Chapter 18. lojbau mekso: Mathematical Expressions in Lojban

lojbau mekso (“Lojbanic mathematical-expression”) is the part of the Lojban language that is tailored for expressing statements of a mathematical character, or for adding numerical information to non-mathematical statements. Its formal design goals include:

Goal 1 requires that mekso not be constrained to a single notation such as Polish notation or reverse Polish notation, but make provision for all forms, with the most commonly used forms the most easily used.

Goal 2 requires the provision of several conversion mechanisms, so that the boundary between mekso and full Lojban can be crossed from either side at many points.

Goal 3 is the most subtle. Written mathematical expression is culturally unambiguous, in the sense that mathematicians in all parts of the world understand the same written texts to have the same meanings. However, international mathematical notation does not prescribe unique forms. For example, the expression

contains omitted multiplication operators, but there are other possible interpretations for the strings 3x and 2y than as mathematical multiplication. Therefore, the Lojban verbal (spoken and written) form of Example 18.1 must not omit the multiplication operators.

The remainder of this chapter explains (in as much detail as is currently possible) the mekso system. This chapter is by intention complete as regards mekso components, but only suggestive about uses of those components – as of now, there has been no really comprehensive use made of mekso facilities, and many matters must await the test of usage to be fully clarified.

The following cmavo are discussed in this section:

The simplest kind of mekso are numbers, which are cmavo or compound cmavo. There are cmavo for each of the 10 decimal digits, and numbers greater than 9 are made by stringing together the cmavo. Some examples:

Therefore, there are no separate cmavo for “ten”, “hundred”, etc.

There is a pattern to the digit cmavo (except for no, 0) which is worth explaining. The cmavo from 1 to 5 end in the vowels a, e, i, o, u respectively; and the cmavo from 6 to 9 likewise end in the vowels a, e, i, and o respectively. None of the digit cmavo begin with the same consonant, to make them easy to tell apart in noisy environments.

The following cmavo are discussed in this section:

A number can be given an explicit sign by the use of ma'u and ni'u, which are the positive and negative signs as distinct from the addition, subtraction, and negation operators. For example:

Grammatically, the signs are part of the number to which they are attached. It is also possible to use ma'u and ni'u by themselves as numbers; the meaning of these numbers is explained in Section 18.8.

Various numerical punctuation marks are likewise expressed by cmavo, as illustrated in the following examples:

(In some cultures, a comma is used instead of a period in the symbolic version of Example 18.6; pi is still the Lojban representation for the decimal point.)

Example 18.7 is the name of the number two-sevenths; it is not the same as “the result of 2 divided by 7” in Lojban, although numerically these two are equal. If the denominator of the fraction is present but the numerator is not, the numerator is taken to be 1, thus expressing the reciprocal of the following number:

Note that the ra'e marks unambiguously where the repeating portion “142857” begins.

(In some cultures, spaces are used in the symbolic representation of Example 18.11; ki'o is still the Lojban representation.)

It is also possible to have less than three digits between successive ki'o s, in which case zeros are assumed to have been elided:

In the same way, ki'o can be used after pi to divide fractions into groups of three:

The following cmavo are discussed in this section:

The last cmavo is the same as the fraction sign cmavo: a fraction sign with neither numerator nor denominator represents the golden ratio.

Numbers can have any of these digit, punctuation, and special-number cmavo of Sections 2, 3, and 4 in any combination:

Note that ka'o is both a special number (meaning “i”) and a number punctuation mark (separating the real and the imaginary parts of a complex number).

The special numbers pai and te'o are mathematically important, which is why they are given their own cmavo:

However, many combinations are as yet undefined:

Example 18.21 is not “1 minus 2”, which is represented by a different cmavo sequence altogether. It is a single number which has not been assigned a meaning. There are many such numbers which have no well-defined meaning; they may be used for experimental purposes or for future expansion of the Lojban number system.

It is possible, of course, that some of these “oddities” do have a meaningful use in some restricted area of mathematics. A mathematician appropriating these structures for specialized use needs to consider whether some other branch of mathematics would use the structure differently.

