- 1 Quantification
- 1.1 What's wrong with this picture?
- 1.2 Existential claims, prenexes, and variables
- 1.3 Universal claims
- 1.4 Restricted claims: da poi
- 1.5 Dropping the prenex
- 1.6 Variables with generalized quantifiers
- 1.7 Grouping of quantifiers
- 1.8 The problem of “any”
- 1.9 Negation boundaries
- 1.10 bridi negation and logical connectives
- 1.11 Using naku outside a prenex
- 1.12 Logical Connectives and DeMorgan's Law
- 1.13 selbri variables
- 1.14 A few notes on variables
- 1.15 Conclusion
What's wrong with this picture?
The following brief dialogue is from Chapter 7 of Through The Looking Glass by Lewis Carroll.
- Example .1:
“Who did you pass on the road?” the King went on, holding out his hand to the Messenger for some more hay.
- Example .2:
“Nobody,” said the Messenger.
- Example .3:
“Quite right,” said the King: “this young lady saw him too. So of course Nobody walks slower than you.”
- Example .4:
“I do my best,” the Messenger said in a sulky tone. “I'm sure nobody walks much faster than I do!”
- Example .5:
“He can't do that,” said the King, “or else he'd have been here first.”
This nonsensical conversation results because the King insists on treating the word “nobody” as a name, a name of somebody. However, the essential nature of the English word “nobody” is that it doesn't refer to somebody; or to put the matter another way, there isn't anybody to which it refers.
The central point of contradiction in the dialogue arises in Example .3, when the King says “... Nobody walks slower than you”. This claim would be plausible if “Nobody” were really a name, since the Messenger could only pass someone who does walk more slowly than he. But the Messenger interprets the word “nobody” in the ordinary English way, and says (in Example .4) “... nobody walks much faster than I do” (i.e., I walk faster, or as fast as, almost everyone), which the King then again misunderstands. Both the King and the Messenger are correct according to their respective understandings of the ambiguous word “nobody/Nobody”.
There are Lojban words or phrases corresponding to the problematic English words “somebody”, “nobody”, “anybody”, “everybody” (and their counterparts “some/no/any/everyone” and “some/no/any/everything”), but they obey rules which can often be surprising to English-speakers. The dialogue above simply cannot be translated into Lojban without distortion: the name “Nobody” would have to be represented by a Lojban name, which would spoil the perfection of the wordplay. As a matter of fact, this is the desired result: a logical language should not allow two conversationalists to affirm “Nobody walks slower than the Messenger” and “Nobody walks faster than the Messenger” and both be telling the truth. (Unless, of course, nobody but the Messenger walks at all, or everyone walks at exactly the same speed.)
This chapter will explore the Lojban mechanisms that allow the correct and consistent construction of sentences like those in the dialogue. There are no new grammatical constructs explained in this chapter; instead, it discusses the way in which existing facilities that allow Lojban-speakers to resolve problems like the above, using the concepts of modern logic. However, we will not approach the matter from the viewpoint of logicians, although readers who know something of logic will discover familiar notions in Lojban guise.
Although Lojban is called a logical language, not every feature of it is
“logical”. In particular, the use of le is incompatible with logical reasoning based on the description selbri, because that selbri may not truthfully apply: you cannot conclude from my statement that
- Example .6:
- mi viska le nanmu
I see the-one-I-refer-to-as-the man.
- I see the man/men.
that there really is a man; the only thing you can conclude is that there is one thing (or more) that I choose to refer to as a man. You cannot even tell which man is meant for sure without asking me (although communication is served if you already know from the context).
In addition, the use of attitudinals (See Chapter ELG-ERROR in Template:Lch) often reduces or removes the ability to make deductions about the bridi to which those attitudinals are applied. From the fact that I hope George will win the election, you can conclude nothing about George's actual victory or defeat.
Existential claims, prenexes, and variables
Let us consider, to begin with, a sentence that is not in the dialogue:
- Example .7:
Something sees me.
There are two plausible Lojban translations of Example .7. The simpler one is:
- Example .8:
- [zo'e] viska mi
Something-unspecified sees me.
The cmavo zo'e indicates that a sumti has been omitted (indeed, even zo'e itself can be omitted in this case, as explained in Section ) and the listener must fill in the correct value from context. In other words, Example .8 means “"You-know-what" sees me.”
However, Example .7 is just as likely to assert simply that there is someone who sees me, in which case a correct translation is:
- Example .9:
- da zo'u da viska mi
There-is-an-X such-that X sees me.
Example .9 does not presuppose that the listener knows who sees the speaker, but simply tells the listener that there is someone who sees the speaker. Statements of this kind are called “existential claims”. (Formally, the one doing the seeing is not restricted to being a person; it could be an animal or – in principle – an inanimate object. We will see in Section .3 how to represent such restrictions.)
Example .9 has a two-part structure: there is the part da zo'u, called the prenex, and the part da viska mi, the main bridi. Almost any Lojban bridi can be preceded by a prenex, which syntactically is any number of sumti followed by the cmavo zo'u (of selma'o ZOhU). For the moment, the sumti will consist of one or more of the cmavo da, de, and di (of selma'o KOhA), glossed in the literal translations as “X”, “Y”, and “Z” respectively. By analogy to the terminology of symbolic logic, these cmavo are called “variables”.
Here is an example of a prenex with two variables:
- Example .10:
- da de zo'u da prami de
There-is-an-X there-is-a-Y such that X loves Y.
- Somebody loves somebody.
