Chapter 18: lojbau mekso: Mathematical Expressions in Lojban
A few other points:
“se” can be used to convert an operator as if it were a selbri, so that its arguments are exchanged. For example:
✥21.1 li ci se vu'u vo du li pa the-number three (inverse) minus four equals the-number one. 3 subtracted from 4 equals 1.
The other converters of selma'o SE can also be used on operators with more than two operands, and they can be compounded to create (probably unintelligible) operators as needed.
Members of selma'o NAhE are also legal on an operator to produce a scalar negation of it. The implication is that some other operator would apply to make the bridi true:
✥21.2 li ci na'e su'i vo du li pare the-number 3 non-plus 4 equals the-number 12
✥21.3 li ci to'e vu'u re du li mu the-number 3 opposite-of-minus 2 equals the-number 5
The sense in which “plus” is the opposite of “minus” is not a mathematical but rather a linguistic one; negated operators are defined only loosely.
“la'e” and “lu'e” can be used on operands with the usual semantics to get the referent of or a symbol for an operand. Likewise, a member of selma'o NAhE followed by “bo” serves to scalar-negate an operand, implying that some other operand would make the bridi true:
✥21.4 li re su'i re du li na'ebo mu the-number 2 plus 2 equals the-number non-5. 2 + 2 = something other than 5.
The digits 0-9 have rafsi, and therefore can be used in making lujvo. Additionally, all the rafsi have CVC form and can stand alone or together as names:
✥21.5 la zel. poi gunta la tebes. pu nanmu those-named “Seven” who attack that-named “Thebes” [past] are-men The Seven Against Thebes were men.
Of course, there is no guarantee that the name “zel.” is connected with the number rafsi: an alternative which cannot be misconstrued is:
✥21.6 la zemei poi gunta la tebes. pu nanmu those-named-the Sevensome who attack Thebes [past] are-men.
Certain other members of PA also have assigned rafsi: “so'a”, “so'e”, “so'i”, “so'o”, “so'u”, “da'a”, “ro”, “su'e”, “su'o”, “pi”, and “ce'i”. Furthermore, although the cmavo “fi'u” does not have a rafsi as such, it is closely related to the gismu “frinu”, meaning “fraction”; therefore, in a context of numeric rafsi, you can use any of the rafsi for “frinu” to indicate a fraction slash.
A similar convention is used for the cmavo “cu'o” of selma'o MOI, which is closely related to “cunso” (probability); use a rafsi for “cunso” in order to create lujvo based on “cu'o”. The cmavo “mei” and “moi” of MOI have their own rafsi, two each in fact: “mem”/“mei” and “mom”/“moi” respectively.
The grammar of mekso as described so far imposes a rigid distinction between operators and operands. Some flavors of mathematics (lambda calculus, algebra of functions) blur this distinction, and Lojban must have a method of doing the same. An operator can be changed into an operand with “ni'enu'a”, which transforms the operator into a matching selbri and then the selbri into an operand.
To change an operand into an operator, we use the cmavo “ma'o”, already introduced as a means of changing a lerfu string such as “fy.” into an operator. In fact, “ma'o” can be followed by any mekso operand, using the elidable terminator “te'u” if necessary.
There is a potential semantic ambiguity in “ma'o fy. [te'u]” if “fy.” is already in use as a variable: it comes to mean “the function whose value is always 'f”'. However, mathematicians do not normally use the same lerfu words or strings as both functions and variables, so this case should not arise in practice.