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Argument Referent Sets#

As stated earlier, a predicate word expresses a relation between the occupants of its cases, and is defined by a referent set consisting of lists of case occupants that are thus related.

To interpret an argument, you start with its predicate's referent set. You retain members consistent with any sub-phrases. From each member you extract the first case occupant, and out of these you make the full referent set of the argument. (The full referent set is empty in important special cases.) The referent subset, which is the set of actual referents of the argument, is a subset of the full set which depends on a prefix word called an article.

Consistent with sub-phrases means this: the n'th case occupant of each main predicate referent set member must be in the referent subset of the n'th case sub-argument. In addition, if the sub-argument has a prefix ve -each, then make equivalence classes of main referent set members (that survive all restrictions simultaneously) which differ only in the n'th case. If the set of n'th case occupants from an equivalence class is not equal to the sub-argument referent subset (i.e. it is not true that each sub-argument referent is in the equivalence class), reject the whole equivalence class. For subordinate clauses, the same procedure applies except the predicate is from the clause and the restricted phrase acts as one of its sub-arguments.

The Articles#

Here are the articles. There are two articles for each meaning; the first unfolds the referent subset so each member is a referent, while the second specifies that the referent is the referent subset as a set. The careful distinction between sets and extensions of their members is characteristic of gua\spi.

xe , xy : The most common article is xe , and it is assumed with most predicates when arguments lack an article. Its English translation is the. The referent subset is whichever members the speaker has in mind to talk about, but generally there are prior context cues to show which out of numerous possibilities are intended as the referents. In particular, if a set of referents has been designated before and if it is the only such set that is a subset of the full referent set of the argument, then those are the referents of the argument. For example,

fkar -car appears three times; in each instances its article is xe -the but the article is unseen, being provided by default. The first instance of fkar -car designates one referent in detail. The other two instances are typical arguments with xe -the: since the prior referent fits this predicate (and in the second sentence its garage does not), the prior referent is being redesignated. Because gua\spi words are so short it is just as efficient to redesignate an argument like this as to use a phrase-relative pronoun, so pronouns are less commonly used in gua\spi than in English.

xa , xu : The subset equals the full referent set. The referents are everything that fits the predicate. Mathematical expressions almost invariably have xu as the article, as in xu -cu = the equivalence class of all pairs = the number two. xi , xr : Only a few members are left out, which the speaker has in mind as being atypical. By judiciously using xi the speaker can prune out excessive special cases from his discussion. But unfortunately listeners have a lot of trouble to identify correctly which members are being left out. Beware. xo , xw : From the full referent set one or more members are selected, and it doesn't matter which ones. For example,

All in the box are equivalent and it doesn't matter which you get. xo is often used for arguments in the serving or portion category, called partitive nouns in English.

xn : No members of the full referent set are in the referent subset. This article is useful for negative statements like

Actually xn makes a statement about the excluded referent set members; were it regarded as a statement about all the members of an empty set, there could be no examples or counterexamples and the sentence would assert nothing. The second example is the contrapositive of the first, and shows what xn really means.

Each case of each predicate has a default prefix, used with any arguments that do not have explicit articles. Most of these defaults are xe ; xu is used with math operators because of the meaning of numbers; xy is used for cases that need sets; xe vo is common for words that deal with events; and xe bi appears in a few cases that need texts. In the latter two cases appearance of any article cancels both xe and vo or bi. xy defaults have a special behavior: they change to xe if the argument is already a set.

When an argument predicate has a default article for its first case other than xe , and when it is in a case with a xe default, the argument predicate's article is used. Otherwise if the defaults are unequal and not xe , the situation is ambiguous and the speaker is required to say the article he wants.

Cartesian Expansion of Arguments#

The reason arguments are in a phrase is to select particular events from the definition of the phrase predicate. When there are several arguments in a phrase their referent sets are multiplied in the sense of Descartes (Cartesian product) to give the events in the phrase referent set: each of the first argument referents is paired with each of the second argument referents, and so on for more arguments. For example,

The second example could easily be interpreted the same as the first, but if one knows that particular events are not in the definition of the predicate --- cows do not crw -eat rats --- one will normally discount such terms in the Cartesian expansion without thought. The arguments then merely select pre-known events from the definition. If the speaker wishes to assert that a cow ate a rat, he should put the unusual event in a separate sentence and should emphasize it.

Often within a sentence --- this applies to top-level sentences, not just phrases --- the same argument appears in several cases, either as a pronoun or explicitly redesignated. In each member of the referent set of the sentence the occupants of the replicated case are equal --- they are not selected independently from the referent sets of the replicated argument, as they would be if the arguments were different. Sub-phrases restrict main phrase events one by one, and the occupant of a replicated case in the sub-phrase is also equal to the one in the main phrase. For example:

There is one exception: if an argument is prefixed by vl , its referent is chosen independently of any other copies of it. vr as a prefix means inter alia or each other. Its meaning is like vl except that in the sentence's referent set the prefixed argument's referent may not duplicate that of other copies of the same argument.

If an argument prefixed by vr or vl is replicated by a pronoun the same referent is used both places, but if the pronoun has its own prefix that one supercedes any in the antecedent and the referent is chosen independently.

In addition, the sentence start words :u and :o mean that the sentence should be interpreted coordinately with the one before it. The referent sets of the two sentence predicates are multiplied in the sense of Descartes and the argument set members are fitted into their respective cases the same as in a single sentence. In particular, arguments replicated in both sentences have the replicated cases equal in each referent set member.

The same statement is shown twice; the first one uses a pronoun while the second repeats the argument ftu =plyw explicitly.

For statements of logic one may produce a nonsentence containing the arguments to be discussed, called prenex arguments, and then state one or more sentences about them, starting each sentence with ^:u and repeating the arguments or using pronouns. Since the Cartesian product is not commutative when negation is involved, or with mixed universal and existential quantification, the speaker may be forced to produce arguments in an inconvenient order, and prenex arguments may sound better. Here is an example of prenex arguments:

A conjunction is a pair or list of similar grammatical structures which act as if they were one unit. We have seen the conjunction word fe in compound words, but it also works with arguments to combine several into one, just as when one argument designates several referents. Here is a tricky example from before:

The two argument pairs are Cartesian multiplied to select referent set members of crw -eat just as in the second example the two arguments designate multiple referents which are Cartesian multiplied. See also Vocabulary: Conjunctions for other kinds of conjunctions.

Miscellany about Arguments#

Formally the articles are described as prefixes, but it makes sense semantically to regard them as predicates with two arguments; a full referent set comes in (as a set) to the second argument and the appropriate referent subset comes out in the first. With this definition, for example, xa (full referent set in extension) would mean exactly the same as xor -set member. It is easier that way to interpret compound articles like xo -xi = any typical X. However, articles have to be prefixes grammatically so the argument's main predicate can have its own arguments before it as well as after.

Most Indo-European languages distinguish between genders and numbers of arguments. Like Chinese and English, gua\spi has no gender, though you may use a subordinate clause like |fmy -female. Number comes from the referent sets, not the grammar. You may specify the exact number of referents with a numeric predicate, like this:

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