In Example .10, the literal interpretation of the two variables da and de as “there-is-an-X” and “there-is-a-Y” tells us that there are two things which stand in the relationship that one loves the other. It might be the case that the supposed two things are really just a single thing that loves itself; nothing in the Lojban version of Example .10 rules out that interpretation, which is why the colloquial translation does not say “Somebody loves somebody else.” The things referred to by different variables may be different or the same. (We use “somebody” here rather than “something” for naturalness; lovers and beloveds are usually persons, though the Lojban does not say so.)
It is perfectly all right for the variables to appear more than once in the main bridi:
- Example .11:
- da zo'u da prami da
There-is-an-X such that X loves X
- Somebody loves himself/herself.
- da prami da is not structurally the same as da prami de. However,
- Example .12:
- de zo'u de prami de
There-is-a-Y such that Y loves Y
means exactly the same thing as Example .11; it does not matter which variable is used as long as they are used consistently.
It is not necessary for a variable to be a sumti of the main bridi directly:
- Example .13:
- da zo'u le da gerku cu viska mi
There-is-an-X such-that the of-X dog sees me
- Somebody's dog sees me
is perfectly correct even though the da is used only in a possessive construction. (Possessives are explained in Section .)
It is very peculiar, however, even if technically grammatical, for the variable not to appear in the main bridi at all:
- Example .14:
- da zo'u la ralf. gerku
- There is something such that Ralph is a dog.
has a variable bound in a prenex whose relevance to the claim of the following bridi is completely unspecified.
What happens if we substitute “everything” for “something” in Example .7? We get:
- Example .15:
Everything sees me.
Of course, this example is false, because there are many things which do not see the speaker. It is not easy to find simple truthful examples of so-called universal claims (those which are about everything), so bear with us for a while. (Indeed, some Lojbanists tend to avoid universal claims even in other languages, since they are so rarely true in Lojban.)
The Lojban translation of Example .15 is
- Example .16:
- ro da zo'u da viska mi
For-every X : X sees me.
When the variable cmavo da is preceded by ro, the combination means “For every X” rather than “There is an X”. Superficially, these English formulations look totally unrelated: Section .5 will bring them within a common viewpoint. For the moment, accept the use of ro da for “everything” on faith.
Here is a universal claim with two variables:
- Example .17:
- ro da ro de zo'u da prami de
For-every X, for-every Y : X loves Y.
- Everything loves everything.
Again, X and Y can represent the same thing, so Example .17 does not mean “Everything loves everything else.” Furthermore, because the claim is universal, it is about every thing, not merely every person, so we cannot use “everyone” or “everybody” in the translation.
Note that ro appears before both da and de. If ro is omitted before either variable, we get a mixed claim, partly existential like those of Section .1, partly universal.
- Example .18:
- ro da de zo'u da viska de
For-every X, there-is-a-Y : X sees Y.
- Everything sees something.
- Example .19:
- da ro de zo'u da viska de
There-is-an-X such-that-for-every-Y : X sees Y.
- Something sees everything.
Example .18 and Example .19 mean completely different things. Example .18 says that for everything, there is something which it sees, not necessarily the same thing seen for every seer. Example .19, on the other hand, says that there is a particular thing which can see everything that there is (including itself). Both of these are fairly silly, but they are different kinds of silliness.
There are various possible translations of universal claims in English: sometimes we use “anybody/anything” rather than “everybody/everything”. Often it makes no difference which of these is used: when it does make a difference, it is a rather subtle one which is explained in Section .7.
Restricted claims: da poi
The universal claims of Section .2 are not only false but absurd: there is really very little to be said that is both true and non-trivial about every object whatsoever. Furthermore, we have been glossing over the distinction between “everything” and “everybody” and the other pairs ending in “-thing” and “-body”. It is time to bring up the most useful feature of Lojban variables: the ability to restrict their ranges.
In Lojban, a variable da, de, or di may be followed by a poi relative clause in order to restrict the range of things that the variable describes. Relative clauses are described in detail in Chapter ELG-ERROR in Template:Lch, but the kind we will need at present consist of poi followed by a bridi (often just a selbri) terminated with ku'o or vau (which can usually be elided). Consider the difference between
- Example .20:
- da zo'u da viska la djim.
There-is-an-X : X sees Jim.
- Something sees Jim.
- Example .21:
- da poi prenu zo'u da viska la djim.
There-is-an-X which is-a-person : X sees Jim.
- Someone sees Jim.
- poi prenu relative clause to persons only, and so
- da poi prenu translates as
“someone.” (The difference between “someone” and “somebody” is a matter of English style, with no real counterpart in Lojban.) If Example .21 is true, then Example .20 must be true, but not necessarily vice versa.
Universal claims benefit even more from the existence of relative clauses. Consider
- Example .22:
- ro da zo'u da vasxu
For-every X : X breathes
- Everything breathes
- Example .23:
- ro da poi gerku zo'u da vasxu
For-every X which is-a-dog : X breathes.
Every dog breathes.
Each dog breathes.
- All dogs breathe.
Example .22 is a silly falsehood, but Example .23 is an important truth (at least if applied in a timeless or potential sense: See Section ). Note the various colloquial translations “every dog”, “each dog”, and “all dogs”. They all come to the same thing in Lojban, since what is true of every dog is true of all dogs. “All dogs” is treated as an English plural and the others as singular, but Lojban makes no distinction.
If we make an existential claim about dogs rather than a universal one, we get:
- Example .24:
- da poi gerku zo'u da vasxu
There-is-an-X which is-a-dog : X breathes.
- Some dog breathes.
Dropping the prenex
It isn't really necessary for every Lojban bridi involving variables to have a prenex on the front. In fact, none of the examples we've seen so far required prenexes at all! The rule for dropping the prenex is simple: if the variables appear in the same order within the bridi as they did in the prenex, then the prenex is superfluous. However, any ro or poi appearing in the prenex must be transferred to the first occurrence of the variable in the main part of the bridi. Thus, Example .9 becomes just:
- Example .25:
- da viska mi
There-is-an-X-which sees me.
