Chapter 18: lojbau mekso: Mathematical Expressions in Lojban

17. Logical and non-logical connectives within mekso

The following cmavo are discussed in this section:

.abu    BY  letter “a”
by  BY  letter “b”
cy  BY  letter “c”
fe'a    VUhU    nth root of (default square root)
lo'o    LOhO    terminator for LI

As befits a logical language, Lojban has extensive provision for logical connectives within both operators and operands. Full details on logical and non-logical connectives are provided in Chapter 14. Operands are connected in afterthought with selma'o A and in forethought with selma'o GA, just like sumti. Operators are connected in afterthought with selma'o JA and in forethought with selma'o GUhA, just like tanru components. This parallelism is no accident.

In addition, A+BO and A+KE constructs are allowed for grouping logically connected operands, and “ke … ke'e” is allowed for grouping logically connected operators, although there are no analogues of tanru among the operators.

Despite the large number of rules required to support this feature, it is of relatively minor importance in the mekso scheme of things. ✥17.1 exhibits afterthought logical connection between operands:

✥17.1    vei ci .a vo ve'o prenu cu klama le zarci
( three or four ) people go-to the market.

✥17.2 is equivalent in meaning, but uses forethought connection:

✥17.2    vei ga ci gi vo ve'o prenu cu klama le zarci
( either 3 or 4 ) people go-to the market.

Note that the mekso here are being used as quantifiers. Lojban requires that any mekso other than a simple number be enclosed in parentheses when used as a quantifier. This rule prevents ambiguities that do not exist when using “li”.

By the way, “li” has an elidable terminator, “lo'o”, which is needed when a “li” sumti is followed by a logical connective that could seem to be within the mekso. For example:

✥17.3    li re su'i re du
    li vo lo'o .onai lo nalseldjuno namcu
the-number two plus two equals
    the-number four or else a non-known number.

Omitting the “lo'o” would cause the parser to assume that another operand followed the “.onai” and reject “lo” as an invalid operand.

Simple examples of logical connection between operators are hard to come by. A contrived example is:

✥17.4    li re su'i je pi'i re du li vo
the-number two plus and times two equals the-number four.
2 + 2 = 4 and 2 × 2 = 4.

The forethought-connection form of ✥17.4 is:

✥17.5    li re ge su'i gi pi'i re
    du li vo
the-number two both plus and times two
    equals the-number four.
Both 2 + 2 = 4 and 2 × 2 = 4.

Here is a classic example of operand logical connection:

✥17.6    go li .abu bi'epi'i vei xy. te'a re ve'o
    su'i by. bi'epi'i xy. su'i cy.
    du li no
gi li xy. du li vei va'a by. ku'e
    su'i ja vu'u fe'a
    vei by. bi'ete'a re vu'u
    vo bi'epi'i .abu bi'epi'i cy. ve'o [ku'e] ve'o
    fe'i re bi'epi'i .abu
if-and-only-if the-number “a”-times-( “x” power two )
    plus “b”-times-“x” plus “c”
    equals the-number zero
then the-number x equals the-number [ the-negation-of( b )
    plus or minus the-root-of
    ( “b”-power-2 minus
    four-times-“a”-times-“c” ) ]
    divided-by two-times-“a”.
Iff ax2  + bx + c = 0,
    then x =-b ± Ö
(b2  - 4ac)
        -----------------------
            2a

Note the mixture of styles in ✥17.6: the negation of b and the square root are represented by forethought and most of the operator precedence by prefixed “bi'e”, but explicit parentheses had to be added to group the numerator properly. In addition, the square root parentheses cannot be removed here in favor of simple “fe'a” and “ku'e” bracketing, because infix operators are present in the operand. Getting ✥17.6 to parse perfectly using the current parser took several tries: a more relaxed style would dispense with most of the “bi'e” cmavo and just let the standard precedence rules be understood.

Non-logical connection with JOI and BIhI is also permitted between operands and between operators. One use for this construct is to connect operands with “bi'o” to create intervals:

✥17.7    li no ga'o bi'o ke'i pa
the-number zero (inclusive) from-to (exclusive) one
[0,1)
the numbers from zero to one, including zero
    but not including one

Intervals defined by a midpoint and range rather than beginning and end points can be expressed by “mi'i”:

✥17.8    li pimu ga'o mi'i ke'i pimu
the-number 0.5 plus-or-minus 0.5

which expresses the same interval as ✥17.7. Note that the “ga'o” and “ke'i” still refer to the endpoints, although these are now implied rather than expressed. Another way of expressing the same thing:

✥17.9    li pimu su'i ni'upimu bi'o ma'upimu
the-number 0.5 plus [-0.5 from-to +0.5]

Here we have the sum of a number and an interval, which produces another interval centered on the number. As ✥17.9 shows, non-logical (or logical) connection of operands has higher precedence than any mekso operator.

You can also combine two operands with “ce'o”, the sequence connective of selma'o JOI, to make a compound subscript:

✥17.10  xy. xi vei by. ce'o dy. [ve'o]
“x” sub ( “b” sequence “d” )
xb,d}