14.2. The Four basic vowels

Lojban regards four of these 16 truth functions as fundamental, and assigns them the four vowels A, E, O, and U. These letters do not represent actual cmavo or selma'o, but rather a component vowel from which actual logical-connective cmavo are built up, as explained in the next section. Here are the four vowels, their truth tables, and rough English equivalents:

A TTTF or, and/or
E TFFF and
O TFFT if and only if
U TTFF whether or not

More precisely:

A is true if either or both sentences are true
E is true if both sentences are true, but not otherwise
O is true if the sentences are both true or both false
U is true if the first sentence is true, regardless of the truth value of the second sentence

With the four vowels, the ability to negate either sentence, and the ability to exchange the sentences, as if their order had been reversed, we can create all of the 16 possible truth functions except TTTT and FFFF, which are fairly useless anyway. The following table illustrates how to create each of the 14 remaining truth functions:

TTFTA with second sentence negated
TFTTA with first sentence negated
TFTFU with sentences exchanged
FTTTA with both sentences negated
FTTFO with either first or second negated (not both)
FTFTU with sentences exchanged and then second negated
FTFFE with second sentence negated
FFTTU with first sentence negated
FFTFE with first sentence negated
FFFTE with both sentences negated

Note that exchanging the sentences is only necessary with U. The three other basic truth functions are commutative; that is, they mean the same thing regardless of the order of the component sentences. There are other ways of getting some of these truth tables; these just happen to be the methods usually employed.