- Something sees me.
and Example .23 becomes:
- Example .26:
- ro da poi gerku cu vasxu
For-every X which is-a-dog, it-breathes.
- Every dog breathes.
You might well suppose, then, that the purpose of the prenex is to allow the variables in it to appear in a different order than the bridi order, and that would be correct. Consider
- Example .27:
- ro da poi prenu ku'o de poi gerku ku'o zo'u de batci da
For-every X which is-a-person, there-is-a-Y which is-a-dog: Y bites X.
The prenex of Example .27 is like that of Example .18 (but with relative clauses): it notes that the following bridi is true of every person with respect to some dog, not necessarily the same dog for each. But in the main bridi part, the de appears before the da. Therefore, the true translation is
- Example .28:
Every person is bitten by some dog (or other).
If we tried to omit the prenex and move the ro and the relative clauses into the main bridi, we would get:
- Example .29:
- de poi gerku cu batci ro da poi prenu
There-is-a-Y which is-a-dog which-bites every X which is-a-person
- Some dog bites everyone.
which has the structure of Example .19: it says that there is a dog (call him Fido) who bites, has bitten, or will bite every person that has ever existed! We can safely rule out Fido's existence, and say that Example .29 is false, while agreeing to Example .27.
Even so, Example .27 is most probably false, since some people never experience dogbite. Examples like 5.3 and 4.4 (might there be some dogs which never have breathed, because they died as embryos?) indicate the danger in Lojban of universal claims even when restricted. In English we are prone to say that
“Everyone says” or that “Everybody does” or that “Everything is” when in fact there are obvious counterexamples which we are ignoring for the sake of making a rhetorical point. Such statements are plain falsehoods in Lojban, unless saved by a context (such as tense) which implicitly restricts them.
How can we express Example .27 in Lojban without a prenex? Since it is the order in which variables appear that matters, we can say:
- Example .30:
- ro da poi prenu cu se batci de poi gerku
Every-X which is-a-person is-bitten-by some-Y which is-a-dog.
using the conversion operator se (explained in Section ) to change the selbri batci ( “bites”) into
- se batci (
“is bitten by”). The translation given in Example .28 uses the corresponding strategy in English, since English does not have prenexes (except in strained “logician's English”). This implies that a sentence with both a universal and an existential variable can't be freely converted with
se; one must be careful to preserve the order of the variables.
If a variable occurs more than once, then any ro or poi decorations are moved only to the first occurrence of the variable when the prenex is dropped. For example,
- Example .31:
- di poi prenu zo'u ti xarci di di
There-is-a-Z which is-a-person : this-thing is-a-weapon for-use-against-Z by-Z
- This is a weapon for someone to use against himself/herself.
(in which di is used rather than da just for variety) loses its prenex as follows:
- Example .32:
- ti xarci di poi prenu ku'o di
This-thing is-a-weapon-for-use-against some-Z which is-a-person by-Z.
As the examples in this section show, dropping the prenex makes for terseness of expression often even greater than that of English (Lojban is meant to be an unambiguous language, not necessarily a terse or verbose one), provided the rules are observed.
Variables with generalized quantifiers
So far, we have seen variables with either nothing in front, or with the cmavo ro in front. Now ro is a Lojban number, and means “all”; thus
- ro prenu means
“all persons”, just as
- re prenu means
“two persons”. In fact, unadorned da is also taken to have an implicit number in front of it, namely su'o, which means “at least one”. Why is this? Consider Example .9 again, this time with an explicit su'o:
- Example .33:
- su'o da zo'u da viska mi
For-at-least-one X : X sees me.
- Something sees me.
From this version of Example .9, we understand the speaker's claim to be that of all the things that there are, at least one of them sees him or her. The corresponding universal claim, Example .16, says that of all the things that exist, every one of them can see the speaker.
Any other number can be used instead of ro or su'o to precede a variable. Then we get claims like:
- Example .34:
- re da zo'u da viska mi
For-two-Xes : X sees me.
- Two things see me.
This means that exactly two things, no more or less, saw the speaker on the relevant occasion. In English, we might take “Two things see me” to mean that at least two things see the speaker, but there might be more; in Lojban, though, that claim would have to be made as:
- Example .35:
- su'ore da zo'u da viska mi
For-at-least-two Xes : X sees me.
which would be false if nothing, or only one thing, saw the speaker, but not otherwise. We note the su'o here meaning “at least”; su'o by itself is short for
- su'opa where
pa means “one”, as is explained in Section .
- Example .36:
- re da viska mi
- Two Xes see me.
- Example .37:
- su'ore da viska mi
At-least-two Xes see me.
respectively, subject to the rules prescribed in Section .4.
Now we can explain the constructions
- ro prenu for “all persons” and :re prenu for “two persons” which were casually mentioned at the beginning of this Section. In fact,
- ro prenu, a so-called “indefinite description”, is shorthand for ro DA poi prenu, where “DA” represents a fictitious variable that hasn't been used yet and will not be used in future. (Even if all three of
da, de, and di have been used up, it does not matter, for there are ways of getting more variables, discussed in Section .13.) So in fact
- Example .38:
- re prenu cu viska mi
- Two persons see me.
is short for
- Example .39:
- re da poi prenu cu viska mi
Two Xes which are-persons see me.
which in turn is short for:
- Example .40:
- re da poi prenu zo'u da viska mi
For-two Xes which are-persons : X sees me.
