Chapter 14. If Wishes Were Horses: The Lojban Connective System#

Logical connection and truth tables#

Lojban is a logical language: the name of the language itself means logical language. The fundamentals of ordinary logic (there are variant logics, which aren't addressed in this book) include the notions of a sentence (sometimes called a statement or proposition), which asserts a truth or falsehood, and a small set of truth functions, which combine two sentences to create a new sentence. The truth functions have the special characteristic that the truth value (that is, the truth or falsehood) of the results depends only on the truth value of the component sentences. For example,

John is a man or James is a woman.

is true if John is a man is true, or if James is a woman is true. If we know whether John is a man, and we know whether James is a woman, we know whether John is a man or James is a woman is true, provided we know the meaning of or. Here John is a man and James is a woman are the component sentences.

We will use the phrase negating a sentence to mean changing its truth value. An English sentence may always be negated by prefixing It is false that ..., or more idiomatically by inserting not at the right point, generally before the verb. James is not a woman is the negation of James is a woman, and vice versa. Recent slang can also negate a sentence by following it with the exclamation Not!

Words like or are called logical connectives, and Lojban has many of them, as befits a logical language. This chapter is mostly concerned with explaining the forms and uses of the Lojban logical connectives. There are a number of other logical connectives in English such as and, and/or, if, only if, whether or not, and others; however, not every use of these English words corresponds to a logical connective. This point will be made clear in particular cases as needed. The other English meanings are supported by different Lojban connective constructs.

The Lojban connectives form a system (as the title of this chapter suggests), regular and predictable, whereas natural-language connectives are rather less systematic and therefore less predictable.

There exist 16 possible different truth functions. A truth table is a graphical device for specifying a truth function, making it clear what the value of the truth function is for every possible value of the component sentences. Here is a truth table for or:

This table means that if the first sentence stated is true, and the second sentence stated is true, then the result of the truth function is also true. The same is true for every other possible combination of truth values except the one where both the first and the second sentences are false, in which case the truth value of the result is also false.

Suppose that John is a man is true (and John is not a man is false), and that James is a woman is false (and James is not a woman is true). Then the truth table tells us that

Note that the kind of or used in this example can also be expressed (in formal English) with and/or. There is a different truth table for the kind of or that means either ... or ... but not both.

To save space, we will write truth tables in a shorter format henceforth. Let the letters T and F stand for True and False. The rows will always be given in the order shown above: TT, TF, FT, FF for the two sentences. Then it is only necessary to give the four letters from the result column, which can be written TTTF, as can be seen by reading down the third column of the table above. So TTTF is the abbreviated truth table for the or truth function. Here are the 16 possible truth functions, with an English version of what it means to assert that each function is, in fact, true (first refers to the first sentence, and second to the second sentence):

Skeptics may work out the detailed truth tables for themselves.

The Four basic vowels#

Lojban regards four of these 16 truth functions as fundamental, and assigns them the four vowels A, E, O, and U. These letters do not represent actual cmavo or selma'o, but rather a component vowel from which actual logical-connective cmavo are built up, as explained in the next section. Here are the four vowels, their truth tables, and rough English equivalents:

More precisely:

With the four vowels, the ability to negate either sentence, and the ability to exchange the sentences, as if their order had been reversed, we can create all of the 16 possible truth functions except TTTT and FFFF, which are fairly useless anyway. The following table illustrates how to create each of the 14 remaining truth functions:

Note that exchanging the sentences is only necessary with U. The three other basic truth functions are commutative; that is, they mean the same thing regardless of the order of the component sentences. There are other ways of getting some of these truth tables; these just happen to be the methods usually employed.

The six types of logical connectives#

In order to remain unambiguous, Lojban cannot have only a single logical connective for each truth function. There are many places in the grammar of the language where logical connection is permitted, and each must have its appropriate set of connectives. If the connective suitable for sumti were used to connect selbri, ambiguity would result.

Consider the English sentence:

Mary went to the window and ...

where the last word could be followed by the door, a noun phrase, or by saw the horses, a sentence with subject omitted, or by John went to the door, a full sentence, or by one of a variety of other English grammatical constructions. Lojban cannot tolerate such grammatical looseness.

Instead, there are a total of five different selma'o used for logical connection: A, GA, GIhA, GUhA, and JA. Each of these includes four cmavo, one based on each of the four vowels, which is always the last vowel in the cmavo. In selma'o A, the vowel is the entire cmavo.

Thus, in selma'o A, the cmavo for the function A is a. (Do not confuse A, which is a selma'o, with A, which is a truth function, or a, which is a cmavo.) Likewise, the cmavo for E in selma'o GIhA is gi'e, and the cmavo for U in selma'o GA is gu. This systematic regularity makes the cmavo easier to learn.

Obviously, four cmavo are not enough to express the 14 truth functions explained in Sec. 14.1. Therefore, compound cmavo must be used. These compound cmavo follow a systematic pattern: each has one cmavo from the five logical connection selma'o at its heart, and may also contain one or more of the auxiliary cmavo se, na, or nai. Which auxiliaries are used with which logical connection cmavo, and with what grammar and meaning, will be explained in the following sections. The uses of each of these auxiliary cmavo relates to its other uses in other parts of Lojban grammar.

For convenience, each of the types of compound cmavo used for logical connection is designated by a Lojban name. The name is derived by changing the final -A of the selma'o name to -ek; the reasons for using -ek are buried deep in the history of the Loglan Project. Thus, compound cmavo based on selma'o A are known as eks, and those based on selma'o JA are known as jeks. (When writing in English, it is conventional to use eks as the plural of ek.) When the term logical connective is used in this chapter, it refers to one or more of these kinds of compound cmavo.

Why does the title of this section refer to six types when there are only five selma'o? A jek may be preceded by i, the usual Lojban cmavo for connecting two sentences. The compound produced by i followed by a jek is known as an ijek. It is useful to think of ijeks as a sixth kind of logical connective, parallel to eks, jeks, geks, giheks, and guheks.

There also exist giks, joiks, ijoiks, and joigiks, which are not logical connectives, but are other kinds of compound cmavo which will be introduced later.

Logical connection of bridi#

Now we are ready to express Example 14.1 in Lojban! The kind of logical connective which is placed between two Lojban bridi to connect them logically is an ijek:

la djan. nanmu .ija la djeimyz. ninmu That-named John is-a-man or that-named James is-a-woman.

Here we have two separate Lojban bridi, la djan. nanmu and la djeimyz. ninmu. These bridi are connected by .i****ja, the ijek for the truth function A. The i portion of the ijek tells us that we are dealing with separate sentences here. Similarly, we can now say:

la djan. nanmu .ije la djeimyz. ninmu That-named John is-a-man and that-named James is-a-woman.

la djan. nanmu .ijo la djeimyz. ninmu That-named John is-a-man if-and-only-if that-named James is-a-woman.

la djan. nanmu .iju la djeimyz. ninmu That-named John is-a-man whether-or-not that-named James is-a-woman.

To obtain the other truth tables listed in Sec. 14.2, we need to know how to negate the two bridi which represent the component sentences. We could negate them directly by inserting na before the selbri, but Lojban also allows us to place the negation within the connective itself.

To negate the first or left-hand bridi, prefix na to the JA cmavo but after the i. To negate the second or right-hand bridi, suffix -nai to the JA cmavo. In either case, the negating word is placed on the side of the connective that is closest to the bridi being negated.

So to express the truth table FTTF, which requires O with either of the two bridi negated (not both), we can say either:

la djan. nanmu .inajo la djeimyz. ninmu That-named John is-not-a-man if-and-only-if that-named James is-a-woman.

la djan. nanmu .ijonai la djeimyz. ninmu That-named John is-a-man if-and-only-if that-named James is-not-a-woman.

The meaning of both Example 14.7 and Example 14.8 is the same as that of:

John is a man or James is a woman, but not both.

Here is another example:

la djan. nanmu .ijanai la djeimyz. ninmu That-named John is-a-man or that-named James is-not-a-woman. John is a man if James is a woman.

