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Mathematical Expressions#

Even uneducated speakers quantify phrases, that is, they say how many or how big some phrase is. It turns out that to support just simple dimensioned quantities the language has to include a complete facility for mathematical expressions.

Numbers, Expressions and Functions#

Cardinal numbers (here exemplified by cu - set of two) are defined as X1 is a set containing so many members X2. The converted predicate means X2 is a member of a set X1 of so many members. Quantifiers are subordinate clauses on an argument, e.g.

How do you say the number two? Any set with two members can be put in 1-1 correspondence with any other such set, but not with a set with different count; this forms an equivalence relation that segregates sets by count. Among the ways to define the number two the one that fits best in gua\spi is xu -cu , designating this equivalence class. All kinds of mathematical objects, such as rational, real, complex and dimensioned numbers, can be produced by various extension maneuvers from these equivalence classes, and can be named in gua\spi by xu -N.

Mathematical functions are defined with such classes as formal parameters, and hence have xu on parameter cases by default --- xu means the entire referent set of an argument, as a set (or class). The first case of a function is its value, and the function is defined as X1 is in the equivalence class that comes from doing (function) on (xu) X2, possibly with several parameters. Thus a function can be used to predicate that something has a particular count or measure. xu recovers the equivalence class. The abbreviation IEC, meaning in equivalence class, is used thus: X1 IEC the result of (whatever). For example,

This syntax for mathematical expressions is neat, compact and unambiguous. No special syntax needs to be added to gua\spi beyond that already in use for ordinary arguments and sentences.

Functions always deliver their value in the first case and take arguments in the second and following cases. For the range and domain of a function F, use xu -F and xu -zu -F respectively.

Ordinal Numbers and List Ends#

An ordinal number, cued by the quasidigit tr , means X1 is N'th in list (xy) X2 starting at X3. For example:

List ends and segments are built with bny -begin and fne -end restricted by a numeric predicate. Note the definition, X1 is the next or previous member of (xy) X2 after X3; restrict with a numeric predicate to change to the N'th next or previous member. Without X3 the list ends are produced, but don't be confused by the polarity: bny -next also means ``beginning or least when the list is ordered by size or degree; fne -previous means end or most. It is clearer to use an ordinal number when you can. For example,

Lists are ordered with smaller or negative numbers first, so the smallest is bny -sty -kqa whereas the largest would be fne -sty -kqa or, sorting the list in reverse order, /bny !sty -spl. See also the discussion of sym -chief under Comparative and Superlative for a better way to do second smallest and the like.

Vectors, Dates and Times#

You express a vector as a stl -list of expressions. Units of measure applied to a vector multiply each component individually. A matrix (by components) is a list of vectors, and so on. A date or time is also a list of expressions.

The date is defined as X1 is the date of event (vo) X2+ starting with unit (xu-jani) X3* in calendar X4 in which auto-conversion lets it restrict a sentence directly, while the unit can still be compounded. The first vector component has that unit, and subsequent components are multiplied by sub-units in the order years, months, days, hours, minutes, seconds. The default unit is jani -years.

Units of Measure#

Units of measure are defined to multiply a number or other expression by the unit. The resulting equivalence class is considered to contain gua\spi events whose degree or measure are that big; hence the unit expression takes the form of a subordinate clause, and the main sentence predicate tells what dimension is being measured. For example,

Scientific notation is used in gua\spi instead of the thousands and millions typical of English and in place of the metric prefixes; it is more compact and much easier to specify syntactically.

This definition of a unit is reasonable mathematically since a physical unit of measure can be interpreted as a basis member of a 1-dimensional vector space of things having that dimension. For example, consider mass. Take the set of all things with mass, and take equivalence classes of things with equal mass. Those equivalence classes occupy, and can be extended to create, a 1-D vector space. Any single member is a basis, and a unit is a member selected by convention, e.g. the standard kilogram. Now for the word, its referent could be the unit, but you have to multiply it by the number (e.g. 2.5 times grams), which makes expressions too wordy. So the unit word is defined as a math function that multiplies by the unit.