Note that when we move more than one variable to the prenex (along with its attached relative clause), we must make sure that the variables are in the same order in the prenex as in the bridi proper.
Grouping of quantifiers
Let us consider a sentence containing two quantifier expressions neither of which is ro or su'o (remembering that su'o is implicit where no explicit quantifier is given):
- Example .41:
- ci gerku cu batci re nanmu
- Three dogs bite two men.
The question raised by Example .41 is, does each of the dogs bite the same two men, or is it possible that there are two different men per dog, for six men altogether? If the former interpretation is taken, the number of men involved is fixed at two; but if the latter, then the speaker has to be taken as saying that there might be any number of men between two and six inclusive. Let us transform Example .41 step by step as we did with Example .38:
- Example .42:
- ci da poi gerku cu batci re de poi nanmu
Three Xes which are-dogs bite two Ys which are-men.
(Note that we need separate variables da and de, because of the rule that says each indefinite description gets a variable never used before or since.)
- Example .43:
- ci da poi gerku ku'o re de poi nanmu zo'u da batci de
For-three Xes which are-dogs, for-two Ys which are-men : X bites Y.
Here we see that indeed each of the dogs is said to bite two men, and it might be different men each time; a total of six biting events altogether.
How then are we to express the other interpretation, in which just two men are involved? We cannot just reverse the order of variables in the prenex to
- Example .44:
- re de poi nanmu ku'o ci da poi gerku zo'u da batci de
For-two Ys which are-men, for-three Xes which are-dogs, X bites Y
for although we have now limited the number of men to exactly two, we end up with an indeterminate number of dogs, from three to six. The distinction is called a “scope distinction”: in Example .42, ci gerku is said to have wider scope than re nanmu, and therefore precedes it in the prenex. In Example .44 the reverse is true.
The solution is to use a termset, which is a group of terms either joined by ce'e (of selma'o CEhE) between each term, or else surrounded by nu'i (of selma'o NUhI) on the front and nu'u (of selma'o NUhU) on the rear. Terms (which are either sumti or sumti prefixed by tense or modal tags) that are grouped into a termset are understood to have equal scope:
- Example .45:
- ci gerku ce'e re nanmu cu batci
- nu'i ci gerku re nanmu [nu'u] cu batci
Three dogs [plus] two men, bite.
which picks out two groups, one of three dogs and the other of two men, and says that every one of the dogs bites each of the men. The second Lojban version uses forethought; note that nu'u is an elidable terminator, and in this case can be freely elided.
What about descriptors, like
- ci lo gerku,
- le nanmu or
- re le ci mlatu? They too can be grouped in termsets, but usually need not be, except for the
lo case which functions like the case without a descriptor. Unless an actual quantifier precedes it,
- le nanmu means
- ro le nanmu, as is explained in
Section . Two sumti with ro quantifiers are independent of order, so:
- Example .46:
- [ro] le ci gerku cu batci [ro] le re nanmu
[All of] the three dogs bite [all of] the two men.
means that each of the dogs specified bites each of the men specified, for six acts of biting altogether. However, if there is an explicit quantifier before le other than ro, the problems of this section reappear.
The problem of “any”
Consider the English sentence
- Example .47:
Anyone who goes to the store, walks across the field.
Using the facilities already discussed, a plausible translation might be
- Example .48:
- ro da poi klama le zarci cu cadzu le foldi
All X such-that-it goes-to the store walks-on the field.
- Everyone who goes to the store walks across the field.
But there is a subtle difference between Example .47 and Example .48. Example .48 tells us that, in fact, there are people who go to the store, and that they walk across the field. A sumti of the type
- ro da poi klama requires that there are things which
klama: Lojban universal claims always imply the corresponding existential claims as well.
Example .47, on the other hand, does not require that there are any people who go to the store: it simply states, conditionally, that if there is anyone who goes to the store, he or she walks across the field as well. This conditional form mirrors the true Lojban translation of
- Example .49:
- ro da zo'u da go klama le zarci gi cadzu le foldi
For-every X: X is-a-goer-to the store if-and-only-if is-a-walker-on the field.
Although Example .49 is a universal claim as well, its universality only implies that there are objects of some sort or another in the universe of discourse. Because the claim is conditional, nothing is implied about the existence of goers-to-the-store or of walkers-on-the-field, merely that any entity which is one is also the other.
There is another use of “any” in English that is not universal but existential. Consider
- Example .50:
- I need any box that is bigger than this one.
Example .50 does not at all mean that I need every box bigger than this one, for indeed I do not; I require only one box. But the naive translation
- Example .51:
- mi nitcu da poi tanxe gi'e bramau ti
I need some-X which is-a-box and is-bigger-than this-one
does not work either, because it asserts that there really is such a box, as the prenex paraphrase demonstrates:
- Example .52:
- da poi tanxe gi'e bramau ti zo'u mi nitcu da
There-is-an-X which is-a-box and is-bigger-than this : I need X.
What to do? Well, the x2 place of nitcu can be filled with an event as well as an object, and in fact Example .51 can also be paraphrased as:
- Example .53:
- mi nitcu lo nu mi ponse lo tanxe poi bramau ti
I need an event-of I possess some box(es) which-are bigger-than this-one.
Rewritten using variables, Example .53 becomes
- Example .54:
- mi nitcu lo nu da zo'u da se ponse mi gi'e tanxe gi'e bramau ti
I need an event-of there-being an-X such-that : X is-possessed-by me and is-a-box and is-bigger-than this-thing.
