All about poi and noi

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This page sets out to explain all you need to now about the relative clause markers poi and noi, how they differ with regards to description sumti, both unquantified and in the presence of quantifiers.

The article deliberately ignores pe, ne, po'u and no'u, because those are nothing more than shortcuts of poi and noi combined with specific predicates. Everything that applies to poi and noi directly applies to those cmavo, too. No separate explanations are required.

The examples on this page will consist of a Lojban phrase or expression, a rendering in logic notation, and an English translation.

Logic symbols used in this article

Symbol Meaning
[math]\exists[/math] existential quantifier
[math]\forall[/math] universal quantifier
logical and
[math]=\lt [/math] is-among, Lojban me
Q any (outer) quantifier
x, y, z singular variables
xx, yy, zz plural variables
c, c0, c1, ... plural constants
[] surrounds prenex
 : corresponds to Lojban poi
 ; separates sentences, like Lojban .i

poi and noi in general

poi and noi introduce relative clauses, restrictive and non-restrictive respectively. They help us narrow down referent sets or allow us to provide additional information about a given sumti. For both poi and noi, their syntactic position and environment make a big difference as to what exactly they achieve in a given sentence. We will look at each and every possible case below.

PA da NOI

PA da poi

PA da noi

NOI and unquantified sumti

With plain sumti, a relative clause can only be placed after the sumti (e.g. mi noi tatpi).

With selbri based descriptions (e.g. lo plise), there are three different places a relative clause (abbreviated <rel>) can go:

lo <rel> broda <rel> ku <rel>

By definition, a relative clause that appears between gadri and selbri is equivalent to one that appears after the ku:

lo <rel> broda

is equivalent to:

lo broda ku <rel>

Because they are equivalent we will only analyse the case where the relative clause follows ku to avoid duplication.

If the relative clause is placed between selbri and ku then the meaning changes compared to the other two cases above.

<sumti> NOI

lo broda NOI

something something

There are only two rules to learn, two rules, which explain every possible combination of quantifiers and relative clauses.

An inner quantifier isn't a true logical quantifier in that it doesn't count true bridi, but merely enumerates the number of referents of a sumti.

An outer quantifier counts how many times or for how many things a bridi is true.

Q da broda
[math][Qx] \: broda(x)[/math]
Q <sumti> cu broda
Q da poi ke'a me <sumti> cu broda

The quantifier Q counts for how many things that are among the referents of <sumti> the bridi is true, or in other words,

it tells you that if you were to substitute each member for , then the bridi would be true exactly Q times.

Since {da} is a singular variable, it always stands in for ...

This is the formula that is evoked any time an outer quantifier occurs in a bridi.

Q da poi ke'a me <sumti> cu broda
[math][Qx : x =\lt c_s ] \: broda(x)[/math]

[...]

<sumti> poi ke'a brode cu broda
[math]c_0 =\lt c_{sumti} \land brode(c_0); \: broda(c_0)[/math]

<sumti> poi ke'a brode creates a new sumti whose referents are those that are both among <sumti> and satisfy brode. It is important to note that all referents that are among <sumti> and satisfy brode are among the referents of the resulting expression, not just some of the referents. This is called maximality.

Looking at the logic notation, we can see that the constant c is only stated to be among c1 and to brode, but not that all referents among c1 that brode are among c0, even though that's what the poi version implies. The following always holds with outer poi:

[math][Axx] \: xx =\lt c_{sumti} \land brode(xx) \to xx =\lt c_0[/math]

So the full sentence would actually be:

[math]c_0 =\lt c_{sumti} \land brode(c_0) \land ([Axx] \: xx =\lt c \land brode(xx) \to xx =\lt c_1); \: broda(c_0)[/math]

However, since this takes up so much space, we will omit the part in the middle with the understanding that maximality is implied in all examples involving outer poi. This is to keep things easier to read.

<sumti> noi ke'a brode cu broda
[math]brode(c); \: broda(c)[/math]

[...]

Q da poi ke'a brode cu broda
[math][Qx : brode(x)] \: broda(x)[/math]

[...]

Q da noi ke'a brode cu broda
[math][Qx] \: broda(x); \: [Ay : brode(y)] \: broda(y)[/math]

Whence the universal quantifier? In simple terms, we know that noi is non-restrictive, which means that its presence must not have any impact on the number of true bridi that the quantifier Q counts. The only logical way in which a relative clause could make a statement without restricting (i.e. selecting a sub-set) is if the predicate of the noi-clause applies to every member of a given referent-set.

Prefixing a na ku to a bridi is always a good way to gain insights about the scope interactions within the bridi. How does such a negation differ between a poi sentence and a noi sentence?

na ku su'o da poi ke'a brode cu broda

[math]![Ex : brode(x)] \: broda(x)[/math]

[math][Ax : brode(x)] \: !broda(x)[/math]

na ku su'o da noi ke'a brode cu broda
[math]![Ex] \: broda(x); \: [Ay : brode(y)] \: broda(y)[/math]

In the noi version, we know that nothing broda, but all things still brode, because noi makes a separate claim.