More information on numbers may be found in Section 18.8 to Section 18.12.

The following cmavo are discussed in this section:

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

Example 18.22, a 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. Outside mathematical contexts, du means “x1 is identical with x2” or “x1 is the same object as x2”.

The cmavo li is the number article. It is required whenever a sentence talks about numbers as numbers, as opposed to using numbers to quantify things. For example:

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

requires li because ci is being used as a sumti. Note that this is the way in which measurements are stated in Lojban: all the predicates for units of length, mass, temperature, and so on have the measured object as the first place and a number as the second place. Using li for le in Example 18.23 would produce

which is grammatical but nonsensical: numbers are not persons.

The cmavo su'i belongs to selma'o VUhU, which is composed of mathematical operators, and means “addition”. As mentioned before, it is distinct from ma'u which means the positive sign as an indication of a positive number:

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

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 Example 18.27), not general sumti. li is used to make the entire mekso into a sumti, which then plays the roles applicable to other sumti: in Example 18.27, filling the places of a bridi

By default, Lojban mathematics is like simple calculator mathematics: there is no notion of “operator precedence”. Consider the following example, where pi'i means “times”, the multiplication operator:

Is the Lojban version of Example 18.28 true? No! “3 + 4 × 5” is indeed 23, because the usual conventions of mathematics state that multiplication takes precedence over addition; that is, the multiplication “4 × 5” is done first, giving 20, and only then the addition “3 + 20”. But VUhU operators by default are done left to right, like other Lojban grouping, and so a truthful bridi would be:

Here we calculate 3 + 4 first, giving 7, and then calculate 7 × 5 second, leading to the result 35. While possessing the advantage of simplicity, this result violates the design goal of matching the standards of 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 Lojban parsing, which totally ignores semantics. This convention 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 even extensible. For Lojban this approach is just not practical. In addition, hard-wired precedence could not be overridden in mathematical systems such as spreadsheets where the conventions are different.)

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 Section 18.20.

The third solution is simple but not very general. When an operator is prefixed with the cmavo bi'e (of selma'o BIhE), it becomes automatically of higher precedence than other operators not so prefixed. Thus,

is a truthful Lojban bridi. If more than one operator has a bi'e prefix, grouping is from the right; multiple bi'e prefixes on a single operator are not allowed.

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 expressions in any way desired:

There are several new usages in Example 18.31: te'a means “raised to the power”, and we also see the use of the lerfu word ny, representing the letter “n”. In mekso, letters stand for just what they do in ordinary mathematics: variables. The parser will accept a string of lerfu words (called a “lerfu string”) as the equivalent of a single lerfu word, in agreement with computer-science conventions; “abc” is a single variable, not the equivalent of “a × b × c”. (Of course, a local convention could state that the value of a variable like “abc”, with a multi-lerfu name, was equal to the values of the variables “a”, “b”, and “c” multiplied together.)

The explicit operator pi'i is required in the Lojban verbal form whereas multiplication is implicit in the symbolic form. Note that ve'o (the right parenthesis) is an elidable terminator: the first use of it in Example 18.31 is required, but the second use (marked by square brackets) could be elided. Additionally, the first bi'e (also marked by square brackets) is not necessary to get the proper grouping, but it is included here for symmetry with the other one.

The following cmavo are discussed in this section:

The infix form explained so far is reasonable for many purposes, but it is limited and rigid. It works smoothly only where all operators have exactly two operands, and where precedences can either be assumed from context or are limited to just two levels, with some help from parentheses.

But there are many operators which do not have two operands, or which have a variable number of operands. The preferred form of expression in such cases is the use of “forethought operators”, also known as Polish notation. In this style of writing mathematics, the operator comes first and the operands afterwards:

Note that the normally elidable number terminator boi is required after pa and re because otherwise the reading would be pareci= 123. It is not required after ci but is inserted here in brackets for the sake of symmetry. The only time boi is required is, as in Example 18.32, when there are two consecutive numbers or lerfu strings.