So we see that a prenex can be attached to a bridi that is within a sentence. By default, a variable always behaves as if it is bound in the prenex which (notionally) is attached to the smallest enclosing bridi, and its scope does not extend beyond that bridi. However, the variable may be placed in an outer prenex explicitly:
- Example .55:
- da poi tanxe gi'e bramau ti zo'u mi nitcu le nu mi ponse da
There-is-an-X which is-a-box and is-bigger-than this-one such-that : I need the event-of my possessing X.
- Example .56:
There's a box, bigger than this one, that I need
which is what Example .52 says, whereas Example .53 turns out to be an effective translation of our original Example .47. So uses of “any” that aren't universal end up being reflected by variables bound in the prenex of a subordinate bridi.
This section, as well as Section .9 through Section .11, are in effect a continuation of Chapter ELG-ERROR in Template:Lch, introducing features of Lojban negation that require an understanding of prenexes and variables. In the examples below, “there is a Y” and the like must be understood as “there is at least one Y, possibly more”.
As explained in Section , the negation of a bridi is usually accomplished by inserting na at the beginning of the selbri:
- Example .57:
- mi na klama le zarci
I [false] go-to the store.
It is false that I go to the store.
- I don't go to the store.
The other form of bridi negation is expressed by using the compound cmavo naku in the prenex, which is identified and compounded by the lexer before looking at the sentence grammar. In Lojban grammar, naku is then treated like a sumti. In a prenex, naku means precisely the same thing as the logician's “it is not the case that” in a similar English context. (Outside of a prenex, naku is also grammatically treated as a single entity – the equivalent of a sumti – but does not have this exact meaning; we'll discuss these other situations in Section .10.)
To represent a bridi negation using a prenex, remove the na from before the selbri and place
- naku at the left end of the prenex. This form is called “external bridi negation”, as opposed to “internal bridi negation” using na. The prenex version of Example .57 is
- Example .58:
- naku zo'u la djan. klama
It is not the case that: John comes.
- It is false that: John comes.
However, naku can appear at other points in the prenex as well. Compare
- Example .59:
- naku de zo'u de zutse
It is not the case that: for some Y, Y sits.
It is false that: for at least one Y, Y sits.
It is false that something sits.
- Nothing sits.
- Example .60:
- su'ode naku zo'u de zutse
For at least one Y, it is false that: Y sits.
- There is something that doesn't sit.
The relative position of negation and quantification terms within a prenex has a drastic effect on meaning. Starting without a negation, we can have:
- Example .61:
- roda su'ode zo'u da prami de
For every X, there is a Y, such that X loves Y.
- Everybody loves at least one thing (each, not necessarily the same thing).
- Example .62:
- su'ode roda zo'u da prami de
There is a Y, such that for each X, X loves Y.
- There is at least one particular thing that is loved by everybody.
The simplest form of bridi negation to interpret is one where the negation term is at the beginning of the prenex:
- Example .63:
- naku roda su'ode zo'u da prami de
It is false that: for every X, there is a Y, such that: X loves Y.
It is false that: everybody loves at least one thing.
- (At least) someone doesn't love anything.
the negation of Example .61, and
- Example .64:
- naku su'ode roda zo'u da prami de
It is false that: there is a Y such that for each X, X loves Y.
It is false that: there is at least one thing that is loved by everybody.
- There isn't any one thing that everybody loves.
the negation of Example .62.
The rules of formal logic require that, to move a negation boundary within a prenex, you must “invert any quantifier” that the negation boundary passes across. Inverting a quantifier means that any ro (all) is changed to su'o (at least one) and vice versa. Thus, Example .63 and Example .64 can be restated as, respectively:
- Example .65:
- su'oda naku su'ode zo'u da prami de
For some X, it is false that: there is a Y such that: X loves Y.
- There is somebody who doesn't love anything.
- Example .66:
- rode naku roda zo'u da prami de
For every Y, it is false that: for every X, X loves Y.
- For each thing, it is not true that everybody loves it.
Another movement of the negation boundary produces:
- Example .67:
- su'oda rode naku zo'u da prami de
There is an X such that, for every Y, it is false that X loves Y.
- There is someone who, for each thing, doesn't love that thing.
- Example .68:
- rode su'oda naku zo'u da prami de
For every Y, there is an X, such that it is false that: X loves Y.
- For each thing there is someone who doesn't love it.
The quantifier no (meaning “zero of”) also involves a negation boundary. To transform a bridi containing a variable quantified with no, we must first expand it. Consider
- Example .69:
- noda rode zo'u da prami de
There is no X, for every Y, such that X loves Y.
- Nobody loves everything.
which is negated by:
- Example .70:
- naku noda rode zo'u da prami de
It is false that: there is no X that, for every Y, X loves Y.
- It is false that there is nobody who loves everything.
We can simplify Example .70 by transforming the prenex. To move the negation phrase within the prenex, we must first expand the no quantifier. Thus “for no x” means the same thing as “it is false for some x”, and the corresponding Lojban noda can be replaced by naku su'oda. Making this substitution, we get:
- Example .71:
- naku naku su'oda rode zo'u da prami de
- It is false that it is false that: for an X, for every Y: X loves Y.
Adjacent pairs of negation boundaries in the prenex can be dropped, so this means the same as:
- Example .72:
- su'oda rode zo'u da prami de
There is an X such that, for every Y, X loves Y.
- At least one person loves everything.
which is clearly the desired contradiction of Example .69.
The interactions between quantifiers and negation mean that you cannot eliminate double negatives that are not adjacent. You must first move the negation phrases so that they are adjacent, inverting any quantifiers they cross, and then the double negative can be eliminated.
bridi negation and logical connectives
A complete discussion of logical connectives appears in Chapter ELG-ERROR in Template:Lch. What is said here is intentionally quite incomplete and makes several oversimplifications.