How's that again? Are those two English sentences in Example 14.10 really equivalent? In English, no. The Lojban TTFT truth function can be glossed A if B, but the if does not quite have its English sense. Example 14.10 is true so long as John is a man, even if James is not a woman; likewise, it is true just because James is not a woman, regardless of John's gender. This kind of if-then is technically known as a material conditional.

Since James is not a woman (by our assertions in Sec. 14.1), the English sentence John is a man if James is a woman seems to be neither true nor false, since it assumes something which is not true. It turns out to be most convenient to treat this if as TTFT, which on investigation means that Example 14.10 is true. Example 14.11, however, is equally true:

la djan. ninmu .ijanai la djeimyz. ninmu That-named John is-a-woman if that-named James is-a-woman.

This can be thought of as a principle of consistency, and may be paraphrased as follows: If a false statement is true, any statement follows from it. All uses of English if must be considered very carefully when translating into Lojban to see if they really fit this Lojban mold.

Example 14.12, which uses the TFTT truth function, is subject to the same rules: the stated gloss of TFTT as only if works naturally only when the right-hand bridi is false; if it is true, the left-hand bridi may be either true or false. The last gloss of Example 14.12 illustrates the use of if ... then as a more natural substitute for only if.

la djan. nanmu .inaja la djeimyz. ninmu That-named John is-not-a-man or that-named James is-a-woman. John is a man only if James is a woman.

The following example illustrates the use of se to, in effect, exchange the two sentences. The normal use of se is to (in effect) transpose places of a bridi, as explained in Sec. 5.11.

la djan. nanmu .iseju la djeimyz. ninmu Whether or not John is a man, James is a woman.

If both na and se are present, which is legal but never necessary, na would come before se.

The full syntax of ijeks, therefore, is:

where the cmavo in brackets are optional.

Forethought bridi connection#

Many concepts in Lojban are expressible in two different ways, generally referred to as afterthought and forethought. Sec. 14.4 discussed what is called afterthought bridi logical connection. The word afterthought is used because the connective cmavo and the second bridi were added, as it were, afterwards and without changing the form of the first bridi. This form might be used by someone who makes a statement and then wishes to add or qualify that statement after it has been completed. Thus,

is a complete bridi, and adding an afterthought connection to make

la djan. nanmu .ija la djeimyz. ninmu John is a man or James is a woman (or both)

provides additional information without requiring any change in the form of what has come before; changes which may not be possible or practical, especially in speaking. (The meaning, however, may be changed by the use of a negating connective.) Afterthought connectives make it possible to construct all the important truth-functional relationships in a variety of ways.

In forethought style the speaker decides in advance, before expressing the first bridi, that a logical connection will be expressed. Forethought and afterthought connectives are expressed with separate selma'o. The forethought logical connectives corresponding to afterthought ijeks are geks:

ga la djan. nanmu gi la djeimyz. ninmu Either John is a man or James is a woman (or both).

ga is the cmavo which represents the A truth function in selma'o GA. The word gi does not belong to GA at all, but constitutes its own selma'o: it serves only to separate the two bridi without having any content of its own. The English translation of ga…gi is either ... or, but in the English form the truth function is specified both by the word either and by the word or: not so in Lojban.

Even though two bridi are being connected, geks and giks do not have any i in them. The forethought construct binds up the two bridi into a single sentence as far as the grammar is concerned.

Some more examples of forethought bridi connection are:

ge la djan. nanmu gi la djeimyz. ninmu (It is true that) both John is a man and James is a woman.

gu la djan. nanmu gi la djeimyz. ninmu It is true that John is a man, whether or not James is a woman.

It is worth emphasizing that Example 14.18 does not assert that James is (or is not) a woman. The gu which indicates that la djeimyz. ninmu may be true or false is unfortunately rather remote from the bridi thus affected.

Perhaps the most important of the truth functions commonly expressed in forethought is TFTT, which can be paraphrased as if ... then ...:

ganai la djan. nanmu gi la djeimyz. ninmu Either that-named John is-not-a-man, or that-named James is-a-woman. If John is a man, then James is a woman.

Note the placement of the nai in Example 14.19. When added to afterthought selma'o such as JA, a following nai negates the second bridi, to which it is adjacent. Since GA cmavo precede the first bridi, a following nai negates the first bridi instead.

Why does English insist on forethought in the translation of Example 14.19? Possibly because it would be confusing to seemingly assert a sentence and then make it conditional (which, as the Lojban form shows, involves a negation). Truth functions which involve negating the first sentence may be confusing, even to the Lojbanic understanding, when expressed using afterthought.

It must be reiterated here that not every use of English if ... then is properly translated by .inaja or ganai…gi; anything with implications of time needs a somewhat different Lojban translation, which will be discussed in Sec. 14.18. Causal sentences like If you feed the pig, then it will grow are not logical connectives of any type, but rather need a translation using rinka as the selbri joining two event abstractions, thus:

le nu do cidja dunda fi le xarju The event-of (you food give to the pig)

Causality is discussed in far more detail in Sec. 9.7.

Example 14.21 and Example 14.22 illustrates a truth function, FTTF, which needs to negate either the first or the second bridi. We already understand how to negate the first bridi:

gonai la djan. nanmu gi la djeimyz. ninmu John is not a man if and only if James is a woman.

How can the second bridi be negated? By adding -nai to the gi.

go la djan. nanmu ginai la djeimyz. ninmu John is a man if and only if James is not a woman.

A compound cmavo based on gi is called a gik; the only giks are gi itself and gi****nai.

Further examples:

ge la djan. nanmu ginai la djeimyz. ninmu John is a man and James is not a woman.

ganai la djan. nanmu ginai la djeimyz. ninmu John is not a man or James is not a woman.

The syntax of geks is:

and of giks (which are not themselves connectives, but part of the machinery of forethought connection) is:

sumti connection#

Geks and ijeks are sufficient to state every possible logical connection between two bridi. However, it is often the case that two bridi to be logically connected have one or more portions in common:

la djan. klama le zarci .ije la .alis. klama le zarci John goes to the market, and Alice goes to the market.

Here only a single sumti differs between the two bridi. Lojban does not require that both bridi be expressed in full. Instead, a single bridi can be given which contains both of the different sumti and uses a logical connective from a different selma'o to combine the two sumti:

la djan .e la .alis. klama le zarci That-named John and that-named Alice go-to the market.

Example 14.26 means exactly the same thing as Example 14.25: one may be rigorously transformed into the other without any change of logical meaning. This rule is true in general for every different kind of logical connection in Lojban; all of them, with one exception (see Sec. 14.12), can always be transformed into a logical connection between sentences that expresses the same truth function.

The afterthought logical connectives between sumti are eks, which contain a connective cmavo of selma'o A. If ijeks were used in Example 14.26, the meaning would be changed:

la djan. -- .ije That-named John [is/does-something]. And

leaving the reader uncertain why John is mentioned at all.

Any ek may be used between sumti, even if there is no direct English equivalent:

la djan. .o la .alis. klama le zarci That-named John if-and-only-if that-named Alice goes-to the market. John goes to the market if, and only if, Alice does.

The second line of Example 14.27 is highly stilted English, but the first line (of which it is a literal translation) is excellent Lojban.

What about forethought sumti connection? As is the case for bridi connection, geks are appropriate. They are not the only selma'o of forethought logical-connectives, but are the most commonly used ones.

ga la djan. gi la .alis. klama le zarci Either John or Alice (or both) goes to the market.

Of course, eks include all the same patterns of compound cmavo that ijeks do. When na or se is part of an ek, a special writing convention is invoked, as in the following example:

la djan. na.a la .alis. klama le zarci That-named John only-if that-named Alice goes-to the market. John goes to the market only if Alice does.

Note the period in na**.a**. The cmavo of A begin with vowels, and therefore must always be preceded by a pause. It is conventional to write all connective compounds as single words (with no spaces), but this pause must still be marked in writing as in speech; otherwise, the na and a would tend to run together.

More than two propositions#

So far we have seen logical connectives used to connect exactly two sentences. How about connecting three or more? Is this possible in Lojban? The answer is yes, subject to some warnings and some restrictions.