Forethought mekso can use any number of operands, in Example 18.32, three. How do we know how many operands there are in ambiguous circumstances? The usual Lojban solution is employed: an elidable terminator, namely ku'e. Here is an example:

where we know that va'a is a forethought operator because there is no operand preceding it.

va'a is the numerical negation operator, of selma'o VUhU. In contrast, vu'u is not used for numerical negation, but only for subtraction, as it always has two or more operands. Do not confuse va'a and vu'u, which are operators, with ni'u, which is part of a number.

In Example 18.33, the operator va'a and the terminator ku'e serve in effect as parentheses. (The regular parentheses vei and ve'o are NOT used for this purpose.) If the ku'e were omitted, the su'i zy would be swallowed up by the va'a forethought operator, which would then appear to have two operands, ny and su'i zy., where the latter is also a forethought expression.

Forethought mekso is also useful for matching standard functional notation. How do we represent “z = f(x)”? The answer is:

Again, no parentheses are used. The construct ma'o fy.boi is the equivalent of an operator, and appears in forethought here (although it could also be used as a regular infix operator). In mathematics, letters sometimes mean functions and sometimes mean variables, with only the context to tell which. Lojban chooses to accept the variable interpretation as the default, and uses the special flag ma'o to mark a lerfu string as an operator. The cmavo xy. and zy. are variables, but fy. is an operator (a function) because ma'o marks it as such. The boi is required because otherwise the xy. would look like part of the operator name. (The use of ma'o can be generalized from lerfu strings to any mekso operand: see Section 18.21.)

When using forethought mekso, the optional marker pe'o may be placed in front of the operator. This usage can help avoid confusion by providing clearly marked pe'o and ku'e pairs to delimit the operand list. Example 18.32 to Example 18.34, respectively, with explicit pe'o and ku'e:

Note: When using forethought mekso, be sure that the operands really are operands: they cannot contain regular infix expressions unless parenthesized with vei and ve'o. An earlier version of the complex Example 18.119 came to grief because I forgot this rule.

So far our examples have been isolated mekso (it is legal to have a bare mekso as a sentence in Lojban) and equation bridi involving du. What about inequalities such as “x < 5”? The answer is to use a bridi with an appropriate selbri, thus:

Here is a partial list of selbri useful in mathematical bridi:

Note the difference between dunli and du; dunli has a third place that specifies the kind of equality that is meant. du refers to actual identity, and can have any number of places:

Lojban bridi can have only one predicate, so the du is not repeated.

Any of these selbri may usefully be prefixed with na, the contradictory negation cmavo, to indicate that the relation is false:

As usual in Lojban, negated bridi say what is false, and do not say anything about what might be true.

The following cmavo are discussed in this section:

Not all the cmavo of PA represent numbers in the usual mathematical sense. For example, the cmavo ro means “all” or “each”. This number does not have a definite value in the abstract: li ro is undefined. But when used to count or quantify something, the parallel between ro and pa is clearer:

Example 18.41 might be true, whereas Example 18.42 is almost certainly false.

The cmavo so'a, so'e, so'i, so'o, and so'u represent a set of indefinite numbers less than ro. As you go down an alphabetical list, the magnitude decreases:

The English equivalents are only rough: the cmavo provide space for up to five indefinite numbers between ro and no, with a built-in ordering. In particular, so'e does not mean “most” in the sense of “a majority” or “more than half”.

Each of these numbers, plus ro, may be prefixed with pi (the decimal point) in order to make a fractional form which represents part of a whole rather than some elements of a totality. piro therefore means “the whole of”:

Similarly, piso'a means “almost the whole of”; and so on down to piso'u, “a tiny part of”. These numbers are particularly appropriate with masses, which are usually measured rather than counted, as Example 18.48 shows.

