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.
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.
^:i !cil |zu -cu ^ji /fi -vyl My twins are male
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''.
!xu -cu -cw -ci The number 2.5 (the class of all sets of ``count'' 2.5)
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,
^:i !xa -ca /plw !co ^cu 3 is the sum of 1 and 2 (all triplets IEC 1+2)
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.
An ordinal number, cued by the quasidigit ``tr'', means ``X1 is N'th in list (xy) X2 starting at X3''. For example:
^:i -brn !junu !qnou =ji |tio /ve -tr -ca Broken is the third claw of my right paw (hand)
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,
^:i |vi -pli ^jo /can -fne !psa -gvu Please go to the end of the line ^:i !ji /bny |cu ^sty -kqa !diu -sui I am the second smallest in the class ^:i !ji /tr -cu !sty -kqa !diu -sui Same thing, better ^:i !bny |te -ca ^sty -bir !kuo |tum =teon ^ji /fi -za -gey |jro ^su -xo -spia The first three people (in order by time) to phone me will be given tickets ^:i !tr -te -ca !sty -bir !kuo |tum =teon ^ji /fi -za -gey |jro ^su -xo -spia Same thing, better
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.
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.
^:i !vnyn /zm -cmu !dman ^dmem !stl !ci ^ca -cy The wind is (5, 30) meters per second (per second meter 5, 30) ^:i !qo -kauai:i /jir !vdei !stl !co -ke ^kl -co -ci -ko Hawaii is at (19, -156) degrees ^:i -tem =jani !su -jn ^stl !co -ke -ka -ke ^cu ^co -ke The date today is 2/19/1989 (order: year, month, day) ^:i !qo -kamleto /zu -fom |tem =qrau !stl !cu -cy ^ca -cy Hamlet will be performed at 20:30 hours
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 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.
^:i !ji /vga |kyam !ku -cy I weigh 70 kilos (I heavy kilo 7 0) ^:i !tor =cenu /cni !ti -kl -co -cw -cu -ka -kn -ku The account balance is about minus 12.8 million dollars ($ about -1.28E7)
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:
^:i !ji /gal |dmem !co-cw-ku-ce I am 1.74 meters high ^:i !ji /gal |xnu -fn -:inca !ko-ke I am 69 inches high
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 <unit>'', as in ``the pound is a provincial unit''. ``xe vo <unit>'' 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.
xa -tara All rats (anywhere, any time) xa -xe -tara All the rats (in an in-mind set) xa -tara |xyn !dowu All the rats in the house xi -tara |xyn !dowu Most of the rats in the house tara |zu -vdu Many rats tara |zu -pqu Few rats tara |gou -sun Enough rats tara |gr -gou -sun Too few rats tara |gou -pqu Few enough rats tara |gr -gou -pqu Too many rats (insufficiently few) tara |zu -ti-ta-cu-cy-cy Almost a hundred rats jmo -vjr Almost vertical
^:i !ji /crw |bir I already ate (something implied) ^:i !ji /daw -crw !jy I want to eat something ^:i !ji /crw !xo -kseo I am eating some cheese ^:i !jw /vdr !xy -jy |kfa /vu -sny Logically, he must have some family (a set) ^:i !xi -jy ^:u -xun !vo !zglo /gr -zu -gul ^vo !zglo /qma -tfa Most things are illegal or fattening ^:i !xa -jy |vdr !xo -sto -fw -kaia . . . For anything in a compact set . . .
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.
A and B = not( (not A) or (not B)) (not A) and (not B) = not( A or B) A or B = not( (not A) and (not B)) (not A) or (not B) = not( A and B)
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.
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.
ge Asserted to be real or factual ^:i !vo -ge -dae !kara !fu -bal !crw |jro ^tara ^kseo If the box is open, which it is, then the rat could eat the cheese gi Potentially true; actual truth is irrelevant ^:i !vo -ge -dae !kara !fu -bal !gi -crw |jro ^tara ^kseo If the box is open then the rat could eat the cheese go Unreal or counter to fact ^:i -go !ji /kio !tara |zey !ju I don't have your rat gu Hypothetical; reality is irrelevant ^:i !ji /gu -fli ^:o -sar !gu -vlw !ji ^qyun If I could fly I would go to the moon
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.