Of the four primitive truth functions A, E, O, and U, all but O have the same truth values no matter how their component sentences are associated in pairs. Therefore,

mi dotco .ije mi ricfu .ije mi nanmu I am-German. And I am-rich. And I am-a-man.

means that all three component sentences are true. Likewise,

mi dotco .ija mi ricfu .ija mi nanmu I am-German. Or I am-rich. Or I am-a-man.

means that one or more of the component sentences is true.

O, however, is different. Working out the truth table for

mi dotco .ijo mi ricfu .ijo mi nanmu I am-German. If-and-only-if I am-rich. If-and-only-if I am-a-man.

shows that Example 14.33 does not mean that either I am all three of these things or none of them; instead, an accurate translation would be:

Of the three properties – German-ness, wealth, and manhood – I possess either exactly one or else all three.

Because of the counterintuitiveness of this outcome, it is safest to avoid O with more than two sentences. Likewise, the connectives which involve negation also have unexpected truth values when used with more than two sentences.

In fact, no combination of logical connectives can produce the all or none interpretation intended (but not achieved) by Example 14.33 without repeating one of the bridi. See Example 14.48.

There is an additional difficulty with the use of more than two sentences. What is the meaning of:

mi nelci la djan. .ije mi nelci la martas. I like that-named John. And I like that-named Martha.

Does this mean:

I like John, and I like either Martha or Mary or both.

Or is the correct translation:

Either I like John and I like Martha, or I like Mary, or both.

Example 14.36 is the correct translation of Example 14.34. The reason is that Lojban logical connectives pair off from the left, like many constructs in the language. This rule, called the left-grouping rule, is easy to forget, especially when intuition pulls the other way. Forethought connectives are not subject to this problem:

ga ge mi nelci la djan. Either (Both I like that-named John

is equivalent in meaning to Example 14.34, whereas

ge mi nelci la djan. Both I like that-named John

is not equivalent to Example 14.34, but is instead a valid translation into Lojban, using forethought, of Example 14.35.

Grouping of afterthought connectives#

There are several ways in Lojban to render Example 14.35 using afterthought only. The simplest method is to make use of the cmavo bo (of selma'o BO). This cmavo has several functions in Lojban, but is always associated with high precedence and short scope. In particular, if bo is placed after an ijek, the result is a grammatically distinct kind of ijek which overrides the regular left-grouping rule. Connections marked with bo are interpreted before connections not so marked. Example 14.39 is equivalent in meaning to Example 14.38:

mi nelci la djan. .ije mi nelci la martas. I like that-named John, and I like that-named Martha

The English translation feebly indicates with a comma what the Lojban marks far more clearly: the I like Martha and I like Mary sentences are joined by .i****ja first, before the result is joined to I like John by .i****je.

Eks can have bo attached in exactly the same way, so that Example 14.40 is equivalent in meaning to Example 14.39:

Forethought connectives, however, never can be suffixed with bo, for every use of forethought connectives clearly indicates the intended pattern of grouping.

What happens if bo is used on both connectives, giving them the same high precedence, as in Example 14.41?

Does this wind up meaning the same as Example 14.34 and Example 14.36? Not at all. A second rule relating to bo is that where several bo-marked connectives are used in succession, the normal Lojban left-grouping rule is replaced by a right-grouping rule. As a result, Example 14.41 in fact means the same as Example 14.39 and Example 14.40. This rule may be occasionally exploited for special effects, but is tricky to keep straight; in writing intended to be easy to understand, multiple consecutive connectives marked with bo should be avoided.

The use of bo, therefore, gets tricky in complex connections of more than three sentences. Looking back at the English translations of Example 14.37 and Example 14.38, parentheses were used to clarify the grouping. These parentheses have their Lojban equivalents, two sets of them actually. tu'e and tu'u are used with ijeks, and ke and ke'e with eks and other connectives to be discussed later. (ke and ke'e are also used in other roles in the language, but always as grouping markers). Consider the English sentence:

I kiss you and you kiss me, if I love you and you love me.

where the semantics tells us that the instances of and are meant to have higher precedence than that of if. If we wish to express Example 14.42 in afterthought, we can say:

mi cinba do .ije[bo] do cinba mi I kiss you and you kiss me,

marking two of the ijeks with bo for high precedence. (The first bo is not strictly necessary, because of the left-grouping rule, and is shown here in brackets.)

But it may be clearer to use explicit parenthesis words and say:

tu'e mi cinba do .ije do cinba mi tu'u ( I kiss you and you kiss me )

where the tu'e…tu'u pairs set off the structure. The cmavo tu'u is an elidable terminator, and its second occurrence in Example 14.44 is bracketed, because all terminators may be elided at the end of a text.

In addition, parentheses are a general solution: multiple parentheses may be nested inside one another, and additional afterthought material may be added without upsetting the existing structure. Neither of these two advantages apply to bo grouping. In general, afterthought constructions trade generality for simplicity.

Because of the left-grouping rule, the first set of tu'e…tu'u parentheses may actually be left off altogether, producing:

mi cinba do .ije do cinba mi I kiss you and you kiss me

What about parenthesized sumti connection? Consider

I walk to either the market and the house, or the school and the office.

Two pairs of parentheses, analogous to Example 14.44, would seem to be the right approach. However, it is a rule of Lojban grammar that a sumti may not begin with ke, so the first set of parentheses must be omitted, producing Example 14.47, which is instead parallel to Example 14.45:

mi dzukla le zarci .e le zdani I walk-to the market and the house

If sumti were allowed to begin with ke, unavoidable ambiguities would result, so ke grouping of sumti is allowed only just after a logical connective. This rule does not apply to tu'e grouping of bridi, as Example 14.44 shows.

Now we have enough facilities to handle the problem of Example 14.33: I am German, rich, and a man – or else none of these. The following paraphrase has the correct meaning:

[tu'e] mi dotco .ijo mi ricfu [tu'u] ( I am-German if-and-only-if I am-rich )

The truth table, when worked out, produces T if and only if all three component sentences are true or all three are false.

Compound bridi#

So far we have seen how to handle two sentences that need have no similarity at all (bridi connection) and sentences that are identical except for a difference in one sumti (sumti connection). It would seem natural to ask how to logically connect sentences that are identical except for having different selbri.

Surprise! Lojban provides no logical connective that is designed to handle selbri and nothing else. Instead, selbri connection is provided as part of a more general-purpose mechanism called compound bridi. Compound bridi result from logically connecting sentences that differ in their selbri and possibly some of their sumti.

The simplest cases result when the x₁ sumti is the only common point:

mi klama le zarci .ije mi nelci la djan. I go-to the market, and I like that-named John.

is equivalent in meaning to the compound bridi:

mi klama le zarci gi'e nelci la djan. I go-to the market and like that-named John.

As Example 14.50 indicates, giheks are used in afterthought to create compound bridi; gi'e is the gihek corresponding to and. The actual phrases klama le zarci and nelci la djan. that the gihek connects are known as bridi-tails, because they represent (in this use) the tail end of a bridi, including the selbri and any following sumti, but excluding any sumti that precede the selbri:

mi ricfu gi'e klama le zarci I am-rich and go-to the market.

In Example 14.51, the first bridi-tail is ricfu, a simple selbri, and the second bridi-tail is klama le zarci, a selbri with one following sumti.

Suppose that more than a single sumti is identical between the two sentences:

mi dunda le cukta do .ije mi lebna lo rupnu do I give the book to-you, and I take some currency-units from-you.

In Example 14.52, the first and last sumti of each bridi are identical; the selbri and the second sumti are different. By moving the final sumti to the beginning, a form analogous to Example 14.50 can be achieved:

fi do fa mi dunda le cukta to/from you - I give the book

where the fi does not have an exact English translation because it merely places do in the third place of both lebna and dunda. However, a form that preserves natural sumti order also exists in Lojban. Giheks connect two bridi-tails, but also allow sumti to be added following the bridi-tail. These sumti are known as tail-terms, and apply to both bridi. The straightforward gihek version of Example 14.52 therefore is:

mi dunda le cukta gi'e lebna lo rupnu vau do I (give the book) and (take some currency-units) - to/from-you.