In addition to these cmavo, there is no'o, meaning “the typical value”, and pino'o, meaning “the typical portion”: Sometimes no'o can be translated “the average value”, but the average in question is not, in general, a mathematical mean, median, or mode; these would be more appropriately represented by operators.

da'a is a related cmavo meaning “all but”:

Example 18.52 is similar in meaning to Example 18.43.

If no number follows da'a, then pa is assumed; da'a by itself means “all but one”, or in ordinal contexts “all but the last”:

(The use of da'a means that Example 18.53 does not require that all rats can eat themselves, but does allow it. Each rat has one rat it cannot eat, but that one might be some rat other than itself. Context often dictates that “itself” is, indeed, the “other” rat.)

As mentioned in Section 18.3, ma'u and ni'u are also legal numbers, and they mean “some positive number” and “some negative number” respectively.

All of the numbers discussed so far are objective, even if indefinite. If there are exactly six superpowers (rairgugde, “superlative-states”) in the world, then ro rairgugde means the same as xa rairgugde. It is often useful, however, to express subjective indefinite values. The cmavo rau (enough), du'e (too many), and mo'a (too few) are then appropriate:

Like the so'a-series, rau, du'e, and mo'a can be preceded by pi; for example, pirau means “a sufficient part of.”

Another possibility is that of combining definite and indefinite numbers into a single number. This usage implies that the two kinds of numbers have the same value in the given context:

Example 18.59 assumes a mostly monogamous culture by stating that three is “many”.

The following cmavo are discussed in this section:

The cmavo ji'i (of selma'o PA) is used in several ways to indicate approximate or rounded numbers. If it appears at the beginning of a number, the whole number is approximate:

If ji'i appears in the middle of a number, all the digits following it are approximate:

If ji'i appears at the end of a number, it indicates that the number has been rounded. In addition, it can then be followed by a sign cmavo (ma'u or ni'u), which indicate truncation towards positive or negative infinity respectively.

Example 18.62 through Example 18.64 are all approximations to te'o (exponential e). ji'i can also appear by itself, in which case it means “approximately the typical value in this context”.

The four cmavo su'e, su'o, me'i, and za'u, also of selma'o PA, express inexact numbers with upper or lower bounds:

Each of these is a subtly different claim: Example 18.66 is true of two or any greater number, whereas Example 18.68 requires three persons or more. Likewise, Example 18.65 refers to zero, one, or two; Example 18.67 to zero or one. (Of course, when the context allows numbers other than non-negative integers, me'i re can be any number less than 2, and likewise with the other cases.) The exact quantifier, “exactly 2, neither more nor less” is just re. Note that su'ore is the exact Lojban equivalent of English plurals.

If no number follows one of these cmavo, pa is understood: therefore,

is a meaningful claim.

Like the numbers in Section 18.8, all of these cmavo may be preceded by pi to make the corresponding quantifiers for part of a whole. For example, pisu'o means “at least some part of”. The quantifiers ro, su'o, piro, and pisu'o are particularly important in Lojban, as they are implicitly used in the descriptions introduced by the cmavo of selma'o LA and LE, as explained in Section 6.7. Descriptions in general are outside the scope of this chapter.

The following cmavo are discussed in this section:

In normal contexts, Lojban assumes that all numbers are expressed in the decimal (base 10) system. However, other bases are possible, and may be appropriate in particular circumstances.

To specify a number in a particular base, the VUhU operator ju'u is suitable:

Here, the final pa no is assumed to be base 10, as usual; so is the base specification. (The base may also be changed permanently by a metalinguistic specification; no standard way of doing so has as yet been worked out.)

Lojban has digits for representing bases up to 16, because 16 is a base often used in computer applications. In English, it is customary to use the letters A-F as the base 16 digits equivalent to the numbers ten through fifteen. In Lojban, this ambiguity is avoided:

Note the pattern in the cmavo: the diphthongs au, ei, ai are used twice in the same order. The digits for A to D use consonants different from those used in the decimal digit cmavo; E and F unfortunately overlap 2 and 4 – there was simply not enough available cmavo space to make a full differentiation possible. The cmavo are also in alphabetical order.

