.

✥7.6 is the correct translation of ✥7.4. The reason is that Lojban logical connectives pair off from the left, like many constructs in the language. This rule, called the left-grouping rule, is easy to forget, especially when intuition pulls the other way. Forethought connectives are not subject to this problem:

✥7.7  ga ge mi nelci la djan. gi mi nelci la martas.
    gi mi nelci la meris.
Either (Both I like John and I like Martha)
    or I like Mary.

is equivalent in meaning to ✥7.4, whereas

✥7.8  ge mi nelci la djan. gi
    ga mi nelci la martas. gi mi nelci la meris.
Both I like John and
    (Either I like Martha or I like Mary).

is not equivalent to ✥7.4, but is instead a valid translation into Lojban, using forethought, of ✥7.5.