# All about poi and noi

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

———Written by selpahi———

This page sets out to explain all you need to now about the relative clause markers poi and noi.

It does not contain separate explanations for 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. For a conversion table see the bottom of this page.

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

Symbol Meaning
$\exists$ existential quantifier
$\forall$ universal quantifier
$\land$ logical and
$\neg$ logical negation
$=$ sameness, the two operands have the same referents.
$\subseteq$ is-among, Lojban me
$\square$ any (outer) quantifier
$x, y, z$ singular variables
$xx, yy, zz$ plural variables
$c, c_0, c_1\dots$ plural constants
$[\cdots]\colon$ surrounds prenex
$\colon$ 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.

# PA 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.

[...]

domain

anything -> anybody

examples

A useful thing to know is that a sentence of the form su'o da poi broda cu brode is truth-functionally equivalent to su'o da ge broda gi brode:

 su'o da poi ke'a broda cu brode
 equals
 su'o da ge broda gi brode

For example:

 su'o da poi ke'a cribe cu blabi $[\exists x\colon \text{cribe}(x)]\colon \text{blabi}(x)$"Some bears are white."

has the same meaning as:

 su'o da ge cribe gi blabi $[\exists x]\colon \text{cribe}(x) \land \text{blabi}(x)$"There exists some X such that X is both a bear and white."

For ro da poi broda cu brode the following equivalence holds instead:

 ro da poi ke'a broda cu brode
 equals
 ro da zo'u ga nai da broda gi da brode

For example:

 Lojban$logic$"English."

which means the same as:

 Lojban$logic$"English."

## PA <selbri>

By definition, PA da poi ke'a broda can be abbreviated PA broda. This is a handy shortcut that saves space, variables, and terminators. Apart from the different grammar, PA broda behaves identical to PA da poi ke'a broda in all contexts.

## PA <sumti>

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

A quantifier $\square$ 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 <sumti>, then the bridi would be true exactly $\square$ times.

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

 Q da poi ke'a me $c$ cu broda$[\square x \colon x \subseteq c ]\colon \text{broda}(x)$

[...]

For example:

 Lojban$logic$"English"

[...]

 Lojban$logic$"English"

We will need this expansion later when we look at PA lo <selbri> and its variations.

# PA da noi

[...]

 Q da broda$[\square x]\colon \text{broda}(x)$
 Q cu broda
 Q da poi ke'a me cu broda

[...]

 Q da poi ke'a brode cu broda$[\square x \colon \text{brode}(x)]\colon \text{broda}(x)$

[...]

 Q da noi ke'a brode cu broda$[\square x]\colon \text{broda}(x); [\forall y \colon \text{brode}(y)]\colon \text{broda}(y)$

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 $\square$ 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

$\neg [\exists x \colon \text{brode}(x)]\colon \text{broda}(x)$

$[\forall x \colon \text{brode}(x)]\colon \neg \text{broda}(x)$

 na ku su'o da noi ke'a brode cu broda$\neg [\exists x]\colon \text{broda}(x); [\forall y \colon \text{brode}(y)]\colon \text{broda}(y)$

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 ($\text{broda}(x) \land \text{brode}(x)$), 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:

 $c$ noi ke'a brode cu broda$\text{brode}(c); \text{broda}(c)$

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"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$?\big[\text{sidju}(\text{do}, \text{mi}); \: \text{certu}(\text{do})\big]$"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$\neg \text{tolstace}(\text{mi}); \text{prami}(\text{mi}, \text{do})$"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$\neg ( \text{prami}(\text{mi}, \text{do}) \land \text{tolstace}(\text{mi}))$"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:

 $c$ poi ke'a brode cu broda$c^\prime \subseteq c \land \text{brode}(c^\prime); \text{broda}(c^\prime)$

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:

$[\forall xx]\colon xx \subseteq c_{\text{sumti}} \land \text{brode}(xx) \implies xx \subseteq c_0$

So the full sentence would actually be:

$c_0 \subseteq c_{\text{sumti}} \land \text{brode}(c_0) \land ([\forall xx]\colon xx \subseteq c \land \text{brode}(xx) \implies xx \subseteq c_1); \text{broda}(c_0)$

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:

 Lojban$logic$"English."

[...]

 Lojban$logic$"English."

[...]

### <sumti> poi with apparently singular <sumti>

Certain kinds of sumti tend to be interpreted as singular, like mi (presupposing a single speaker here), or perhaps a sumti formed with zo. What effect does using poi on a sumti like this have, and what does this tell us about poi?