The base point pi is used in non-decimal bases just as in base 10:

Since ju'u is an operator of selma'o VUhU, it is grammatical to use any operand as the left argument. Semantically, however, it is undefined to use anything but a numeral string on the left. The reason for making ju'u an operator is to allow reference to a base which is not a constant.

There are some numerical values that require a “base” that varies from digit to digit. For example, times represented in hours, minutes, and seconds have, in effect, three “digits”: the first is base 24, the second and third are base 60. To express such numbers, the compound base separator pi'e is used:

Each digit sequence separated by instances of pi'e is expressed in decimal notation, but the number as a whole is not decimal and can only be added and subtracted by special rules:

Of course, only context tells you that the first part of the numbers in Example 18.74 and Example 18.75 is hours, the second minutes, and the third seconds.

The same mechanism using pi'e can be used to express numbers which have a base larger than 16. For example, base-20 Mayan mathematics might use digits from no to paso, each separated by pi'e:

Carefully note the difference between:

which is equal to ten, and:

which is equal to twenty.

Both pi and pi'e can be used to express large-base fractions:

pi'e is also used where the base of each digit is vague, as in the numbering of the examples in this chapter:

The following cmavo are discussed in this section:

Lojban possesses a special category of selbri which are based on mekso. The simplest kind of such selbri are made by suffixing a member of selma'o MOI to a number. There are five members of MOI, each of which serves to create number-based selbri with specific place structures.

The cmavo mei creates cardinal selbri. The basic place structure is:

A cardinal selbri interrelates a set with a given number of members, the mass formed from that set, and the individuals which make the set up. The mass argument is placed first as a matter of convenience, not logical necessity.

Some examples:

Here, the mass of my rats is said to have three components; that is, I have three rats.

Another example, with one element this time:

In Example 18.82, mi refers to a mass, “the mass consisting of me”. Personal pronouns are vague between masses, sets, and individuals.

However, when the number expressed before -mei is an objective indefinite number of the kind explained in Section 18.8, a slightly different place structure is required:

An example:

In Example 18.83, the x2 and x3 places are vacant, and the x4 place is filled by lo'i ratcu, which (because no quantifiers are explicitly given) means “the whole of the set of all those things which are rats”, or simply “the set of all rats.”

In Example 18.84, the conversion cmavo se swaps the x1 and the x2 places, so that the new x1 is the set. The x4 set is unspecified, so the implication is that the rats are “many” with respect to some unspecified comparison set.

More explanations about the interrelationship of sets, masses, and individuals can be found in Section 6.3.

The cmavo moi creates ordinal selbri. The place structure is:

Some examples:

Example 18.87 means, in the appropriate context, that my position in line is sufficiently far to the front that I will get a seat for the movie.

The cmavo si'e creates portion selbri. The place structure is:

Some examples:

The cmavo cu'o creates probability selbri. The place structure is:

The number must be between 0 and 1 inclusive. For example:

The cmavo va'e creates a scale selbri. The place structure is:

If the scale is granular rather than continuous, a form like cifi'uxa (3/6) may be used; in this case, 3/6 is not the same as 1/2, because the third position on a scale of six positions is not the same as the first position on a scale of two positions. Here is an example:

When the quantifier preceding any MOI cmavo includes the subjective numbers rau, du'e, or mo'a (enough, too many, too few) then an additional place is added for “by standard”. For example:

The extra place (which for -mei is the x4 place labeled by fo) is provided rather than using a BAI tag such as ma'i because a specification of the standard for judgment is essential to the meaning of subjective words like “enough”.

This place is not normally explicit when using one of the subjective numbers directly as a number. Therefore, du'e ratcu means “too many rats” without specifying any standard.

It is also grammatical to substitute a lerfu string for a number:

More complex mekso cannot be placed directly in front of MOI, due to the resulting grammatical ambiguities. Instead, a somewhat artificial form of expression is required.