A logical connective is a cmavo or compound cmavo. In this chapter, we will make use of the logical connectives “and” and “or” (where “or” really means “and/or”, “either or both”). The following simplified recipes explain how to make some logical connectives:
- To logically connect two Lojban sumti with “and”, put them both in the bridi and separate them with the cmavo e.
- To logically connect two Lojban bridi with “and”, replace the regular separator cmavo i with the compound cmavo .ije.
- To logically connect two Lojban sumti with “or”, put them both in the bridi and separate them with the cmavo a.
- To logically connect two Lojban bridi with “or”, replace the regular separator cmavo i with the compound cmavo .ija.
More complex logical connectives also exist; in particular, one may place
na before e or a, or between i and je or ja; likewise, one may place nai at the end of a connective. Both na and nai have negative effects on the sumti or bridi being connected. Specifically, na negates the first or left-hand sumti or bridi, and nai negates the second or right-hand one.
Whenever a logical connective occurs in a sentence, that sentence can be expanded into two sentences by repeating the common terms and joining the sentences by a logical connective beginning with i. Thus the following sentence:
- Example .73:
- mi .e do klama ti
- I and you come here.
can be expanded to:
- Example .74:
- mi klama ti .ije do klama ti
- I come here, and, you come here.
The same type of expansion can be performed for any logical connective, with any valid combination of na or nai attached. No change in meaning occurs under such a transformation.
Clearly, if we know what negation means in the expanded sentence forms, then we know what it means in all of the other forms. But what does negation mean between sentences?
The mystery is easily solved. A negation in a logical expression is identical to the corresponding bridi negation, with the negator placed at the beginning of the prenex. Thus:
- Example .75:
- mi .enai do prami roda
- I, and not you, love everything.
- Example .76:
- mi prami roda .ijenai do prami roda
I love everything, and-not, you love everything.
and then into prenex form as:
- Example .77:
- roda zo'u mi prami da .ije naku zo'u do prami da
- For each thing: I love it, and it is false that you love (the same) it.
By the rules of predicate logic, the ro quantifier on da has scope over both sentences. That is, once you've picked a value for da for the first sentence, it stays the same for both sentences. (The da continues with the same fixed value until a new paragraph or a new prenex resets the meaning.)
Thus the following example has the indicated translation:
- Example .78:
- su'oda zo'u mi prami da .ije naku zo'u do prami da
For at least one thing: I love that thing. And it is false that: you love that (same) thing.
- There is something that I love that you don't.
If you remember only two rules for prenex manipulation of negations, you won't go wrong:
- Within a prenex, whenever you move naku past a bound variable (da, de, di, etc.), you must invert the quantifier.
- A na before the selbri is always transformed into a naku at the left-hand end of the prenex, and vice versa.
Using naku outside a prenex
Let us consider the English sentence
- Example .79:
- Some children do not go to school.
We cannot express this directly with na; the apparently obvious translation
- Example .80:
- su'oda poi verba na klama su'ode poi ckule
At-least-one X which-are child(ren) [false] go-to at-least-one Y which-are school(s).
when converted to the external negation form produces:
- Example .81:
- naku zo'u su'oda poi verba cu klama su'ode poi ckule
It is false that some which are children go-to some which are schools.
- All children don't go to some school (not just some children).
Lojban provides a negation form which more closely emulates natural language negation. This involves putting naku before the selbri, instead of a na. naku is clearly a contradictory negation, given its parallel with prenex bridi negation. Using naku, Example .79 can be expressed as:
- Example .82:
- su'oda poi verba naku klama su'ode poi ckule
Some which-are children don't go-to some which-are schools.
- Some children don't go to a school.
Although it is not technically a sumti, naku can be used in most of the places where a sumti may appear. We'll see what this means in a moment.
When you use naku within a bridi, you are explicitly creating a negation boundary. As explained in Section .8, when a prenex negation boundary expressed by naku moves past a quantifier, the quantifier has to be inverted. The same is true for naku in the bridi proper. We can move naku to any place in the sentence where a sumti can go, inverting any quantifiers that the negation boundary crosses. Thus, the following are equivalent to Example .82 (no good English translations exist):
- Example .83:
- su'oda poi verba cu klama rode poi ckule naku
- For some children, for every school, they don't go to it.
- Example .84:
- su'oda poi verba cu klama naku su'ode poi ckule
- Some children don't go to (some) school(s).
- Example .85:
- naku roda poi verba cu klama su'ode poi ckule
- It is false that all children go to some school(s).
In Example .83, we moved the negation boundary rightward across the quantifier of de, forcing us to invert it. In Example .85 we moved the negation boundary across the quantifier of da, forcing us to invert it instead. Example .84 merely switched the selbri and the negation boundary, with no effect on the quantifiers.
The same rules apply if you rearrange the sentence so that the quantifier crosses an otherwise fixed negation. You can't just convert the selbri of Example .82 and rearrange the sumti to produce
- Example .86:
- su'ode poi ckule ku'o naku se klama roda poi verba
Some schools aren't gone-to-by every child.
- naku negation is not symmetric; not all sumti are treated identically, and some sumti are not invariant under conversion. Thus, internal negation with
- naku is considered an advanced technique, used to achieve stylistic compatibility with natural languages.
It isn't always easy to see which quantifiers have to be inverted in a sentence. Example .82 is identical in meaning to:
- Example .87:
- su'o verba naku klama su'o ckule
Some children don't go-to some school.
but in Example .87, the bound variables da and de have been hidden.