In the poi version, we know have a much weaker claim, namely that nothing satisfies both brode and broda together (broda(x) \land brode(x)), while nothing is said about how many things satisfy broda, and how many satisfy brode.

lo PA broda ku is a case of <sumti>, and we already know how <sumti> interacts with quantifiers and relative clauses. There isn't anything new to understand here, but we'll look at an example regardless:

lo mu plise ku poi xunre cu kukte
[math]plise(c_p) \land mumei(c_p); \: c =\lt c_p \land xunre(c); \: kukte(c)[/math]

Quite a different case is lo broda poi brode, where the relative clause is placed between selbri and ku. Here, poi acts even before lo to define the initial predicate of which lo then extracts the x1.

lo [broda poi brode]

A poi in this position is exactly equivalent to je poi'i:

lo broda je poi'i ke'a brode
lo poi'i ke'a broda gi'e brode

Since the inner poi merely defines the initial referents of the sumti, it has no bearing on outer quantifiers.

lo broda poi brode ku is really just another way of writing lo <selbri> ku.

lo mu broda poi brode = lo mu brodi

Five [brodas that brode]

mi lebna re lo mu plise poi xunre
re da poi ke'a me lo mu plise poi xunre zo'u mi da lebna
[math]plise(c) \land xunre(c) \land mumei(c); \: [2x : x =\lt c] \: lebna(mi, x)[/math]
"I take two of the five red apples."

What happens, when poi is outside the ku? Then we get a case we've already covered: <sumti> poi broda

<sumti> poi broda is the same as lo me <sumti> je broda, which is the same as lo poi'i ke'a me <sumti> gi'e broda.

We shall now examine the most complicated case, an outer quantifier paired with an outer poi, and then noi.

re lo mu plise ku poi xunre cu kukte

Interpretation 1:

re da poi ke'a me lo mu plise gi'e xunre cu kukte
[math]plise(c) \land mumei(c); \: [2x : x =\lt c \land xunre(c)] \: kukte(c)[/math]
"Two things that are among the five apples and are red are delicious."

Interpretation 2:

re lo poi'i ke'a me lo mu plise gi'e xunre cu kukte
[math]plise(c_{plise}) \land mumei(c_{plise}); \: c =\lt c_{plise} \land xunre(c); \: [2x : x =\lt c] \: kukte(x)[/math]
"Two things that are among those things which are among the five apples and red are delicious."

With this example, the two interpretations give the same result, the same number of delicious apples. However, the next example shows the inadequacy of interpretation 1:

lo pulji cu arrest reno lo panono demonstrators ku poi sruri lo dinju

(Note by the way, that the same sentence with inner poi would translate to English as "The police arrested 20 of the 100 building-surrounding demonstrators", i.e., all 100 both demonstrated and surrounded the building)

Let us again examine the two possible interpretations:

Interpretation 1:

lo pulji cu arrest reno da poi ke'a me lo panono demonstrators gi'e sruri lo dinju
[math]pulji(c_{pulji}); \: demonstrate(c_{demonstrators}) \land panonomei(c_{demonstrator}); \: dinju(c_{dinju}); \: [20x : x =\lt c_{demonstrators} \land sruri(x,c_{dinju}] \: arrest(c_{pulji}, x)[/math]
"The police arrested 20 things that were among the one hundred demonstrators and that surrounded the building."

Interpretation 2:

lo pulji cu arrest reno lo poi'i ke'a me lo panono demonstrators gi'e sruri lo dinju
[math]logic[/math]
"The police arrested 20 of those among the 100 demonstrators that surrounded the building"

(100 people were demonstrating, some of them surrounded the building, and 20 of those got arrested)

Sentence 1 actually means that the police didn't arrest anyone!

This is because interpretation 1 does not account for non-distributive plural predication, while interpretation 2 does.

What if the outer poi is replaced with outer noi?

mi lebna ci lo mu plise ku noi xunre

Interpretation 1:

mi lebna ci lo mu plise .i ro plise poi mi lebna ke'a cu xunre
[math]plise(c) \land mumei(c); \: [3x : x =\lt c] \: lebna(mi, x); \: [Ay : plise(y) \land lebna(mi, y)] \: xunre(y)[/math]
"I take three of the five apples. Each apple that I take is red."

(Note that the noi version differs significantly from a hypothetical poi version. With noi we know that exactly three apples are taken. With poi we know that three red apples are taken, but not whether any non-red apples are taken)

Interpretation 2:

lo mu plise cu xunre .i mi lebna ci ri
[math]plise(c) \land mumei(c); \: xunre(c); \: [3x : x =\lt c] \: lebna(mi, x)[/math]
"The five apples are red. I take three of them."

Again, both interpretations yield the same number of red apples taken by the speaker (three red apples), but they get to that result by different mechanisms. Which one is correct? Interpretation 1 parallels Interpretation 1 of the poi, and interpretation 2 parallels interpretation 2. We saw that interpretation two of poi was better, does the same apply to noi?

Let's consider noi in the context of the demonstrators that surrounded the building:

lo pulji cu arrest reno lo panono demonstrators ku noi sruri lo dinju
[math][/math]
"The police arrested 20 of the 100 demonstrators, who surrounded the building."