The cmavo me (of selma'o ME) has the function of making a sumti into a selbri. A whole me construction can have a member of MOI added to the end to create a complex mekso selbri:

Here the mekso ny. su'i pa is made into a sumti (with li) and then changed into a mekso selbri with me and me'u moi. The elidable terminator me'u is required here in order to keep the pa and the moi separate; otherwise, the parser will combine them into the compound pamoi and reject the sentence as ungrammatical.

It is perfectly possible to use non-numerical sumti after me and before a member of MOI, producing strange results indeed:

Note: the elidable terminator boi is not used between a number and a member of MOI. As a result, the me'u in Example 18.93 could also be replaced by a boi, which would serve the same function of preventing the pa and moi from joining into a compound.

The following cmavo is discussed in this section:

The cmavo xo, a member of selma'o PA, is used to ask questions whose answers are numbers. Like most Lojban question words, it fills the blank where the answer should go. (See Section 19.5 for more on Lojban questions.)

xo can also be combined with other digits to ask questions whose answers are already partly specified. This ability could be very useful in writing tests of elementary arithmetical knowledge:

to which the correct reply would be mu, or 5. The ability to utter bare numbers as grammatical Lojban sentences is primarily intended for giving answers to xo questions. (Another use, obviously, is for counting off physical objects one by one.)

The following cmavo is discussed in this section:

Subscripting is a general Lojban feature, not used only in mekso; there are many things that can logically be subscripted, and grammatically a subscript is a free modifier, usable almost anywhere. In particular, of course, mekso variables (lerfu strings) can be subscripted:

Subscripts always begin with the flag xi (of selma'o XI). xi may be followed by a number, a lerfu string, or a general mekso expression in parentheses:

Note that subscripts attached directly to lerfu words (variables) generally need a boi terminating the variable. Free modifiers, of which subscripts are one variety, generally require the explicit presence of an otherwise elidable terminator.

There is no standard way of handling superscripts (other than those used as exponents) or for subscripts or superscripts that come before the main expression. If necessary, further cmavo could be assigned to selma'o XI for these purposes.

The elidable terminator for a subscript is that for a general number or lerfu string, namely boi. By convention, a subscript following another subscript is taken to be a sub-subscript:

See Example 18.123 for the standard method of specifying multiple subscripts on a single object.

More information on the uses of subscripts may be found in Section 19.6.

The following cmavo are discussed in this section:

The infix operators presented so far have always had exactly two operands, and for more or fewer operands forethought notation has been required. However, it is possible to use an operator in infix style even though it has more or fewer than two operands, through the use of a pair of tricks: the null operand tu'o and the null operator ge'a. The first is suitable when there are too few operands, the second when there are too many. For example, suppose we wanted to express the numerical negation operator va'a in infix form. We would use:

The tu'o fulfills the grammatical requirement for a left operand for the infix use of va'a, even though semantically none is needed or wanted.

Finding a suitable example of ge'a requires exhibiting a ternary operator, and ternary operators are not common. The operator gei, however, has both a binary and a ternary use. As a binary operator, it provides a terse representation of scientific (also called “exponential”) notation. The first operand of gei is the exponent, and the second operand is the mantissa or fraction:

Why are the arguments to gei in reverse order from the conventional symbolic notation? So that gei can be used in forethought to allow easy specification of a large (or small) imprecise number:

Note, however, that although 10 is far and away the most common exponent base, it is not the only possible one. The third operand of gei, therefore, is the base, with 10 as the default value. Most computers internally store so-called “floating-point” numbers using 2 as the exponent base. (This has nothing to do with the fact that computers also represent all integers in base 2; the IBM 360 series used an exponent base of 16 for floating point, although each component of the number was expressed in base 2.) Here is a computer floating-point number with a value of 40:

The following cmavo are discussed in this section:

A mathematical vector is a list of numbers, and a mathematical matrix is a table of numbers. Lojban considers matrices to be built up out of vectors, which are in turn built up out of operands.

jo'i, the only cmavo of selma'o JOhI, is the vector indicator: it has a syntax reminiscent of a forethought operator, but has very high precedence. The components must be simple operands rather than full expressions (unless parenthesized). A vector can have any number of components; te'u is the elidable terminator. An example:

Vectors can be combined into matrices using either pi'a, the matrix row operator, or sa'i, the matrix column operator. The first combines vectors representing rows of the matrix, and the second combines vectors representing columns of the matrix. Both of them allow any number of arguments: additional arguments are tacked on with the null operator ge'a.