It is trivial to export an internal bridi negation expressed with
na to the prenex, as we saw in Section .8; you just move it to the left end of the prenex. In comparison, it is non-trivial to export a naku to the prenex because of the quantifiers. The rules for exporting naku require that you export all of the quantified variables (implicit or explicit) along with naku, and you must export them from left to right, in the same order that they appear in the sentence. Thus Example .82 goes into prenex form as:
- Example .88:
- su'oda poi verba ku'o naku su'ode poi ckule zo'u da klama de
For some X which is a child, it is not the case that there is a Y which is a school such that: X goes to Y.
We can now move the naku to the left end of the prenex, getting a contradictory negation that can be expressed with na:
- Example .89:
- naku roda poi verba su'ode poi ckule zo'u da klama de
It is not the case that for all X's which are children, there is a Y which is a school such that: X goes to Y.
from which we can restore the quantified variables to the sentence, giving:
- Example .90:
- naku zo'u roda poi verba cu klama su'ode poi ckule
- It is not the case that all children go to some school.
or more briefly
- Example .91:
- ro verba cu na klama su'o ckule
All children [false] go-to some school(s).
As noted in Section .4, a sentence with two different quantified variables, such as Example .91, cannot always be converted with se without first exporting the quantified variables. When the variables have been exported, the sentence proper can be converted, but the quantifier order in the prenex must remain unchanged:
- Example .92:
- roda poi verba su'ode poi ckule zo'u de na se klama da
It is not the case that for all X's which are children, there is a Y which is a school such that: Y is gone to by X.
While you can't freely convert with se when you have two quantified variables in a sentence, you can still freely move sumti to either side of the selbri, as long as the order isn't changed. If you use na negation in such a sentence, nothing special need be done. If you use naku negation, then quantified variables that cross the negation boundary must be inverted.
Clearly, if all of Lojban negation was built on naku negation instead of na negation, logical manipulation in Lojban would be as difficult as in natural languages. In Section .11, for example, we'll discuss DeMorgan's Law, which must be used whenever a sumti with a logical connection is moved across a negation boundary.
Since naku has the grammar of a sumti, it can be placed almost anywhere a sumti can go, including be and bei clauses; it isn't clear what these mean, and we recommend avoiding such constructs.
You can put multiple naku s in a sentence, each forming a separate negation boundary. Two adjacent naku s in a bridi are a double negative and cancel out:
- Example .93:
- mi naku naku le zarci cu klama
Other expressions using two naku s may or may not cancel out. If there is no quantified variable between them, then the naku s cancel.
Negation with internal naku is clumsy and non-intuitive for logical manipulations, but then, so are the natural language features it is emulating.
Logical Connectives and DeMorgan's Law
DeMorgan's Law states that when a logical connective between terms falls within a negation, then expanding the negation requires a change in the connective. Thus (where “p” and “q” stand for terms or sentences) “not (p or q)” is identical to “not p and not q”, and “not (p and q)” is identical to “not p or not q”. The corresponding changes for the other two basic Lojban connectives are: “not (p equivalent to q)” is identical to “not p exclusive-or not q”, and “not (p whether-or-not q)” is identical to both “not p whether-or-not q” and “not p whether-or-not not q”. In any Lojban sentence having one of the basic connectives, you can substitute in either direction from these identities. (These basic connectives are explained in Chapter ELG-ERROR in Template:Lch.)
The effects of DeMorgan's Law on the logical connectives made by modifying the basic connectives with nai, na and se can be derived directly from these rules; modify the basic connective for DeMorgan's Law by substituting from the above identities, and then, apply each nai, na and se modifier of the original connectives. Cancel any double negatives that result.
When do we apply DeMorgan's Law? Whenever we wish to “distribute” a negation over a logical connective; and, for internal naku negation, whenever a logical connective moves in to, or out of, the scope of a negation – when it crosses a negation boundary.
Let us apply DeMorgan's Law to some sample sentences. These sentences make use of forethought logical connectives, which are explained in
Section . It suffices to know that ga and gi, used before each of a pair of sumti or bridi, mean “either” and “or” respectively, and that ge and gi used similarly mean “both” and “and”. Furthermore, ga, ge, and gi can all be suffixed with nai to negate the bridi or sumti that follows.
We have defined na and naku zo'u as, respectively, internal and external bridi negation. These forms being identical, the negation boundary always remains at the left end of the prenex. Thus, exporting or importing negation between external and internal bridi negation forms never requires DeMorgan's Law to be applied.
- Example .94:
- la djan. na klama ga la paris. gi la rom.
John [false] goes-to either Paris or Rome.
- Example .95:
- naku zo'u la djan. klama ga la paris. gi la rom.
It-is-false that: John goes-to either Paris or Rome.
It is not an acceptable logical manipulation to move a negator from the bridi level to one or more sumti. However, Example .94 and related examples are not sumti negations, but rather expand to form two logically connected sentences. In such a situation, DeMorgan's Law must be applied. For instance,
Example .95 expands to:
- Example .96:
- ge la djan. la paris. na klama gi la djan. la rom. na klama
[It is true that] both John, to-Paris, [false] goes, and John, to-Rome, [false] goes.
The ga and gi, meaning “either-or”, have become ge and gi, meaning “both-and”, as a consequence of moving the negators into the individual bridi.
Here is another example of DeMorgan's Law in action, involving bridi-tail logical connection (explained in
- Example .97:
- la djein. le zarci na ge dzukla gi bajrykla
Jane to-the market [false] both walks and runs.
- Example .98:
- la djein. le zarci ganai dzukla ginai bajrykla
Jane to-the market either [false] walks or [false] runs.
Jane to-the market if walks then ([false] runs).
(Placing le zarci before the selbri makes sure that it is properly associated with both parts of the logical connection. Otherwise, it is easy to erroneously leave it off one of the two sentences.)