The vau (of selma'o VAU) serves to separate the bridi-tail from the tail-terms. Every bridi-tail is terminated by an elidable vau, but only in connection with compound bridi is it ever necessary to express this vau. Thus:

mi klama le zarci [vau] I go-to the market.

has a single elided vau, and Example 14.50 is equivalent to:

where the double vau at the end of Example 14.56 terminates both the right-hand bridi-tail and the unexpressed tail-terms.

A final use of giheks is to combine bridi-tails used as complete sentences, the Lojban observative:

klama le zarci gi'e dzukla le briju A-goer to-the market and a-walker to-the office.

Since x₁ is omitted in both of the bridi underlying Example 14.57, this compound bridi does not necessarily imply that the goer and the walker are the same. Only the presence of an explicit x₁ (other than zo'e, which is equivalent to omission) can force the goer and the walker to be identical.

A strong argument for this convention is provided by analysis of the following example:

klama la nu,IORK. A-goer to-that-named New-York

If the rule were that the x₁ places of the two underlying bridi were considered identical, then (since there is nothing special about x₁), the unspecified x₄ (route) and x₅ (means) places would also have to be the same, leading to the absurd result that the route from Phoenix to New York is the same as the route from Rome to New York. Inserting da, meaning roughly something, into the x₁ place cures the problem:

da klama la nu,IORK. la finyks. Something is-a-goer to-that-named New-York from-that-named Phoenix

The syntax of giheks is:

which is exactly parallel to the syntax of eks.

Multiple compound bridi#

Giheks can be combined with bo in the same way as eks:

mi nelci la djan. gi'e nelci la martas. gi'abo nelci la meris. I like John and ( like Martha or like Mary ).

is equivalent in meaning to Example 14.39 and Example 14.40. Likewise, ke…ke'e grouping can be used after giheks:

mi dzukla le zarci I walk-to the market

is the gihek version of Example 14.47. The same rule about using ke…ke'e bracketing only just after a connective applies to bridi-tails as to sumti, so the first two bridi-tails in Example 14.61 cannot be explicitly grouped; implicit left-grouping suffices to associate them.

Each of the pairs of bridi-tails joined by multiple giheks can have its own set of tail-terms:

- mi dejni lo rupnu la djan. [If] I owe some currency-units to-that-named John,

is equivalent in meaning to:

- mi dejni lo rupnu nagi'a dunda [If] I owe some currency-units then (give

The literal English translation in Example 14.63 is almost unintelligible, but the Lojban is perfectly grammatical. mi fills the x₁ place of all three selbri; lo rupnu is the x₂ of dejni, whereas le cukta is a tail-term shared between dunda and lebna; la djan. is a tail-term shared by dejni and by dunda gi'abo lebna. In this case, greater clarity is probably achieved by moving la djan. to the beginning of the sentence, as in Example 14.53:

fi la djan. - fa mi dejni lo rupnu To/from that-named John, [if] - I owe some currency-units

Finally, what about forethought logical connection of bridi-tails? There is no direct mechanism for the purpose. Instead, Lojban grammar allows a pair of forethought-connected sentences to function as a single bridi-tail, and of course the sentences need not have terms before their selbri. For example:

mi ge klama le zarci gi nelci la djan. I both go-to the market and like that-named John.

is equivalent in meaning to Example 14.50.

Of course, either of the connected sentences may contain giheks:

mi ge klama le zarci gi'e dzukla le zdani I both (go to-the market and walk to-the house)

The entire gek-connected sentence pair may be negated as a whole by prefixing na:

- mi na ge klama le zarci gi dzukla le zdani [False!] I - both go-to the market and walk-to the house.

Since a pair of sentences joined by geks is the equivalent of a bridi-tail, it may be followed by tail terms. The forethought equivalent of Example 14.54 is:

mi ge dunda le cukta I both (give the book)

Here is a pair of gek-connected observatives, a forethought equivalent of Example 14.57:

ge klama le zarci gi dzukla le briju Both a-goer to-the market and a-walker to-the office.

Finally, here is an example of gek-connected sentences with both shared and unshared terms before their selbri:

mi gonai le zarci cu klama gi le bisli cu dansu I either-but-not-both to-the office - go or on-the ice - dance. I either go to the office or dance on the ice (but not both).

Termset logical connection#

So far we have seen sentences that differ in all components, and require bridi connection; sentences that differ in one sumti only, and permit sumti connection; and sentences that differ in the selbri and possibly one or more sumti, and permit bridi-tail connection. Termset logical connectives are employed for sentences that differ in more than one sumti but not in the selbri, such as:

I go to the market from the office and to the house from the school.

The Lojban version of Example 14.71 requires two termsets joined by a logical connective. A term is either a sumti or a sumti preceded by a tense or modal tag such as pu or bai. Afterthought termsets are formed by linking terms together by inserting the cmavo ce'e (of selma'o CEhE) between each of them. Furthermore, the logical connective (which is a jek) must be prefixed by the cmavo pe'e (of selma'o PEhE). (We could refer to the combination of pe'e and a jek as a pehejek, I suppose.)

mi klama le zarci ce'e le briju I go to-the market [plus] from-the office

The literal translation uses [plus] to indicate the termset connective, and [joint] to indicate the position of the logical connective joint. As usual, there is an equivalent bridi-connection form:

mi klama le zarci le briju I go to-the market from-the office,

which illustrates that the two bridi differ in the x₂ and x₃ places only.

What happens if the two joined sets of terms are of unequal length? Expanding to bridi connection will always make clear which term goes in which place of which bridi. It can happen that a sumti may fall in the x₂ place of one bridi and the x₃ place of another:

mi pe'e ja do ce'e le zarci cu klama le briju I [joint] or you [plus] to-the market - go to/from-the office.

can be clearly understood by expansion to:

mi klama le briju .ija do le zarci cu klama I go to-the office, or you to-the market - go

So le briju is your origin but my destination, and thus falls in the x₂ and x₃ places of klama simultaneously! This is legal because even though there is only one selbri, klama, there are two distinct bridi expressed here. In addition, mi in Example 14.74 is serving as a termset containing only one term. An analogous paradox applies to compound bridi with tail-terms and unequal numbers of sumti within the connected bridi-tails:

mi -- klama le zarci gi'e dzukla vau le briju I ( go to-the market and walk ) to/from-the office.

means that I go to the market from the office, and I walk to the office; le briju is the x₃ place of klama and the x₂ place of dzukla.

Forethought termsets also exist, and use nu'i of selma'o NUhI to signal the beginning and nu'u of selma'o NUhU (an elidable terminator) to signal the end. Nothing is inserted between the individual terms: they simply sit side-by-side. To make a logical connection in a forethought termset, use a gek, with the gek just after the nu'i, and an extra nu'u just before the gik:

mi klama nu'i ge le zarci le briju I go [start-termset] both to-the market from-the office

Note that even though two termsets are being connected, only one nu'i is used.

The grammatical uses of termsets that do not contain logical connectives are explained in Sec. 9.8, Sec. 10.25, and Sec. 16.7.

Logical connection within tanru#

As noted at the beginning of Sec. 14.9, there is no logical connective in Lojban that joins selbri and nothing but selbri. However, it is possible to have logical connectives within a selbri, forming a kind of tanru that involves a logical connection. Consider the simple tanru blanu zdani, blue house. Now anything that is a blue ball, in the most ordinary understanding of the phrase at least, is both blue and a ball. And indeed, instead of blanu bolci, Lojbanists can say blanu je bolci, using a jek connective within the tanru. (We saw jeks used in Sec. 14.11 also, but there they were always prefixed by pe'e; in this section they are used alone.) Here is a pair of examples:

ti blanu zdani This is-a-blue-type-of house.

ti blanu je zdani This is-blue and is-a-house.

But of course Example 14.78 and Example 14.79 are not necessarily equivalent in meaning! It is the most elementary point about Lojban tanru that Example 14.78 might just as well mean

This is a house for blue inhabitants.

and Example 14.79 certainly is not equivalent in meaning to Example 14.80.