(i.e., 100 people were demonstrating, 20 of those surrounded the building, and all those 20 got arrested)

Interpretation 1:

lo pulji cu arrest reno da poi ke'a menre lo panono demonstrators .i ro demonstrator poi lo pulji cu arrest ke'a cu sruri lo dinju
[math]pulji(c_{pulji}); \: demonstrate(c_{demonstrators}) \land panonomei(c_{demonstrator}); \: [20x : x =\lt c_{demonstrators}] \: arrest(c_{pulji}, x); \: dinju(c_{dinju}); \: [\forall y : demonstrator(y) \land arrest(c_{pulji}, y)] \: sruri(y, c_{dinju}[/math]
"The police arrested 20 of the 100 demonstrators. Each demonstrator that the police arrested was surrounding the building."

Note that this does not mean that it's necessarily the case that nobody else among the demonstrators surrounded the building. It does however say that each single arrested demonstrator surrounded the building individually, which is not the intended meaning, and likely a very bizarre situation. Maybe that's why the police felt the need to arrest them.

Interpretation 2:

lo panono demonstrators cu sruri lo dinju .i lo pulji cu arrest reno lo go'i
</math>
"The 100 demonstrators were surrounding the building. The police arrested 20 of them."

This says that in fact all 100 demonstrators were building-surrounders. Thus, any 20 that the police could arrest automatically are building-surrounders also. Additionally, interpretation 2 again does what interpretation 1 does not: account for plural quantification.

But what if we do want to say that what interpretation 1 said (minus the distributivity)?. We might not want to claim each demonstrater a building-surrounder. We might want to say that each arrested demonstrator was a building-surrounder, no, that each arrested demonstrator was among building-surrounders. Well, we can:

lo pulji cu arrest reno lo panono demonstrators ku poi sruri lo dinju
lo pulji cu arrest reno lo poi'i ke'a me lo panono demonstrators gi'e sruri lo dinju

What about inner noi?

lo broda noi brode
lo poi'i ke'a broda (to ri brode toi)

[...]

lo plise noi xunre cu kukte
[math]plise(c); \: xunre(c); \: kukte(c)[/math]

At first glance this looks identical to (5.2) (lo plise poi xunre cu kukte)

lo plise poi xunre cu kukte
[math]plise(c) \land xunre(c); \: kukte(c)[/math]

(not even a negation could cause a difference here, because the sentence contains nothing but constants)

As we've seen above, an inner poi clause attaches directly to the selbri, in a sense restricting the referent set of the predicate by replacing the single predicate with a conjunction of it and another predicate.

In other words, the poi in lo broda poi brode creates a new predicate [math]H[/math] that has the same referent set as [math]F \land G[/math].

In this sense, poi performs a restrictive roll, even in the absence of any logical quantifiers. noi, however, is the non-restrictive counterpart. It, too, will try to attach directly to the predicate, but without restricting the referent set of the resulting predicate. This means that it does not create a new predicate, or, perhaps more accurately, the referent set of [math]F \land G[/math] in the case of noi is equivalent to that of the lone predicate [math]F[/math].

The nuance here is more subtle than with quantifiers and applies directly to the universe of discourse. Let's build an analogue between the quantified noi/poi distinction we already understand, and the inner noi/poi distinction to see the parallels:

Q da poi ke'a brode cu broda
[math][Qx : brode(x)] \: broda(x)[/math]

[...]

Q da noi ke'a brode cu broda
[math][Qx] \: broda(x); \: [Ay : brode(y)] \: broda(y)[/math]

As covered much earlier, the noi implies a universal quantifier, because that's the only logical way that a conjunction can have the same referent set as the predicates in isolation. A similar thing could be said about inner noi, let's compare:

lo broda poi brode cu brodi (replace with actual words!!)
lo poi'i ke'a broda gi'e brode cu brodi

But in keeping with the theme of creating a new predicate, let's introduce a me'au:

lo me'au lo ka broda je lo ka brode cu brodi
[math]broda(c) \land brode(c); \: brodi(c)[/math]

The sumti is now very clearly one whose referent(s) satisfy the conjunction of two predicates. What about noi?

lo broda noi brode cu brodi
lo me'au lo ka broda cu brodi .i ro me'au lo ka broda cu me'au lo ka brode
[math]ka(c_F, broda); \: ka(c_G, brode); \: ckaji(c, c_F) \land ckaji(c, c_G); \: brodi(c)[/math]

This could be written simpler if Lojban had a non-restrictive version of je (something I've proposed it should have), but it doesn't.

Thus, we can conclude that the previous example implies a universe of discourse where [math]F[/math] and [math]F \land G[/math] have the same referent set, or put another way, in which for all things, being broda entails being brode.

No such implication occurs in the poi version.

With this, we have covered every possible case of relative clauses.

Conversion formulas

su'o da poi ke'a brode cu broda su'o da ge brode gi broda
ro da poi ke'a brode cu broda ro da ga nai brode cu broda
<sumti> poi ke'a broda lo me <sumti> je broda
lo poi'i ke'a me <sumti> gi'e broda