[...]

# Selbri-based descriptions

## 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 broda ku

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

 lo broda

is equivalent to:

 lo broda ku

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 relative clauses - 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 semantically 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 lo mu plise poi xunre$\text{plise}(c) \land \text{xunre}(c) \land \text{mumei}(c); \text{lebna}(\text{mi}, c)$"I take the five red apples."

Now the same example with an additional outer quantifier:

 mi lebna re lo mu plise poi xunre
 re da poi ke'a me lo mu plise poi xunre zo'u mi da lebna$\text{plise}(c) \land \text{xunre}(c) \land \text{mumei}(c); \: [2x \colon x \subseteq c]\colon \: \text{lebna}(\text{mi}, x)$"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). But we will consider this case in more detail later.

### lo <selbri> noi

[...]

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

[...]

 lo plise noi xunre cu kukte$\text{plise}(c); \: \text{xunre}(c); \: \text{kukte}(c)$

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

 lo plise poi xunre cu kukte$\text{plise}(c) \land \text{xunre}(c); \: \text{kukte}(c)$

(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 $H$ that has the same referent set as $F \land G$.

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 $F \land G$ in the case of noi is equivalent to that of the lone predicate $F$.

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$[\square x \colon \text{brode}(x)]\colon \: \text{broda}(x)$"English"

[...]

 Q da noi ke'a brode cu broda$[\square x]\colon \: \text{broda}(x); \: [\forall y \colon \text{brode}(y)]\colon \: \text{broda}(y)$"English"

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"English"

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$\text{broda}(c) \land \text{brode}(c); \: \text{brodi}(c)$"English"

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'oi me'au lo ka broda cu me'au lo ka brode$\text{ka}(c_F, \text{broda}); \: \text{ka}(c_G, \text{brode}); \: \text{ckaji}(c, c_F) \land \text{ckaji}(c, c_G); \: \text{brodi}(c)$"English"

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 $F$ and $F \land G$ 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 relative clauses - 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$\text{plise}(c_p) \land \text{mumei}(c_p); \: c \subseteq c_p \land \text{xunre}(c); \: \text{kukte}(c)$"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$\text{plise}(c_p) \land \text{mumei}(c_p); \: \text{kukte}(c_p); \: \text{xunre}(c_p)$"The five apples, which are red, are delicious."

[...]

## PA 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$\text{plise}(c) \land \text{mumei}(c); \: [2x \colon x \subseteq c \land \text{xunre}(c)]\colon \: \text{kukte}(c)$"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$\text{plise}(c_{\text{plise ) \land \text{mumei}(c_{\text{plise}}); \: c \subseteq c_{\text{plise}} \land \text{xunre}(c); \: [2x \colon x \subseteq c]\colon \: \text{kukte}(x)$|"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 ro lo panono panja'o ku poi sruri lo dinju

(Note by the way, that the same sentence with inner poi would translate to English as "The police arrested each 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 ro da poi ke'a me lo panono panja'o gi'e sruri lo dinju$\text{pulji}(c_{\text{pulji ); \: \text{panjaho}(c_{\text{panjaho}}) \land \text{panonomei}(c_{\text{panjaho}}); \: \text{dinju}(c_{\text{dinju}}); \: [\forall x \colon x \subseteq c_{\text{panjaho}} \land \text{sruri}(x,c_{\text{dinju}})]\colon \: \text{arrest}(c_{\text{pulji}}, x)$|"The police arrested each thing that was among the one hundred demonstrators and that surrounded the building."}}

Interpretation 2:

 lo pulji cu arrest ro lo poi'i ke'a me lo panono panja'o gi'e sruri lo dinju$logic$"The police arrested each of those among the 100 demonstrators that surrounded the building."

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

With interpretation 1 the police didn't actually arrest anyone! (unless we're dealing with relatives of Mister Fantastic or Plastic Man)

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

Therefore, interpretation 2 is the way to go. As a bonus, it also happens to result in fewer rules, which is always good.