A full explanation of logical connection within tanru belongs rather to a discussion of selbri structure than to logical connectives in general. Why? Because although Example 14.79 happens to mean the same as

and therefore as

the rule of expansion into separate bridi simply does not always work for tanru connection. Supposing Alice to be a person who lives in blue houses, then

la .alis. cu - blanu je zdani - prenu That-named Alice - is-a-( blue and house ) type-of-person.

would be true, because tanru grouping with a jek has higher precedence than unmarked tanru grouping, but:

la .alis. cu - blanu prenu That-named Alice - is-a blue person,

is probably false, because the blueness is associated with the house, not with Alice, even leaving aside the question of what it means to say Alice is a blue person. (Perhaps she belongs to the Blue team, or is wearing blue clothes.) The semantic ambiguity of tanru make such logical manipulations impossible.

It suffices to note here, then, a few purely grammatical points about tanru logical connection. bo may be appended to jeks as to eks, with the same rules:

la teris. cu ricfu je nakni jabo fetsi That-named Terry - is-rich and (male or female).

The components of tanru may be grouped with ke both before and after a logical connective:

la .teris. cu [ke] ricfu ja pindi [ke'e] That-named Terry - ( is-rich or is-poor )

where the first ke…ke'e pair may be omitted altogether by the rule of left-grouping, but is optionally permitted. In any case, the last instance of ke'e may be elided.

The syntax of jeks is:

parallel to eks and giheks.

Forethought tanru connection does not use geks, but uses guheks instead. Guheks have exactly the same form as geks:

Using guheks in tanru connection (rather than geks) resolves what would otherwise be an unacceptable ambiguity between bridi-tail and tanru connection:

la .alis. gu'e ricfu gi fetsi That-named Alice is-both rich and female.

Note that giks are used with guheks in exactly the same way they are used with geks. Like jeks, guheks bind more closely than unmarked tanru grouping does:

la .alis. gu'e blanu gi zdani prenu That-named Alice is-a-(both blue and a-house) type-of-person.

is the forethought version of Example 14.83.

A word of caution about the use of logically connected tanru within descriptions. English-based intuition can lead the speaker astray. In correctly reducing

mi viska pa nanmu .ije mi viska pa ninmu I see a man, and I see a woman.

to

mi viska pa nanmu .e pa ninmu I see a man and a woman.

there is a great temptation to reduce further to:

mi viska pa nanmu je ninmu I see a man and woman.

But Example 14.91 means that you see one thing which is both a man and a woman simultaneously! A nanmu je ninmu is a manwoman, a presumably non-existent creature who is both a nanmu and a ninmu.

Truth questions and connective questions#

So far we have addressed only sentences which are statements. Lojban, like all human languages, needs also to deal with sentences which are questions. There are many ways of asking questions in Lojban, but some of these (like questions about quantity, tense, and emotion) are discussed in other chapters.

The simplest kind of question is of the type Is it true that ... where some statement follows. This type is called a truth question, and can be represented in English by Example 14.92:

Is it true that Fido is a dog? Is Fido a dog?

Note the two formulations. English truth questions can always be formed by prefixing Is is true that to the beginning of a statement; there is also usually a more idiomatic way involving putting the verb before its subject. Is Fido a dog? is the truth question corresponding to Fido is a dog. In Lojban, the equivalent mechanism is to prefix the cmavo xu (of selma'o UI) to the statement:

xu la faidon. gerku Is-it-true-that that-named Fido is-a-dog?

Example 14.92 and Example 14.93 are equivalent in meaning.

A truth question can be answered yes or no, depending on the truth or falsity, respectively, of the underlying statement. The standard way of saying yes in Lojban is go'i and of saying no is na****go'i. (The reasons for this rule are explained in Sec. 7.6.) In answer to Example 14.93, the possible answers are:

go'i Fido is a dog.

and

nago'i Fido is not a dog.

Some English questions seemingly have the same form as the truth questions so far discussed. Consider

Is Fido a dog or a cat?

Superficially, Example 14.96 seems like a truth question with the underlying statement:

Fido is a dog or a cat.

By translating Example 14.97 into Lojban and prefixing xu to signal a truth question, we get:

xu la faidon. gerku gi'onai mlatu Is-it-true-that that-named Fido is-a-dog or is-a-cat (but--not--both)?

Given that Fido really is either a dog or a cat, the appropriate answer would be go'i; if Fido were a fish, the appropriate answer would be na****go'i.

But that is not what an English-speaker who utters Example 14.96 is asking! The true significance of Example 14.96 is that the speaker desires to know the truth value of either of the two underlying bridi (it is presupposed that only one is true).

Lojban has an elegant mechanism for rendering this kind of question which is very unlike that used in English. Instead of asking about the truth value of the connected bridi, Lojban users ask about the truth function which connects them. This is done by using a special question cmavo: there is one of these for each of the logical connective selma'o, as shown by the following table:

(This list unfortunately departs from the pretty regularity of the other cmavo for logical connection. The two-syllable selma'o, GIhA and GUhA, make use of the cmavo ending in -i which is not used for a truth function, but gi and i were not available, and different cmavo had to be chosen. This table must simply be memorized, like most other non-connective cmavo assignments.)

One correct translation of Example 14.96 employs a question gihek:

la .alis. gerku gi'i mlatu That-named Alice is-a-dog [truth-function?] is-a-cat?

Here are some plausible answers:

nagi'e Alice is not a dog and is a cat.

gi'enai Alice is a dog and is not a cat.

nagi'enai Alice is not a dog and is not a cat.

nagi'o Alice is a dog or is a cat but not both (I'm not saying which).

Example 14.103 is correct but uncooperative.

As usual, Lojban questions are answered by filling in the blank left by the question. Here the blank is a logical connective, and therefore it is grammatical in Lojban to utter a bare logical connective without anything for it to connect.

The answer gi'e, meaning that Alice is a dog and is a cat, is impossible in the real world, but for:

do djica tu'a loi ckafi You desire something-about a-mass-of coffee

the answer e, meaning that I want both, is perfectly plausible, if not necessarily polite.

The forethought questions ge'i and gu'i are used like the others, but ambiguity forbids the use of isolated forethought connectives as answers – they sound like the start of forethought-connected bridi. So although Example 14.105 is the forethought version of Example 14.104:

do djica tu'a ge'i loi ckafi You desire something-about [truth-function?] a-mass-of coffee

the answer must be in afterthought form.

There are natural languages, notably Chinese, which employ the Lojbanic form of connective question. The Chinese sentence

means Do you walk or run?, and is exactly parallel to the Lojban:

do cadzu gi'i bajra You walk [or?] run?

However, Chinese does not use logical connectives in the reply to such a question, so the resemblance, though striking, is superficial.

Truth questions may be used in bridi connection. This form of sentence is perfectly legitimate, and can be interpreted by using the convention that a truth question is true if the answer is yes and false if the answer is no. Analogously, an imperative sentence (involving the special pro-sumti ko, which means you but marks the sentence as a command) is true if the command is obeyed, and false otherwise. A request of Abraham Lincoln's may be translated thus:

ganai ti ckafi gi ko bevri loi tcati mi If this is-coffee then [you!] bring a-mass-of tea to-me,

In logical terms, however, but is the same as and; the difference is that the sentence after a but is felt to be in tension or opposition to the sentence before it. Lojban represents this distinction by adding the discursive cmavo ku'i (of selma'o UI), which is explained in Sec. 13.12, to the logical .i****je.)

Non-logical connectives#

Way back in Sec. 14.1, the point was made that not every use of English and, if ... then, and so on represents a Lojban logical connective. In particular, consider the and of:

John and Alice carried the piano.

Given the nature of pianos, this probably means that John carried one end and Alice the other. So it is not true that:

John carried the piano, and Alice carried the piano.

which would mean that each of them carried the piano by himself/herself. Lojban deals with this particular linguistic phenomenon as a mass. John and Alice are joined together into a mass, John-and-Alice, and it is this mass which carried the piano, not either of them separately. The cmavo joi (of selma'o JOI) is used to join two or more components into a mass:

la djan. joi la .alis. cu bevri le pipno That-named John massed-with that-named Alice - carry the piano.