Therefore, the “magic square” matrix

can be represented either as:

or as

The regular mekso operators can be applied to vectors and to matrices, since grammatically both of these are expressions. It is usually necessary to parenthesize matrices when used with operators in order to avoid incorrect groupings. There are no VUhU operators for the matrix operators of inner or outer products, but appropriate operators can be created using a suitable symbolic lerfu word or string prefixed by ma'o.

Matrices of more than two dimensions can be built up using either pi'a or sa'i with an appropriate subscript numbering the dimension. When subscripted, there is no difference between pi'a and sa'i.

The following cmavo is discussed in this section:

So far, the Lojban notational conventions have mapped fairly familiar kinds of mathematical discourse. The use of forethought operators may have seemed odd when applied to “+”, but when applied to “f” they appear as the usual functional notation. Now comes a sharp break. Reverse Polish (RP) notation represents something completely different; even mathematicians don't use it much. (The only common uses of RP, in fact, are in some kinds of calculators and in the implementation of some programming languages.)

In RP notation, the operator follows the operands. (Polish notation, where the operator precedes its operands, is another name for forethought mekso of the kind explained in Section 18.6.) The number of operands per operator is always fixed. No parentheses are required or permitted. In Lojban, RP notation is always explicitly marked by a fu'a at the beginning of the expression; there is no terminator. Here is a simple example:

The operands are re and ci; the operator is su'i.

Here is a more complex example:

Here the operands of the first pi'i are re and ci; the operands of the second pi'i are vo and mu (with boi inserted where needed), and the operands of the su'i are reboi ci pi'i, or 6, and voboi mu pi'i, or 20. As you can see, it is easy to get lost in the world of reverse Polish notation; on the other hand, it is especially easy for a mechanical listener (who has a deep mental stack and doesn't get lost) to comprehend.

The operands of an RP operator can be any legal mekso operand, including parenthesized mekso that can contain any valid syntax, whether more RP or something more conventional.

In Lojban, RP operators are always parsed with exactly two operands. What about operators which require only one operand, or more than two operands? The null operand tu'o and the null operator ge'a provide a simple solution. A one-operand operator like va'a always appears in a reverse Polish context as tu'o va'a. The tu'o provides the second operand, which is semantically ignored but grammatically necessary. Likewise, the three-operand version of gei appears in reverse Polish as ge'a gei, where the ge'a effectively merges the 2nd and 3rd operands into a single operand. Here are some examples:

The following cmavo are discussed in this section:

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. Example 18.114 exhibits afterthought logical connection between operands:

Example 18.115 is equivalent in meaning, but uses forethought connection:

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:

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:

The forethought-connection form of Example 18.117 is:

Here is a classic example of operand logical connection:

Note the mixture of styles in Example 18.119: 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 Example 18.119 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:

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

which expresses the same interval as Example 18.120. 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:

Here we have the sum of a number and an interval, which produces another interval centered on the number. As Example 18.122 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:

The following cmavo are discussed in this section:

One of the mekso design goals requires the ability to make use of Lojban's vocabulary resources within mekso to extend the built-in cmavo for operands and operators. There are three relevant constructs: all three share the elidable terminator te'u (which is also used to terminate vectors marked with jo'i)

The cmavo na'u makes a selbri into an operator. In general, the first place of the selbri specifies the result of the operator, and the other unfilled places specify the operands:

tanjo is the gismu for “x1 is the tangent of x2”, and the na'u here makes it into an operator which is then used in forethought