It is wise, before freely doing transformations such as the one from Example .97 to Example .98, that you become familiar with expanding logical connectives to separate sentences, transforming the sentences, and then recondensing. Thus, you would prove the transformation correct by the following steps. By moving its na to the beginning of the prenex as a naku, Example .97 becomes:
- Example .99:
- naku zo'u la djein. le zarci ge dzukla gi bajrykla
It is false that : Jane to-the market (both walks and runs).
And by dividing the bridi with logically connected selbri into two bridi,
- Example .100:
- naku zo'u ge la djein. le zarci cu dzukla gi la djein. le zarci cu bajrykla
It-is-false-that: both (Jane to-the market walks) and (Jane to-the market runs).
is the result.
At this expanded level, we apply DeMorgan's Law to distribute the negation in the prenex across both sentences, to get
- Example .101:
- ga la djein. le zarci na dzukla gi la djein. le zarci na bajrykla
Either Jane to-the market [false] walks, or Jane to-the market [false] runs.
which is the same as
- Example .102:
- ganai la djein. le zarci cu dzukla ginai la djein. le zarci cu bajrykla
If Jane to-the market walks, then Jane to-the market [false] runs.
- If Jane walks to the market, then she doesn't run.
which then condenses down to Example .98.
DeMorgan's Law must also be applied to internal naku negations:
- Example .103:
- ga la paris. gi la rom. naku se klama la djan.
(Either Paris or Rome) is-not gone-to-by John.
- Example .104:
- la djan. naku klama ge la paris. gi la rom.
John doesn't go-to both Paris and Rome.
In addition to the variables da, de, and di that we have seen so far, which function as sumti and belong to selma'o KOhA, there are three corresponding variables bu'a, bu'e, and bu'i which function as selbri and belong to selma'o GOhA. These new variables allow existential or universal claims which are about the relationships between objects rather than the objects themselves. We will start with the usual silly examples; the literal translation will represent bu'a, bu'e and bu'i with F, G, and H respectively.
- Example .105:
- su'o bu'a zo'u la djim. bu'a la djan.
For-at-least-one relationship-F : Jim stands-in-relationship-F to-John.
- There's some relationship between Jim and John.
The translations of Example .105 show how unidiomatic selbri variables are in English; Lojban sentences like
Example .105 need to be totally reworded in English. Furthermore, when a selbri variable appears in the prenex, it is necessary to precede it with a quantifier such as su'o; it is ungrammatical to just say bu'a zo'u. This rule is necessary because only sumti can appear in the prenex, and su'o bu'a is technically a sumti – in fact, it is an indefinite description like re nanmu, since bu'a is grammatically equivalent to a brivla like nanmu. However, indefinite descriptions involving the bu'a-series cannot be imported from the prenex.
When the prenex is omitted, the preceding number has to be omitted too:
- Example .106:
- la djim. bu'a la djan.
Jim stands-in-at-least-one-relationship to-John.
As a result, if the number before the variable is anything but su'o, the prenex is required:
- Example .107:
- ro bu'a zo'u la djim. bu'a la djan.
For-every relationship-F : Jim stands-in-relationship-F to-John.
- Every relationship exists between Jim and John.
Example .107 is palpably false, however; if Jim and John were related by every possible relationship, then they would have to be both brothers and father-and-son, which is impossible.
A few notes on variables
A variable may have a quantifier placed in front of it even though it has already been quantified explicitly or implicitly by a previous appearance, as in:
- Example .108:
- ci da poi mlatu cu blabi .ije re da cu barda
Three Xs which-are cats are white, and two Xs are big.
What does Example .108 mean? The appearance of ci da quantifies da as referring to three things, which are restricted by the relative clause to be cats. When re da appears later, it refers to two of those three things – there is no saying which ones. Further uses of da alone, if there were any, would refer once more to the three cats, so the requantification of da is purely local.
In general, the scope of a prenex that precedes a sentence extends to following sentences that are joined by ijeks (explained in Section ) such as the .ije in Example .108. Theoretically, a bare i terminates the scope of the prenex. Informally, however, variables may persist for a while even after an i, as if it were an .ije. Prenexes that precede embedded bridi such as relative clauses and abstractions extend only to the end of the clause, as explained in Section .7. A prenex preceding tu'e…tu'u long-scope brackets persists until the tu'u, which may be many sentences or even paragraphs later.
If the variables da, de, and di (or the selbri variables
bu'a, bu'e, and
bu'i) are insufficient in number for handling a particular problem, the Lojban approach is to add a subscript to any of them. Each possible different combination of a subscript and a variable cmavo counts as a distinct variable in Lojban. Subscripts are explained in full in
Section , but in general consist of the cmavo xi (of selma'o XI) followed by a number, one or more lerfu words forming a single string, or a general mathematical expression enclosed in parentheses.
A quantifier can be prefixed to a variable that has already been bound either in a prenex or earlier in the bridi, thus:
- Example .109:
- ci da poi prenu cu se ralju pa da
Three Xs which are-persons are-led-by one-of X
- Three people are led by one of them.
The pa da in Example .109 does not specify the number of things to which da refers, as the preceding ci da does. Instead, it selects one of them for use in this sumti only. The number of referents of da remains three, but a single one (there is no way of knowing which one) is selected to be the leader.
This chapter is incomplete. There are many more aspects of logic that I neither fully understand nor feel competent to explain, neither in abstract nor in their Lojban realization. Lojban was designed to be a language that makes predicate logic speakable, and achieving that goal completely will need to wait for someone who understands both logic and Lojban better than I do. I can only hope to have pointed out the areas that are well-understood (and by implication, those that are not).