Example 14.111 covers the case mentioned, where John and Alice divide the labor; it also could mean that John did all the hauling and Alice did the supervising. This possibility arises because the properties of a mass are the properties of its components, which can lead to apparent contradictions: if John is small and Alice is large, then John-and-Alice is both small and large. Masses are also discussed in Sec. 6.3.

Grammatically, joi can appear between two sumti (like an ek) or between two tanru components (like a jek). This flexibility must be paid for in the form of occasional terminators that cannot be elided:

le nanmu ku joi le ninmu [ku] cu klama le zarci The man - massed-with the woman - - go-to the market.

The cmavo ku is the elidable terminator for le, which can almost always be elided, but not in this case. If the first ku were elided here, Lojban's parsing rules would see le nanmu joi and assume that another tanru component is to follow; since the second le cannot be part of a tanru, a parsing error results. No such problem can occur with logical connectives, because an ek signals a following sumti and a jek a following tanru component unambiguously.

Single or compound cmavo involving members of selma'o JOI are called joiks, by analogy with the names for logical connectives. It is not grammatical to use joiks to connect bridi-tails.

In tanru, joi has the connotation mixed with, as in the following example:

ti blanu joi xunre bolci This is-a-(blue mixed-with red) ball. This is a blue and red ball.

Here the ball is neither wholly blue nor wholly red, but partly blue and partly red. Its blue/redness is a mass property. (Just how blue something has to be to count as wholly blue is an unsettled question, though. A blanu zdani may be so even though not every part of it is blue.)

There are several other cmavo in selma'o JOI which can be used in the same grammatical constructions. Not all of them are well-defined as yet in all contexts. All have clear definitions as sumti connectives; those definitions are shown in the following table:

The cmavo se is grammatical before any JOI cmavo, but only useful with those that have inherent order. Here are some examples of joiks:

mi cuxna la .alis. la frank. I choose that-named Alice from-that-named Frank

The x₃ place of cuxna is a set from which the choice is being made. A set is an abstract object which is determined by specifying its members. Unlike those of a mass, the properties of a set are unrelated to its members' properties: the set of all rats is large (since many rats exist), but the rats themselves are small. This chapter does not attempt to explain set theory (the mathematical study of sets) in detail: explaining propositional logic is quite enough for one chapter!

In Example 14.114 we specify that set by listing the members with ce joining them.

ti liste mi ce'o do ce'o la djan. This is-a-list-of me and-sequence you and-sequence that-named John. This is a list of you, me, and John.

The x₂ place of liste is a sequence of the things which are mentioned in the list. (It is worth pointing out that lo liste means a physical object such as a grocery list: a purely abstract list is lo porsi, a sequence.) Here the three sumti connected by ce'o are in a definite order, not just lumped together in a set or a mass.

So joi, ce, and ce'o are parallel, in that the sumti connected are taken to be individuals, and the result is something else: a mass, a set, or a sequence respectively. The cmavo jo'u serves as a fourth element in this pattern: the sumti connected are individuals, and the result is still individuals – but inseparably so. The normal Lojban way of saying that James and George are brothers is:

la djeimyz. bruna la djordj. That-named James is-the-brother-of that-named George.

possibly adding a discursive element meaning and vice versa. However, James and George are brothers cannot be correctly translated as:

la djeimyz. .e la djordj. bruna That-named James and that-named George is-a-brother.

since that expands to two bridi and means that James is a brother and so is George, but not necessarily of each other. If the e is changed to jo'u, however, the meaning of Example 14.116 is preserved:

la djeimyz. jo'u That-named James in-common-with that-named

The tanru remei bruna is not strictly necessary in this sentence, but is used to make clear that we are not saying that James and George are both brothers of some third person not specified. Alternatively, we could turn the tanru around: the x₁ place of re****mei is a mass with two components, leading to:

la djeimyz. joi That-named James massed-with

where joi is used to create the necessary mass.

Likewise, fa'u can be used to put two individuals together where order matters. Typically, there will be another fa'u somewhere else in the same bridi:

la djeimyz. fa'u la djordj. That-named James jointly-in-order-with that-named George

Here the information carried by the English adverb respectively, namely that James loves Mary and George loves Martha, is divided between the two occurrences of fa'u. If both uses of fa'u were to be changed to e, we would get:

la djeimyz. .e la djordj. prami That-named James and that-named George love

which can be transformed to four bridi:

la djeimyz. prami la meris. .ije la djordj. prami That-named James loves that-named Mary, and that-named George loves

which represents quite a different state of affairs from Example 14.120. The meaning of Example 14.120 can also be conveyed by a termset:

la djeimyz. ce'e la meris. pe'e That-named James [plus] that-named Mary [joint]

at the expense of re-ordering the list of names so as to make the pairs explicit. This option is not available when one of the lists is only described rather than enumerated:

la djeimyz. fa'u la djordj. prami re mensi That-named James and-respectively that-named George love two sisters.

which conveys that James loves one sister and George the other, though we are not able to tell which of the sisters is which.

More about non-logical connectives#

The final three JOI cmavo, jo'e, ku'a, and pi'u, are probably only useful when talking explicitly about sets. They represent three standard set operators usually called union, intersection, and cross product (also known as Cartesian product). The union of two sets is a set containing all the members that are in either set; the intersection of two sets is a set containing all the members that are in both sets. The cross product of two sets is the set of all possible ordered pairs, where each ordered pair contains a single element from the first set followed by a single element from the second. This may seem very abstract; hopefully, the following examples will help:

lo'i ricfu ku jo'e lo'i dotco cu barda The-set-of rich-things - union the-set-of German-things - is-large.

lo'i ricfu ku ku'a lo'i dotco cu cmalu The-set-of rich-things - intersection the-set-of German-things - is-small.

There is a parallelism between logic and set theory that makes Example 14.125 and Example 14.126 equivalent respectively to:

lo'i ricfu ja dotco cu barda The-set-of (rich-things or German-things) - is-large.

and

lo'i ricfu je dotco cu cmalu The-set-of (rich-things and German-things) - is-small.

The following example uses se remei, which is a set (not a mass) of two elements:

la djeimyz. ce[bo] la djordj. pi'u That-named James and-set that-named George cross-product

means that each of the pairs James/Mary, George/Mary, James/Martha, and George/Martha love each other. Therefore it is similar in meaning to Example 14.121; however, that example speaks only of the men loving the women, not vice versa.

Joiks may be combined with bo or with ke in the same way as eks and jeks; this allows grouping of non-logical connections between sumti and tanru units, in complete parallelism with logical connections:

mi joibo do ce la djan. joibo la djein. (I massed-with you) and (that-named John massed-with that-named Jane)

asserts that there is a set of two items each of which is a mass.

Non-logical connection is permitted at the joint of a termset; this is useful for associating more than one sumti or tagged sumti with each side of the non-logical connection. The place structure of casnu is:

so the x₁ place must be occupied by a mass (for reasons not explained here); however, different components of the mass may discuss in different languages. To associate each participant with his or her language, we can say:

mi ce'e bau la lojban. pe'e joi (I [plus] in-language that-named Lojban [joint] massed-with

Like all non-logical connectives, the usage shown in Example 14.131 cannot be mechanically converted into a non-logical connective placed at another location in the bridi. The forethought equivalent of Example 14.131 is:

Non-logical forethought termsets are also useful when the things to be non-logically connected are sumti preceded with tense or modal (BAI) tags:

la djan. fa'u la frank. cusku That-named John respectively-with that-named Frank express

Example 14.133 associates speaking in Lojban with John, and speaking under George's compulsion with Frank. We do not know what language Frank uses, or whether John speaks under anyone's compulsion.

Joiks may be prefixed with i to produce ijoiks, which serve to non-logically connect sentences. The ijoik .i****ce'o indicates that the event of the second bridi follows that of the first bridi in some way other than a time relationship (which is handled with a tense):

mi ba gasnu la'e di'e .i I [future] do the-referent-of the-following: -

Example 14.134 represents a list of things to be done in priority order. The order is important, hence the need for a sequence connective, but does not necessarily represent a time order (the dog may end up getting walked first). Note the use of tu'e and tu'u as general brackets around the whole list. This is related to, but distinct from, their use in Sec. 14.8, because there is no logical connective between the introductory phrase mi ba gasnu la'edi'e and the rest. The brackets effectively show how large an utterance the word di'e, which means the following utterance, refers to.