In units of measure, the first argument occupants are not things but properties, e.g. masses of things, which are events, e.g. something is massive. The need for a predicate to go with the thing being measured is easiest to see in 3-D, e.g. the argument could be high, wide or deep but all are measured by the single dimension of meters. Then the unit becomes a modal case of the predicate. These examples show how to use MKS and provincial units:

In particular, no quantifiable relation (e.g. heavy or exceeds in dimension vo X3) has an explicit case for how much it is, relying instead on the modal case of units. There is one exception: kun -quantity is like a unit in providing a modal case for quantity, but provides an identity transformation, so that a question word can be dropped into the multiplicand argument without forcing a specific unit.

To talk about the unit rather than to use it, use xu vo , as in the pound is a provincial unit. xe vo will deliver the standard unit, if there is one, given suitable context cues.

Compound units, like ohms, require a product or quotient of several units. One may use the personal name units (ohm, volt, pascal, celsius) in the same manner as provincial units.

Quantification and Negation#

Some Important Quantifiers#

Words for Something#

Nine Varieties of Negation#

^:i -sfa !kio !ji ^tara |zey !ju It is false that I have your rat. This is the prototype of negation, and it is the policy in gua\spi to use predicates when possible rather than prefixes or other structure words. However, the negated sentence is an extra level down, a problem for speakers. ^:i -go !ji /kio !tara |zey !ju I don't have your rat. go is a mood prefix which means that the asserted sentence is counter to fact. It is simpler and more familiar to natural language speakers than sfa -false is, and it works in subordinate clauses where sfa doesn't. ^:i !ji /kio !tara |go -zey !ju I have a rat which isn't yours. go can equally be used in subordinate clauses, or even in argument predicates. ^:i !ji /kio !xn -kseo I have no cheese. xn means that of the members of the full referent set of the argument, none fit in the predicated relation. Unlike the rest of the articles, this is actually a statement about the excluded members, and means the same as ^:i !ji /go -kio !xa -kseo --- freely translated, for all pieces of cheese, I don't have it. (See De Morgan's rules below.) ^:i !ji /kio !kseo |zu -cy I have zero pieces of cheese. This is the most natural form of argument negation in Loglan , but gua\spi looks strictly at referent sets, and if you say you have all the members of the null set, it isn't a cheesy null set --- there is only one null set. The statement is a tautology, and says nothing about cheese. Many logical fallacies, such as St. Anselm's ontological proof of the existence of God, are like this example in that they prove a statement about the members of a set which may not have any members. In gua\spi use xn as above. ^:i !ji /kio !ple !tara I have something which isn't the rat. The full referent set of ple !xe -tara (and therefore its referent subset) is in the complement of the referent subset of xe -tara. ^:i !jw |kseo /fi -stu -zao This cheese is bad in flavor. In George Orwell's 1984, the language newspeak was designed to destroy the ability of people to think, and one of its design features was that negative words were eliminated; bad became ungood. Gua\spi (imitating Loglan) offers specific negated words for major predicates when the negations are used frequently. Nonetheless, most negations will have to be done with compound words as in the next examples. Be alert for creative expression possibilities such as ^:i !jw |kseo !fu -zu -dyi --- this cheese is disgusting . ^:i !jw |kseo !fu /gl -zao This cheese is flavor less. Many dimensions are quantifiable (more or less) but unsigned, so their degree ranges from zero to larger values. This is how to assert that the degree is zero or negligible. ^:i !jw |kseo !fu /gr -ksi This cheese is not fresh. When the dimension ranges from positive to negative values, gr interchanges positive and negative. On occasion, gl will also apply to indicate the zero point, though it is meaningless with ksi -fresh. For extremes of unfreshness one can use fpu -rotten. ^:i !jw |kseo /fi -vry -can -psl This cheese is de solidifying. When a process occurs in the reverse of the usual order, vry -reverse indicates this.

De Morgan's Rules in Quantification#

Negation interacts with and and or, which necessarily occur in sentences which are quantified or whose arguments have multiple referents. Therefore it is advisable to digress into some elementary symbolic logic. Here is De Morgan's rule for negation, stated four ways: (A and B are sentences)

Remember that in logic, A or B is true if one or both of the statements is true, unlike in English where the or generally excludes both being true.