## PA lo <selbri> ku noi

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$\text{plise}(c) \land \text{mumei}(c); \: [3x \colon x \subseteq c]\colon \: \text{lebna}(\text{mi}, x); \: [\forall y \colon \text{plise}(y) \land \text{lebna}(\text{mi}, y)]\colon \: \text{xunre}(y)$"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$\text{plise}(c) \land \text{mumei}(c); \: \text{xunre}(c); \: [3x \colon x \subseteq c]\colon \: \text{lebna}(\text{mi}, x)$"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 panja'o ku noi sruri lo dinju

Interpretation 1:

 lo pulji cu arrest reno da poi ke'a menre lo panono panja'o .i ro panja'o poi lo pulji cu arrest ke'a cu sruri lo dinju$\text{pulji}(c_{\text{pulji ); \: \text{panjaho}(c_{\text{panjaho}}) \land \text{panonomei}(c_{\text{panjaho}}); \: [20x \colon x \subseteq c_{\text{panjaho}}]\colon \: \text{arrest}(c_{\text{pulji}}, x); \: \text{dinju}(c_{\text{dinju}}); \: [\forall y \colon \text{panjaho}(y) \land \text{arrest}(c_{\text{pulji}}, y)]\colon \: \text{sruri}(y, c_{\text{dinju}})$|"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 panja'o cu sruri lo dinju .i lo pulji cu arrest reno lo go'i$logic$"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 ro reno lo panono panja'o ku poi sruri lo dinju
 lo pulji cu arrest ro reno lo poi'i ke'a me lo panono panja'o gi'e sruri lo dinju$logic$"The police arrested all 20 of those that were among the 100 demonstrators and that surrounded the building."

lo==Relative clauses and la==

la broda NOI

la NOI broda ku'o broda

When to put la first, and when to put it last?

[...]

 Lojban"English"

[...]

## la and relative clauses

"President Obama", "Teris the tiger".

[...]

la broda poi la broda ku poi

la broda noi la broda ku noi

po'u la broda

[...]

# ma NOI

First, let's recap how the question word ma (as well as all the other question words) behave. ma always has top-scope, no matter how deeply nested it appears in a sentence. (1) and (2) are equivalent:

(1) do djica lo nu mi zukte ma
"You want me to do what?"
"What do you want me to do?"
(2) ma poi'i do djica lo nu mi zukte ke'a
"What is such that you want me to do it?"
"What is it you want me to do?"


However, if we wish to restrict the referent pool of the question word, the familiar method of using poi (as in (3)) may not be semantically accurate -- the desired expansion of (3) is (4), but the actual meaning is (5). The difference might be subtle and difficult to spot, but it's there, and it means that we should find a new way to say "which/what".

 (3) do pu kanpe lo nu mi cuxna ma poi taxfu
(4) ma taxfu gi'e poi'i do pu kanpe lo nu mi cuxna ke'a
(5) ma poi'i do pu kanpe lo nu mi cuxna ke'a poi taxfu
ma poi'i do pu kanpe lo nu mi cuxna lo me ke'a je taxfu


One option might be to use noi instead of poi (as in (6)), another would be to use mo (as in (7)).

(6) do pu kanpe lo nu mi cuxna ma noi taxfu
(7) do pu kanpe lo nu mi cuxna lo mo taxfu
"What kind of clothes did you expect me to choose?"
"[The] what-clothes did you expect me to choose?"


Both methods leave something to be desired. Even if noi has the right semantics, it still requires a terminator and a ke'a, while using mo can be vague (since it's a tanru) and it can be awkward to use.

Therefore, it might be a good idea to create a new gadri X (in selma'o LO) that has the semantics of (4) but keeps the wh in-situ capability of (3). In other words, ko'a broda X brode would be defined as ma brode gi'e poi'i ko'a broda ke'a.

It remains to be decided how that new gadri should be spelled.

# Summary

[...]

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 PA PA da poi ke'a PA PA da poi ke'a me PA da noi ke'a broda cu brode PA da broda .i ro broda cu brode noi ke'a broda cu brode brode .i broda poi ke'a broda lo me je poi'i ke'a broda lo poi'i ke'a me gi'e broda lo broda poi brode lo broda je poi'i ke'a brode lo poi'i ke'a broda gi'e brode lo broda noi brode lo broda (to ro'oi broda cu brode toi) lo broda ku poi brode lo poi'i ke'a me lo broda gi'e brode lo broda ku noi brode lo broda ku (to ri brode toi) lo NOI ke'a brode ku'o broda lo broda ku NOI brode

## Conversion to and from relative phrases

 pe poi ke'a co'e ne noi ke'a co'e po'u poi ke'a du no'u noi ke'a du goi noi ca'e ke'a du