Similarly, .i****joi is used to connect sentences that represent the components of a joint event such as a joint cause: the Lojban equivalent of Fran hit her head and fell out of the boat, so that she drowned would join the events Fran hit her head and Fran fell out of the boat with .i****joi.

The following nai, if present, does not negate either of the things to be connected, but instead specifies that some other connection (logical or non-logical) is applicable: it is a scalar negation:

mi jo'u nai do cu remei I in-common-with [not!] you - are-a-twosome

The result of mi jo'u do would be two individuals, not a mass, therefore jo'u is not applicable; joi would be the correct connective.

There is no joik question cmavo as such; however, joiks and ijoiks may be uttered in isolation in response to a logical connective question, as in the following exchange:

do djica tu'a loi ckafi You desire something-about a-mass-of coffee

joi Mixed-mass-and. Both as a mass (i.e, mixed together).

Ugh. (Or in Lojban: .a'unaisai****ro'o.)

Interval connectives and forethought non-logical connection#

In addition to the non-logical connectives of selma'o JOI explained in Sec. 14.14 and Sec. 14.15, there are three other connectives which can appear in joiks: bi'i, bi'o, and mi'i, all of selma'o BIhI. The first two cmavo are used to specify intervals: abstract objects defined by two endpoints. The cmavo bi'i is correct if the endpoints are independent of order, whereas bi'o or se****bi'o are used when order matters.

An example of bi'i in sumti connection:

mi ca sanli I [present] stand-on-surface

In Example 14.138, it is all the same whether I am standing between Dresden and Frankfurt or between Frankfurt and Dresden, so bi'i is the appropriate interval connective. The sumti la drezdn. bi'i la frankfurt. falls into the x₂ place of sanli, which is the surface I stand on; the interval specifies that surface by its limits. (Obviously, I am not standing on the whole of the interval; the x₂ place of sanli specifies a surface which is typically larger in extent than just the size of the stander's feet.)

mi cadzu ca la pacac. I walk simultaneous-with - First-hour

In Example 14.139, on the other hand, it is essential that la pacac. comes before la recac.; otherwise we have an 11-hour (or 23-hour) interval rather than a one-hour interval. In this use of an interval, the whole interval is probably intended, or at least most of it.

Example 14.139 is equivalent to:

mi cadzu ca la recac. I walk simultaneous-with - Second-hour

English cannot readily express se****bi'o, but its meaning can be understood by reversing the two sumti.

The third cmavo of selma'o BIhI, namely mi'i, expresses an interval seen from a different viewpoint: not a pair of endpoints, but a center point and a distance. For example:

le jbama pu daspo la .uacintyn. The bomb [past] destroys - Washington

Here we have an interval whose center is Washington and whose distance, or radius, is fifty miles.

In Example 14.138, is it possible that I am standing in Dresden (or Frankfurt) itself? Yes. The connectives of selma'o BIhI are ambiguous about whether the endpoints themselves are included in or excluded from the interval. Two auxiliary cmavo ga'o and ke'i (of cmavo GAhO) are used to indicate the status of the endpoints: ga'o means that the endpoint is included, ke'i that it is excluded:

mi ca sanli la drezdn. ga'o I [present] stand that-named Dresden [inclusive]

mi ca sanli la drezdn. ga'o I [present] stand that-named Dresden [inclusive]

mi ca sanli la drezdn. ke'i I [present] stand that-named Dresden [exclusive]

mi ca sanli la drezdn. ke'i I [present] stand that-named Dresden [exclusive]

As these examples should make clear, the GAhO cmavo that applies to a given endpoint is the one that stands physically adjacent to it: the left-hand endpoint is referred to by the first GAhO, and the right-hand endpoint by the second GAhO. It is ungrammatical to have just one GAhO.

(Etymologically, ga'o is derived from ganlo, which means closed, and ke'i from kalri, which means open. In mathematics, inclusive intervals are referred to as closed intervals, and exclusive intervals as open ones.)

BIhI joiks are grammatical anywhere that other joiks are, including in tanru connection and (as ijoiks) between sentences. No meanings have been found for these uses.

Negated intervals, marked with a -nai following the BIhI cmavo, indicate an interval that includes everything but what is between the endpoints (with respect to some understood scale):

do dicra .e'a mi ca la daucac. You disturb (allowed) me at that-named 10

The complete syntax of joiks is:

• [se] JOI [nai]
• [se] BIhI [nai]
• GAhO [se] BIhI [nai] GAhO

Notice that the colloquial English translations of bi'i and bi'o have forethought form: between ... and for bi'i, and from ... to for bi'o. In Lojban too, non-logical connectives can be expressed in forethought. Rather than using a separate selma'o, the forethought logical connectives are constructed from the afterthought ones by suffixing gi. Such a compound cmavo is not unnaturally called a joigik; the syntax of joigiks is any of:

• [se] JOI [nai] GI
• [se] BIhI [nai] GI
• GAhO [se] BIhI [nai] GAhO GI

Joigiks may be used to non-logically connect bridi, sumti, and bridi-tails; and also in termsets.

Example 14.111 in forethought becomes:

joigi la djan. gi la .alis. bevri le pipno [Together] that-named John and that-named Alice carry the piano.

The first gi is part of the joigik; the second gi is the regular gik that separates the two things being connected in all forethought forms.

Example 14.143 can be expressed in forethought as:

mi ca sanli ke'i bi'i I [present] stand [exclusive] between

In forethought, unfortunately, the GAhOs become physically separated from the endpoints, but the same rule applies: the first GAhO refers to the first endpoint.

Logical and non-logical connectives within mekso#

Lojban has a separate grammar embedded within the main grammar for representing mathematical expressions (or mekso in Lojban) such as 2 + 2. Mathematical expressions are explained fully in Ch. 18. The basic components of mekso are operands, like 2, and operators, like +. Both of these may be either logically or non-logically connected.

Operands are connected in afterthought with eks and in forethought with geks, just like sumti. Operators, on the other hand, are connected in afterthought with jeks and in forethought with guheks, just like tanru components. (However, jeks and joiks with bo are not allowed for operators.) This parallelism is no accident.

In addition, eks with bo and with ke…ke'e are allowed for grouping logically connected operands, and ke…ke'e is allowed for grouping logically connected operators, although there is no analogue of tanru among the operators.

Only a few examples of each kind of mekso connection will be given. Despite the large number of rules required to support this feature, it is of relatively minor importance in either the mekso or the logical-connective scheme of things. These examples are drawn from Sec. 18.17, and contain many mekso features not explained in this chapter.

Example 14.149 exhibits afterthought logical connection between operands:

vei ci .a vo [ve'o] prenu cu klama le zarci ( Three or four ) people - go-to the market.

Example 14.150 is equivalent in meaning, but uses forethought connection:

vei ga ci gi vo [ve'o] prenu cu klama le zarci ( Either 3 or 4 ) people - go-to the market.

Note that the mekso in Example 14.149 and Example 14.150 are being used as quantifiers. Lojban requires that any mekso other than a simple number be enclosed in vei and ve'o parentheses when used as a quantifier. The right parenthesis mark, ve'o, is an elidable terminator.

Simple examples of logical connection between operators are hard to come by. A contrived example is:

li re su'i je pi'i re du li vo The-number 2 plus and times 2 equals the-number 4. 2 + 2 = 4 and 2 x 2 = 4.

The forethought form of Example 14.151 is:

li re gu'e su'i gi pi'i re du li vo The-number two both plus and times two equals the-number four. Both 2 + 2 = 4 and 2 x 2 = 4.

Non-logical connection with joiks or joigiks is also permitted between operands and between operators. One use for this construct is to connect operands with bi'i to create mathematical intervals:

li no ga'o bi'i ke'i pa the-number zero (inclusive) from-to (exclusive) one the numbers from zero to one, including zero but not including one

You can also combine two operands with ce'o, the sequence connective of selma'o JOI, to make a compound subscript:

xy. boi xi vei by. ce'o dy. [ve'o] x - sub (b sequence d)

Note that the boi in Example 14.154 is not elidable, because the xi subscript needs something to attach to.