Universal quantification means a statement is true when applied to all members of a set, of the form S1 and S2 and S3 and . . ., where S1 is the statement applied to member 1 and so on. Existential quantification means that a statement is true about at least one set member, in form S1 or S2 or S3 or . . . When such statements are negated, De Morgan's rule applies. Here are some more specific examples.

^:i -kio !ji ^kseo I have the cheese. This will be the basic example sentence. Let us make the existential quantification more explicit: ^:i -kio !ji ^kseo |zu -to I have at least one piece of cheese. Existential quantification like this means the same as I have piece 1 or I have piece 2 or . . . for all pieces of cheese. Now the simplest negation of this sentence is simply: ^:i -sfa !kio !ji ^kseo |zu -to It is false that I have at least one piece of cheese. This form does not suit typical speakers; we want to negate the relation word kio -possess, not the whole sentence, like this: ^:i !ji /go -kio !xa -kseo I don't have any cheese --- I don't have piece 1 and I don't have piece 2 and . . . To negate (or de-negate) a disjunction (compound sentence with or), we had to change or to and, producing a universal quantification. The same principle applies when you start with a universal: ^:i !xa -xe -cil /jir All the children are here --- Child 1 is here and child 2 is here and . . . Rather than negating the whole sentence with sfa -false, let us negate the predicate jir -here: ^:i -go -jir !cil |zu -to At least one of the children is not here --- Child 1 is not here or child 2 is not here or . . . In general, when you negate the predicate of a sentence involving quantification or multiple argument referents of any kind, you will also have to reverse the type of quantification or conjunction used.

Sentence Forms#

Moods and Imperatives#

These are the mood prefixes in gua\spi , which indicate the manner of assertion of a phrase. A top level sentence has ge on it by default unless another mood prefix appears.

Closely related to the mood prefixes is the aspect operator tri -ritual, a sign of a performative phrase. Performative means that by uttering the words the speaker makes something true, as in a marriage vow or the illustrated naming ceremony. Note that auto-conversion is suppressed by zo ; without it, the sentence would merely be the topic of a ceremony, not the ceremony itself.

In English there is an imperative mood; however, in gua\spi you make a sentence imperative by using jo -you or ja -we in the case for the actor, generally the first. These pronouns are distinguished from the non-imperative ju -you and je -we. A decoration pli -please softens the command. For example,

Special Features of Infinitives#

In an infinitive the previous argument is replicated by default as the infinitive's first argument, while the first argument of a subordinate clause comes normally from the restricted phrase. Hence numbered cases skip over the first argument, and you must use the caselink so for any explicit first case in an infinitive or subordinate clause. In an infinitive with vo a predicate is made out of the sub-sentence that follows, including arguments and clauses. In the rare case where a sub-phrase (like a subordinate clause) must go on the infinitive predicate rather than into the sub-sentence, you can put a prefix before vo , like an article, and put the clause between the article and vo.

When an infinitive with vo is an argument, the main sentence asserts the relation of arguments to the infinitive's events, but does not make a separate assertion of those events. To additionally assert or deny the sub-phrase, use ge or go respectively. For example:

Comparative and Superlative#

Natural languages have various complicated arrangements to change a simple property to become comparative or superlative. Gua\spi does it with a predicate.

In the case of sym -superlative it is possible for several members to be equally green, each being greener than the remaining members. Also, a numeric predicate modifying sym produces the N'th greenest member. Here are some sentences with comparatives and superlatives:

Causal Sentence Connectives#

^:i !tara /crw !kseo ^:o -kau !gai -tuol !kseo The rat eats the cheese, and that causes it to be dirty. A cause is rather mechanical. Actors with free will are rarely caused to do anything, despite their protestations. Here the rat may have free will, but the cheese, caused to be dirty, certainly does not. ^:i !ji /gri !tara ^kai |kei ^kseo ^:o -kmo !qma -qtu !ji ^tara I am angry at the rat for stealing the cheese, which motivates me to kill the rat. The theft motivates the anger and the anger motivates the planned killing. When a free agent acts it is usually because of a motivation. Here the speaker includes kei -crime in the sentence as a justification for his action. The definition of this word reminds you that it has the modal case tue -culture, which presumably includes the speaker --- but not the rat. ^:i !xi -tara /qai -crw |jro ^kseo ^:o -zu -zni !vel !klo ^kseo So that rats cannot eat the cheese, is the reason the cheese is in a closed container. A reason is an end (ending event) or consequence that motivates someone to make a starting event happen, such as keeping the cheese in the box, that will cause the consequence. The concept of zni -reason is rather slippery. First, the desired or planned consequence should be stated, not its inverse; ^:i !xo -tara /gi -crw !kseo = A rat might eat the cheese is the negative of the correct consequence. Second, we say in English past event Y is the reason for action Z where the gua\spi definition of zni -reason requires vengeance for past event Y --- a future consequence of action Z. vou = vengeance. Third, a gul -rule can be said to cause its reason, provided the obligees obey it. ^:i -dae !kara ^:o -sny !pwo -cyr -xyn !xi -tara ^kara The box being open implies that a rat can go into it. The relation of logical entailment has to do with definitions and theorems, not with the arrangement of the real world or the will of its actors. zny -imply is the corresponding set operator: X1 is the union of X2 and the complement of X3, where X2 and X3 can be infinitives with vo. Perhaps the distinction between sny and zny is merely an artifact of old Loglan and English usage. We shall see if this is true as gua\spi matures. ^:i -dae !kara ^:o -bal !crw |jro ^tara ^kseo If the box is open then maybe the rat will eat the cheese. This kind of fuzzy inference based on real-world consequences is what people use most often, rather than pure logic. ^:i !ji /gu -fli ^:o -sar !gu -vlw !ji ^qyun If I could fly I would go to the moon. Necessary conditions are very commonly expressed and the logical if-then catches their true meaning poorly. Related is sno -sufficient.

These are the sentence connectives most often seen. But the speaker may connect sentences with any useful word having suitable cases. And like all gua\spi words, the sentence connectives can also be useful as arguments and as modal caselinks.

Logical Sentence Connectives#

Old Loglan was intended to be a logical language, thereby to differ as much as possible from English. Therefore, one of its key features is support for what amounts to spoken symbolic logic. This feature is de-emphasized in gua\spi ; in practice, what language users encounter most often, and stumble over, are Cartesian expansion of multiple arguments, non-commutative quantification, and complicated negations. These topics are well-supported in gua\spi. Nonetheless, set arithmetic can be performed on infinitives and the result is a set of events to which the listener's attention is drawn, just as with a more normal sentence. The logician's if-then can be realized through zny. Here are some examples of logical sentence connectives:

Features of Thesaurus Categories#

The gua\spi words have been put into groups with related meanings, for ease of learning and for ease of creation. The dictionary includes a thesaurus of these categories. Many categories have closely related cases, or certain special features, which are described below.

Abstract Comparisons#

Many abstract comparisons (1.1.1) and set member words (1.1.3) include a dimension on which comparison occurs. In a compound with the dimension as sub-word, its cases merge in an unusual manner. Considering the dimension to be single-ended (e.g. a color, as opposed to a directional property), its first case is applied to several arguments as noted in the definitions, e.g.

xgi -green is applied to both X1 and X2 in the first and second sentence. This is described as a dual merge. In the last sentence, xgi -green is applied to X1 and to members of X2. The dictionary indicates all these special merges.

stl -list involves a dimension which is applied pairwise to members of the list, indicating the ordering.

qaw -equally has a very unusual definition: the first case is an infinitive into whose first case the rest of the cases are copied in turn; the predicate means that all the arguments fit in the infinitive equally. Normally the predicate of this infinitive is provided by compounding, as in the example above.

Sets#

For several words in category 1.1.2 (sets) of the form (set) X1 is a (whatever), you can make a compound vdr =W to get the members.

When xy (in-mind set) is the default article for a case, then if the referents are sets the default changes automatically to xe (in-mind in extension). But xu (whole set) does not change to xa (same in extension) because in math functions the usual occupant of such a case is supposed to be a set of equal-count sets.

The predicates tla -set and stl -list have a special arrangement of cases. They mean X1 is a set (in extension) or list (ordered) consisting of members X2, X3, X4, . . ., as many cases as needed. If X2 etc. have multiple referents in extension (which must be ordered for stl), all referents go in the set or list. Five or six words have this as many as needed argument list.