The cmavo ni'e makes a selbri into an operand. The x1 place of the selbri generally represents a number, and therefore is often a ni abstraction, since ni abstractions represent numbers. The ni'e makes that number available as a mekso operand. A common application is to make equations relating pure dimensions:

The cmavo mo'e operates similarly to ni'e, but makes a sumti (rather than a selbri) into an operand. This construction is useful in stating equations involving dimensioned numbers:

Another use is in constructing Lojbanic versions of so-called “folk quantifiers”, such as “a pride of lions”:

The following cmavo are discussed in this section:

So far we have seen mekso used as sumti (with li), as quantifiers (often parenthesized), and in MOI and ME-MOI selbri. There are a few other minor uses of mekso within Lojban.

The cmavo me'o has the same grammatical use as li but slightly different semantics. li means “the number which is the value of the mekso ...”, whereas me'o just means “the mekso ...” So it is true that:

but false that:

since the expressions “2 + 2” and “4” are not the same. The relationship between li and me'o is related to that between la djan., the person named John, and zo .djan., the name “John”

The cmavo nu'a is the inverse of na'u, and allows a mekso operator to be used as a normal selbri, with the place structure:

for as many places as may be required. For example:

uses nu'a to make the operator va'a into a two-place bridi

Used together, nu'a and na'u make it possible to ask questions about mekso operators, even though there is no specific cmavo for an operator question, nor is it grammatical to utter an operator in isolation. Consider Example 18.131, to which Example 18.132 is one correct answer:

In Example 18.131, na'u mo is an operator question, because mo is the selbri question cmavo and na'u makes the selbri into an operator. Example 18.132 makes the true answer su'i into a selbri (which is a legal utterance) with the inverse cmavo nu'a. Mechanically speaking, inserting Example 18.132 into Example 18.131 produces:

where the na'u nu'a cancels out, leaving a truthful bridi

Numerical free modifiers, corresponding to English “firstly”, “secondly”, and so on, can be created by suffixing a member of selma'o MAI to a digit string or a lerfu string. (Digit strings are compound cmavo beginning with a cmavo of selma'o PA, and containing only cmavo of PA or BY; lerfu strings begin with a cmavo of selma'o BY, and likewise contain only PA or BY cmavo.) Here are some examples:

The difference between mai and mo'o is that mo'o enumerates larger subdivisions of a text. Each mo'o subdivision can then be divided into pieces and internally numbered with mai. If this chapter were translated into Lojban, each section would be numbered with mo'o. (See Section 19.7 for more on these words.)

A numerical tense can be created by suffixing a digit string with roi. This usage generates tenses corresponding to English “once”, “twice”, and so on. This topic belongs to a detailed discussion of Lojban tenses, and is explained further in Section 10.9.

Note: the elidable terminator boi is not used between a number and a member of MAI or ROI.

As mentioned earlier, Lojban does provide a way for the precedences of operators to be explicitly declared, although current parsers do not understand these declarations.

The declaration is made in the form of a metalinguistic comment using ti'o, a member of selma'o SEI. sei, the other member of SEI, is used to insert metalinguistic comments on a bridi which give information about the discourse which the bridi comprises. The format of a ti'o declaration has not been formally established, but presumably would take the form of mentioning a mekso operator and then giving it either an absolute numerical precedence on some pre-established scale, or else specifying relative precedences between new operators and existing operators.

In future, we hope to create an improved machine parser that can understand declarations of the precedences of simple operators belonging to selma'o VUhU. Originally, all operators would have the same precedence. Declarations would have the effect of raising the specified cmavo of VUhU to higher precedence levels. Complex operators formed with na'u, ni'e, or ma'o would remain at the standard low precedence; declarations with respect to them are for future implementation efforts. It is probable that such a parser would have a set of “commonly assumed precedences” built into it (selectable by a special ti'o declaration) that would match mathematical intuition: times higher than plus, and so on.

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:

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:

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:

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:

Of course, there is no guarantee that the name zel. is connected with the number rafsi: an alternative which cannot be misconstrued is:

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.