Tenses, modals, and logical connection#

The tense and modal systems of Lojban interact with the logical connective system. No one chapter can explain all of these simultaneously, so each chapter must present its own view of the area of interaction with emphasis on its own concepts and terminology. In the examples of this chapter, the many tenses of various selma'o as well as the modals of selma'o BAI are represented by the simple time cmavo pu, ca, and ba (of selma'o PU) representing the past, the present, and the future respectively. Preceding a selbri, these cmavo state the time when the bridi was, is, or will be true (analogous to English verb tenses); preceding a sumti, they state that the event of the main bridi is before, simultaneous with, or after the event given by the sumti (which is generally a le nu abstraction; see Sec. 11.2).

The two types of interaction between tenses and logical connectives are logically connected tenses and tensed logical connections. The former are fairly simple. Jeks may be used between tense cmavo to specify two connected bridi that differ only in tense:

la .artr. pu nolraitru That-named Arthur [past] is-a-noblest-governor.

can be reduced to:

la .artr. pu je ba nolraitru That-named Arthur [past] and [future] is-a-noblest-governor. Arthur was and will be king.

Example 14.155 and Example 14.156 are equivalent in meaning; neither says anything about whether Arthur is king now.

Non-logical connection with joiks is also possible between tenses:

mi pu bi'o ba vasxu I [past] from-...-to [future] breathe. I breathe from a past time until a future time.

The full tense system makes more interesting tense intervals expressible, such as from a medium time ago until a long time from now.

No forethought connections between tenses are permitted by the grammar, nor is there any way to override the default left-grouping rule; these limitations are imposed to keep the tense grammar simpler. Whatever can be said with tenses or modals can be said with subordinate bridi stating the time, place, or mode explicitly, so it is reasonable to try to remove at least some complications.

Tensed logical connections are both more complex and more important than logical connections between tenses. Consider the English sentence:

I went to the market, and I bought food.

The verbatim translation of Example 14.158, namely:

mi pu klama le zarci .ije mi pu tervecnu lo cidja I [past] go-to the market. And I [past] buy items-of food.

fails to fully represent a feature of the English, namely that the buying came after the going. (It also fails to represent that the buying was a consequence of the going, which can be expressed by a modal that is discussed in Ch. 9.) However, the tense information – that the event of my going to the market preceded the event of my buying food – can be added to the logical connective as follows. The .i****je is replaced by .ijebo, and the tense cmavo ba is inserted between .i****je and bo:

mi pu klama le zarci I [past] go-to the market.

Here the pu cmavo in the two bridi-tails express the time of both actions with respect to the speaker: in the past. The ba relates the two items to one another: the second item is later than the first item. The grammar does not permit omitting the bo; if it were omitted, the ba and the second pu would run together to form a compound tense bapu applying to the second bridi-tail only.

Adding tense or modal information to a logical connective is permitted only in the following situations:

Between an ek (or joik) and bo, as in:

la .djan .e cabo la .alis. klama le zarci That-named John and [simultaneous] that-named Alice go-to the market. John and Alice go to the market simultaneously.

Between an ek (or joik) and ke, as in:

mi dzukla le zarci .e pu I walk-to the market and [earlier]

Between a gihek and bo, as in:

mi dunda le cukta gi'e babo I give the book and [later]

Between a gihek and ke, as in:

mi dzukla le zarci gi'e ca I walk-to the market and [simultaneous]

Between an ijek (or ijoik) and bo, as in:

mi viska pa nanmu .ije babo mi viska pa ninmu I see a man. And [later] I see a woman. I see a man, and then I see a woman.

Between an ijek (or ijoik) and tu'e, as in:

mi viska pa nanmu .ije batu'e mi viska pa ninmu [tu'u] I see a man. And [later] I see a woman. I see a man, and then I see a woman.

And finally, between a jek (or joik) and bo, as in:

mi mikce jebabo ricfu I am-a-doctor and-[later] rich I am a doctor and future rich person.

As can be seen from Example 14.165 and Example 14.166, the choice between bo and ke (or tu'e) is arbitrary when there are only two things to be connected. If there were no tense information to include, of course neither would be required; it is only the rule that tense information must always be sandwiched between the logical connective and a following bo, ke, or tu'e that requires the use of one of these grouping cmavo in Example 14.161 and Example 14.163 through Example 14.167.

Non-logical connectives with bo and ke can include tense information in exactly the same way as logical connectives. Forethought connectives, however (except as noted below) are unable to do so, as are termsets or tense connectives. Mathematical operands and operators can also include tense information in their logical connectives as a result of their close parallelism with sumti and tanru components respectively:

vei ci .ebabo vo [ve'o] tadni cu zvati le kumfa ( 3 and-[future] 4 ) students - are-at the room. Three and, later, four students were in the room.

is a simple example. There is a special grammatical rule for use when a tense applies to both of the selbri in a forethought bridi-tail connection: the entire forethought construction can just be preceded by a tense. For example:

mi pu ge klama le zarci gi tervecnu lo cidja I [past] both go-to the market and buy some food I went to the market and bought some food.

Example 14.169 is similar to Example 14.159. There is no time relationship specified between the going and the buying; both are simply set in the past.

Abstractor connection and connection within abstractions#

Last and (as a matter of fact) least: a logical connective is allowed between abstraction markers of selma'o NU. As usual, the connection can be expanded to a bridi connection between two bridi which differ only in abstraction marker. Jeks are the appropriate connective. Example 14.170 and Example 14.171 are equivalent in meaning:

le ka la frank. ciska cu xlali The quality-of that-named Frank's writing - is-bad,

le ka je ni la frank. ciska cu xlali The quality and quantity of that-named Frank's writing - is-bad.

As with tenses and modals, there is no forethought and no way to override the left-grouping rule.

Logical connectives and abstraction are related in another way as well, though. Since an abstraction contains a bridi, the bridi may have a logical connection inside it. Is it legitimate to split the outer bridi into two, joined by the logical connection? Absolutely not. For example:

mi jinvi le du'u loi jmive I opine the fact-that a-mass-of living-things

is true, since the embedded sentence is a tautology, but:

mi jinvi le du'u loi jmive cu zvati la .iupiter. I opine the fact-that a-mass-of living-things - is-at that-named Jupiter

is false, since I have no evidence one way or the other (jinvi requires some sort of evidence, real or fancied, unlike krici).

Constructs and appropriate connectives#

The following table specifies, for each kind of construct that can be logically or non-logically connected in Lojban, what kind of connective is required for both afterthought and (when possible) forethought modes. An asterisk (*) indicates that tensed connection is permitted.

A dash indicates that connection of the specified type is not possible.

Truth functions and corresponding logical connectives#

The following table specifies, for each truth function, the most-often used cmavo or compound cmavo which expresses it for each of the six types of logical connective. (Other compound cmavo are often possible: for example, se**.a** means the same as a, and could be used instead.)

Note: ijeks are exactly the same as the corresponding jeks, except for the prefixed i.

Rules for making logical and non-logical connectives#

The full set of rules for inserting na, se, and nai into any connective is:

Afterthought logical connectives (eks, jeks, giheks, ijeks):

• Negate first construct: Place na before the connective cmavo (but after the i of an ijek).
• Negate second construct: Place nai after the connective cmavo.
• Exchange constructs: Place se before the connective cmavo (after na if any).

Forethought logical connectives (geks, guheks):

• Negate first construct: Place nai after the connective cmavo.
• Negate second construct: Place nai after the gi.
• Exchange constructs: Place se before the connective cmavo.

Non-logical connectives (joiks, joigiks):

• Negate connection: Place nai after the connective cmavo (but before the gi of a joigik).
• Exchange constructs: Place se before the connective cmavo.

Locations of other tables#

Sec. 14.1: a table explaining the meaning of each truth function in English.

Sec. 14.2: a table relating the truth functions to the four basic vowels.

Sec. 14.13: a table of the connective question cmavo.

Sec. 14.14: a table of the meanings of JOI cmavo when used to connect sumti.