Properties#

Noncomparative Properties are distinguished in Loglan from the Comparative Properties in that it is not useful to say that X is more than Y; for example, X is more dead than Y. For this reason Loglan Comparative Properties each have a case for the compared item and Noncomparative Properties do not. Nonetheless, many of its members may actually be used comparatively (like ksu -delicious) and the distinction is rather artificial. In gua\spi , Properties do not have comparative arguments.

Directional Properties (1.5): These are often compounded with motion words, in which the moving case is related to the destination. (Special case: tai -outside merges with the start point. Examples in Compound Words.) Note that the polarity (e.g. up/down) in such compounds is often backwards from English.

Timelike Directional Properties (1.5.3): These are the relation words for the tense modal case.

Behaviors#

Abstract Behaviors(2.1): These have the form X1 does (vo) X2+1, in which X1 is automatically replicated as the first case of the infinitive vo X2.

Double Actor Transitive Activities (2.1.3): These have the prototype X1 makes X2 do (vo) X3+2, in which X2 is automatically replicated as the first case of vo X3.

Games for Two Players (2.1.4): Generally you will want to use a reciprocal construction like this, unless the relation really is unilateral:

Motion Words (2.2): The prototype is X1 goes to X2 (destination) from X3 (start point) via X4 (route). Since motion words are complicated, effort has been put in to make them all regular. They are very frequently combined with directional properties, q.v.

Transitive Motion Words (2.2.3): The prototype is X1 makes X2 go to X3 from X4 via X5, and again they are all regular. Directional properties relate X2, the mover, with X3, the start point.

Quasi-motions and Routes (2.2.4): The routes are set up as regular motion words. The quasi-motions can profitably be compounded with motion words.

Communication and Mental Activity (2.3): The pattern X1 knows that X2 is (vo) X3+2 is common, with X2 merging as the first case of X3. However, quite a few predicates in this category have different patterns, so watch out.

Transitive Activities with an Object (2.4): A number of these words have an X3 case for a tool or means which is typically filled by a transitive compound, as in:

Things and Materials#

Animals and Plants (3.1): These have just one argument. The animals and plants category has been extended to include a primitive for each phylum, or at least most of them.

Body Parts (3.2): These have the prototype X1 is a (part) of creature X2*.

Materials (3.3): Almost all of these are of the form (xo) X1 is a serving/portion of (material). The xo appears by default when the word is used as an argument, unless the containing sentence provides a default article other than the usual xe.

Places, Seasons and Weather (3.5): Places mostly have the form X1 is a (place) of locality or superset X2.

Containers (4.1.1) and Cooking and Eating (4.1.2): These have the form X1 is a container containing (xo) X2*. Constructions like spoonful are handled with ful -contained quantity, like this:

Transport (4.1.4), Machines (4.1.5), and Parts of Structures (4.1.7): Many of these are like body parts: X1 is a (part) of structure X2*.

Houses (4.1.8): House parts are as above. Houses themselves have the form X1 is a (house) of resident X2*.

Cloth and Parts of Clothes (4.2.2): Parts are as above. Cloth has the form (xo) X1 is a portion of (cloth).

Food (4.3): Mostly of the form (xo) X1 is a serving/portion of (food).

Works of Art (4.4.1): All have the form X1 is a (thing) about X2 created by X3 and performed by X4. X2 may be an event or a thing; there is no vo default. X4 is present only on relevant words such as jiul -drama.

Miscellaneous Categories#

Nationalities (4.7.1): Loglan has words for nationalities, for the languages spoken there, and for the basis money unit of the nation. But only about fifteen arbitrarily chosen nations are supported, mainly European ones. Gua\spi uses foreign names for these concepts, through zina -nation, gua -language, and cni -money. spi -person translates the usual self-referential word in primitive languages for ethnic members of that culture.

Business (4.7.3): A number of these words have the form X1 (sells) goods or services X2 to other trader X3 for amount of money (xu) X4.

Most Frequent Words#

So far, the corpus of gua\spi text available for analysis consists of 3140 words of fiction representing a teenager setting up a small business and interacting with younger children, parents, customers and girlfriend. I originally wrote this story in Loglan to test various features, and it is known that word frequencies will differ in other topics. However, this text gives some guidance about which words a beginner should be sure to learn.

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