All about poi and noi

From Lojban
Revision as of 21:23, 11 August 2015 by Selpahi (talk | contribs)
Jump to: navigation, search


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.

A general note about inner quantifiers

It should be noted that so-called inner quantifiers (the ones between gadri and selbri) aren't true logical quantifiers in that they don't count true bridi, but merely enumerate the number of referents of a sumti. The presence or absence of an inner quantifier has no effect on the behaviour of relative clauses, be it outer or inner relative clauses. What a relative clause means is solely dependent on whether it is a restrictive (poi) or non-restrictive (noi) one, and whether it is an inner (lo broda NOI) or outer (lo broda ku NOI) relative clause.

Q da poi


What does a quantifier do? A quantifier counts how many times or for how many things a bridi is true, or, put another way, a quantifier counts true bridi.



poi AND


Q <sumti>

poi also appears "invisibly" any time a sumti carries an outer quantifier.

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.

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]


For example:



PA da noi


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


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.

<sumti> NOI

<sumti> noi

Let's begin this section with the simplest case first, <sumti> noi. This is noi in its most basic habitat. noi makes an incidental claim about the head noun which is separate from the main bridi. The general syntax of <sumti> noi is this:

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

Logically speaking this is the simplest possible use of a relative clause; two simple claims, no quantification. For example:

mi pilno ti noi ke'a plixau
[math]c = ti; \: pilno(mi, c); \: plixau(c)[/math]
"I'm using this, which is useful." / "I'm using this. It's useful."

What does it mean for the noi clause to constitute a separate claim? It means that any operators that operate on the main bridi will not affect the content of the noi clause. For example:

xu do noi ke'a certu ba sidju mi
[math]? \: sidju(do, mi); \: certu(do)[/math]
"Will you, who are an expert, help me?"

The question word xu only affects the main bridi, keeping the noi clause an assertion.

Similarly so for negation:

na ku mi noi prami do cu tolstace
[math]! \: tolstace(mi); \: prami(mi, do)[/math]
"It's not the case that I, who love you, am dishonest." (i.e., "I love you, and I'm not dishonest."

While it's possible to translate mi noi prami do cu tolstace into English as "I love you and am being dishonest", this is not recommended, because it could suggest that noi can be translated into gi'e, which it cannot, as becomes apparent when the na ku is added back in:

na ku mi prami do gi'e tolstace
"It's not the case that I both love you and am being dishonest." (i.e., at least one of those two is false)

Therefore it's best to always consider the noi clause as entirely independent from the main claim, lest one could make wrong assumptions.

<sumti> poi

The previous section that <sumti> noi is very straightforward. It has an obvious meaning and its logic is easy to grasp. <sumti> poi on the other hand might be less obvious. In the section about da poi we saw poi restricting the domain of a quantifier. But here we have no quantifier, so what is the poi, the restrictive relative clause marker, to restrict in <sumti> poi?

The general syntax of <sumti> poi is this:

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

What does this mean? It means that <sumti> poi ke'a brode creates a new sumti whose referents are those that are both among <sumti> and satisfy brode. Since there is no quantifier whose domain could be restricted, the poi has to perform a restriction directly on the sumti. It does this by selecting referents from among the sumti which satisfy a certain predicate, namely the predicate inside the relative clause.

It is important to note that precisely 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][\forall xx] \: 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 ([\forall xx] \: 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.

Let's look at some examples:





Selbri-based descriptions and relative clause placements

A selbri-based description is a sumti made from a gadri plus a selbri, e.g. lo plise. With selbri-based descriptions 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.

Below we will examine the different possibilities, lo broda poi, lo broda noi, lo broda ku poi and lo broda ku noi in detail.

Inner NOI - lo <selbri> NOI

lo <selbri> poi


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. The resulting predicate has the same referent set as the conjunction of the initial predicate and the relative clause.

lo [broda poi brode]

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

lo broda je poi'i ke'a brode

And this is equivalent to:

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 a specific case of lo <selbri> ku.

For example:

lo plise poi xunre

Yields a sumti that refers to red apples. Let's examine the logic in a full sentence:

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 (we recall that <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).

Now, what about inner noi?

lo <selbri> 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.

Again: inner poi creates a new predicate that has the same referent set as the conjunction of the two conjoined predicates.

Inner noi does not create a conjunction, but rather comments on the predicate saying that being broda also means being brode (in the current universe of discourse).

Outer NOI - lo <selbri> ku NOI

lo <selbri> ku poi

If you have read everything above, then this section contains no truly new information.

lo <selbri> 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]
"The red ones among the five apples are delicious."


lo <selbri> ku noi

Again, lo <selbri> ku noi is a case of <sumti> noi, which was covered earlier.

lo mu plise ku noi xunre cu kukte
[math]plise(c_p) \land mumei(c_p); \: kukte(c_p); \: xunre(c_p)[/math]
"The five apples, which are red, are delicious."


Q lo <selbri> ku poi

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_{demonstrators}); \: 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
"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 in the previous example 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

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_{demonstrators}); \: [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
"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



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


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
Q <sumti> Q da poi ke'a me <sumti>
Q da noi ke'a broda cu brode Q da broda .i ro broda cu brode
<sumti> noi ke'a broda cu brode <sumti> brode .i <sumti> broda