^:i |zo -tri ^qo -ben /zu -xim !jw |cil (Performative:) Ben is the name of this child
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,
^:i |faw ^vu -qnu !qo -josefo /jo /qma -duw !gunu !ju Josepho, move your ass! ^:i |vi -pli ^jo /pin -dwo Please be patient.
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:
^:i !qo -kira /juy -xna !do ^qnou !xgno Kira allows it to hold his hand (offers --- but instead it swims away --- infinitive not asserted) ^:i !do /qou !qo -kira ^ge -qma -za -pai !cana ^ve -tum =tuen It watches as Kira bails (drains) the boat with a bucket (infinitive is also asserted)
Natural languages have various complicated arrangements to change a simple property to become comparative or superlative. Gua\spi does it with a predicate.
^:i !X1 /qaw -xgi !X2 X1 is equally green as X2 ^:i !X1 /gre -xgi !X2 X1 is more green than X2 ^:i !X1 /sym -xgi !X2 X1 is (one of) the greenest member(s) of set X2
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:
^:i !star -fn -siriu ^qo -prosyon /gre -xgm The star Sirius is brighter than Procyon ^:i !qo -jupiter /sym -kqa !stel Jupiter is the largest planet ^:i !qo -siriu /sym -xgm |cu ^xu -star |vu -sen !zu -jrer \hfil
^:i !qo -siriu /fne |cu ^sty -xgm !xu -star
Sirius is the second brightest of all stars, as seen from Earth (two ways) ^:i !qo -siriu /sym -xgm !tei !star ^qo -sol Sirius is the brightest star except for the Sun
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.
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:
^:i !xun !vo !ji /crw !ftu =plyw /vo !ji /crw !ftu =peir I eat an apple or I eat a pear (or both, per logic) ^:i !ji /crw !ftu !xun !plyw ^peir I eat a fruit of the apple or pear tree (better sentence) ^:i !zny !vo !xa -fma /zu -bor !cy /vo !fma /bor !jy If a shape has void boundary then it is itself the boundary of something
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.
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.
^:i !X1 /qaw -xgi !X2 X1 is equally green as X2 ^:i !X1 /gre -xgi !X2 X1 is more green than X2 ^:i !X1 /sym -xgi !X2 X1 is (one of) the greenest member(s) of set X2
``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.
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.
Noncomparative Properties are distinguished in Loglan from the Comparative Properties in that it is not useful to say that X is more <whatever> 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.
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:
^:i !qo -jan ^fe -qo -mery /kul !vr -zdmo John and Mary kissed each other
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:
^:i !ji /fey =cuem !kliw I pound on the nail (hammer hit)
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:
^:i !ji /crw !ze -kme |ful =spun I take a spoonful of medicine
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''.
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''.
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.
Word Count Meaning Structure Words :i 259 Sentence start zu 89 2nd case conversion ql 55 Speaker \leftrightarrow listener fi 43 Grammar to level zero va 39 Subordinate assertion sa 35 3rd caselink qo 33 Foreign name qa 31 Pop modal stack fe 30 Conjunction :a 29 Next sent in sequence qe 28 Stack modal default xo 28 Article ``any'' :e 26 Sentence conjunction :o 24 Retroactive downjump vo 23 Infinitive za 21 3rd case conversion gr 18 Linear negation gl 17 Polar negation xi 17 Article ``typical'' qi 14 Replace modal default vi 14 Attitude indicator fy 13 Retroactive downjump vu 11 Restrictive clause xa 11 Article ``all''
179 Variables ji 132 Me ju 77 You jo 48 You (imperative) zgly 24 Previous sentence jw 20 Object being shown po 17 Yes-no question vgry 15 Question sentence jy 10 Anonymous variable
jai 56 Speaker and listener xim 25 Name bir 23 Past tense jro 21 Future tense gza 18 Paragraph bwy 16 Different cnu 13 Present tense pli 13 Please qnu 13 Pay attention faw 12 Emphatic
Words in Compounds
qma 45 Transitive conversion can 25 Change to become co 25 Various digits fto 14 Such (more than usual) cyr 13 Go tai 12 Outside gre 11 More (comparative) kau 11 Cause (sent. conn.) pql 11 Only (less than usual) jur 10 Turn xyn 8 Inside zni 8 Reason (sent. conn.)
Story Topic Words
tye 16 Adhere dowu 15 House kmaw 15 Shop cil 14 Child vem 14 Trouble crw 13 Eat jaiw 13 Eye tlme 12 Metal cun 11 Connect tiri 10 Tiger