MEX paper, draft 1.0, part #2 of 5

[cowan (John Cowan)]

3. Simple Infix Expressions and Equations

Let us begin at the beginning:  one plus one equals two.  In Lojban,
that sentence translates to:

	li pa su'i pa du li re
	the-number one plus one equals the-number two.

This mekso sentence is a regular Lojban bridi that exploits mekso features.
"du" is the predicate meaning "x1 is mathematically equal to x2".
It is a cmavo for conciseness, but it has the same grammatical uses as
any brivla.

"li" is the number article.  It is required whenever a sentence talks
about numbers as numbers, as opposed to the use of numbers to quantify
things.  For example:

	le ci prenu
	the three persons

requires no "li" article, because the "ci" is being used to specify the
number of "prenu".  However, the sentence

	li ci cu kancu le'i prenu
	the-number three counts the-set-of persons [I have in mind]

requires "li" because "ci" is being used as a sumti.

"su'i" is the first purely mekso cmavo.  It belongs to selma'o VUhU,
which is composed of mathematical operators, and means "plus".  As
mentioned before, it is distinct from "ma'u" which means the plus
sign as an indication of a positive number:

	li ma'u pa su'i ni'u pa du li no
	the-number plus-sign one plus minus-sign one equals the-number zero
	+1 + -1 = 0

Of course, it is legal to have complex mekso on both sides of "du":

	li mu su'i pa du li ci su'i ci
	the-number five plus one equals the-number three plus three
	5 + 1 = 3 + 3

Why don't we say "li mu su'i li pa" rather than just "li mu su'i pa"?
The answer is that VUhU operators connect mekso operands (numbers,
in this case), not general sumti.  "li" is used to convert the entire
mekso into a sumti, which then plays the roles applicable to other sumti:
in this case, filling the place structures of a bridi.

By default, Lojban mathematics is like calculator mathematics: there is
no notion of "priority of operators".  Consider the following example
("pi'i" means "times", the multiplication operator):

	li ci su'i vo pi'i mu du li reci
	the-number three plus four times five equals the-number two-three
	3 + 4 x 5 = 23

Is this Lojban statement true?  No!  "3 + 4 x 5" is indeed 23, because the
conventions of mathematics state that multiplication takes precedence over
addition; that is, the multiplication "4 x 5" is done first, and then
the addition "3 + 20".  But VUhU operators by default are done left to
right, like other Lojban grouping, and so the correct statement would be:

	li ci su'i vo pi'i mu du li cimu
	the-number three plus four times five equals the-number three-five
	3 + 4 x 5 = 35

Here we do the "3 + 4" first, and then the "7 x 5" second, leading to the
result 35.  While possessing the advantage of simplicity, this violates
the design goal of matching the standards of international mathematics.
What can be done?

There are three solutions, all of which will probably be used to some degree.
The first solution is to ignore the problem.  People will say "li ci
su'i vo pi'i mu" and mean 23 by it, because the notion that multiplication
takes precedence over addition is too deeply ingrained to be eradicated
by the Lojban parser with its total ignorance of semantics.  This essentially
allows semantics to dominate syntax in this one area.

(Why not hard-wire the precedences into the grammar, as is done in computer
programming languages?  Essentially because there are too many operators,
known and unknown, with levels of precedence that vary according to usage.
The programming language 'C' has 13 levels of precedence, and its list
of operators is not extensible.  For Lojban this approach is just not
practical.)

The second solution is to use explicit means to specify the precedence
of operators.  This approach is fully general, but clumsy, and will be
explained in a later section.

The third solution is simple but not very general.  It builds on a
mechanism used elsewhere in the language: the short-scope cmavo "bo".
When an operator is prefixed with "bo", it becomes automatically of
higher precedence than other operators not so prefixed.  Thus,

	li ci su'i vo bo pi'i mu du li reci
	the-number three plus four-times-five equals the-number two-three
	3 + 4 x 5 = 23

is a truthful Lojban bridi.

In addition, of course, Lojban has the mathematical parentheses "vei" and
"ve'o", which can be used just like their written equivalents "(" and ")"
to group things in any way desired:

	li vei ny. su'i pa ve'o pi'i ve'i ny. su'i pa
		du li ny. bo te'a re su'i re bo pi'i ny. su'i pa
	the-number ( n plus one ) times ( n plus one
		equals the-number n-power-two plus two-times-n plus 1
	                  2
	(n + 1)(n + 1) = n  + 2n + 1

There are several new things here:  "te'a" means "raised to the power",
and we also see the use of the lerfu "ny", representing the letter "n".
In mekso, letters mean just what they mean in ordinary mathematics:
variables.  The parser will accept a string of letters as the equivalent
of a single letter, in agreement with computer science; "abc" is a single
variable, not the equivalent of "a x b x c".

Note that the explicit operator "pi'i" is required in the Lojban verbal
form whereas multiplication is implicit in the symbolic form.  Note also
that "ve'o" is an elidable terminator, and is actually elided here before
the "du".