A Simpler Quantifier Logic

This article is part of the Simpler series on simplifying certain aspects of Lojban.
See also A Simpler Connective System and A Simpler Morphology [coming soon].

In this article I will take a closer look at the quantifier logic currently present in (post-xorlo) Lojban. I will identify logical problems as well as practical disadvantages. Finally, I will offer a solution that addresses both.

Not only is the current system both impractical and unintuitive, its introduction also left behind one major logical flaw as Lojban made the move from singular logic to plural logic.

Read on to find out what singular logic and plural logic are, how they are related to xorlo, and why the current situation is not tenable.

Let’s begin by looking at one of the most common types of mistakes in Lojban.

simxu: x1 are doing x2 to each other
tavla: x1 talks to x2

? ro da simxu lo ka tavla
“Everyone is talking to each other.”

Seems correct? Actually, it’s complete gibberish. What the Lojban actually means is:

ro da simxu lo ka tavla
“Each one is talking to each other.”

No single person can talk to each other, and the result is a nonsensical claim.

Similarly:

verba: x1 is a child …
jmaji: x1 are gathering at x2

ze verba cu jmaji lo jinto
“Seven children are each gathering at the well.”

This sentence claims that this child gathered, and that child gathered, etc. But gathering is not something a person can do alone. It takes multiple people to gather. Even if gathering could be done alone, the meaning would likely be very different: “The child gathered” sounds like the child was in pieces and is now having its parts move back together. So maybe a zombie can gather, but a normal person certainly cannot.

Why is it that the Lojban sentences above make those claims instead of the intended ones?

The answer to this question leads us directly to singular logic.

Singular logic

In singular logic, which is what was used by CLL Lojban, a variable (e.g. Lojban da) can only take on singular values, and a constant can only refer to a singular individual. A sentence like

ci da nenri lo tanxe
“Three things are inside the box.”

really says that for “X is in the box”, if we substitute everything in the universe for X one by one, we would get a true sentence exactly three of those times. The sentence is evaluated with a singular value for X (da) each time, and yields “true” as a result three times.

Singular logic is not capable of expressing something like “everyone talked to each other” (the example we began with) directly. Instead, it has to help itself by introducing imaginary sets, and then making claims about the sets instead. This is because it cannot talk about multiple things at once (hence the “singular“). In order to cope, singular logic obsessively singularizes the world, using sets as the singularization mechanism. Instead of talking about “people” (plural), it talks about a set of people” (singular). It always talks about one thing rather than many, and in doing so it also introduces a new metaphysical object (the set).

lo’i prenu cu simxu lo ka tavla
A = {x : prenu(x)}, simxu(A, lo ka tavla)
“The set of [all] people is such that its members are in a reciprocal relation of talking.”

but not, for example

 * ci prenu cu simxu lo ka tavla
[3x : prenu(x)] simxu(x, lo ka tavla)
“There exist exactly three people such that each one talks to each other.” (nonsensical)

My personal gripes with the set approach are that I would rather talk about things directly than talk about a set which has those things as its elements. But many people seem to enjoy set logic, and this is roughly what CLL Lojban had to put up with for a long time, until xorlo came along and changed the very fabric of our little logical language — or did it?

Singular variables meet plural constants

So, what changed when xorlo happened? xorlo changed the semantics of description sumti (really, any sumti without an explicit outer quantifier) so that all of them are now plural constants. One could assume that things ended there, but the introduction of plural constants necessarily carried with it a change in the way predicates are used. When everything is singular, predicates must be defined in such a way that singular things can satisfy them. But one of the points of xorlo is that the referents of lo broda can satisfy a predicate together as well as distributively. Therefore, it no longer makes much sense for predicates to be defined exclusively for singular referents because this would render one of the main points of xorlo completely void.

Therefore, xorlo not only changed the meaning of description sumti, but also reframed the entire lexicon of predicates to account for plural reference. For instance, simxu went from taking a set in the x1 to taking multiple things. Really, any set place was replaced by a plural place. (Note that it is not possible for a predicate to accept either of the two, as that would introduce a true ambiguity: lo selcmi cu simxu lo ka broda would be ambiguous between  a set or sets having its/their members in a reciprocal relation and multiple sets being in a reciprocal relation to each other. The same ambiguity applies to any other such argument place.)

As a result of all this, Lojban moved from pure singular logic to a form of plural logic. I’m saying “form of plural logic”, because, as I will show next, xorlo only put one leg into plural territory but got stuck on the fence with the other.

xorlo introduced plural constants (plural reference), but made no changes to variables and quantification, which were, and still are, singular. This makes post-xorlo Lojban a hybrid between singular logic and plural logic. Unfortunately, as it turns out, this sort of hybrid is logically highly problematic.

I now find myself in the rare situation where quoting John Clifford actually supports one of my points. Buried in the archives of the numerous discussions that took place in 2004 during the BPFK’s prime, one can find the following conversation between John Clifford and xorxes:

xorxes: All constants can be plural, it’s only da that is singular.
Clifford: No can do: if the constants are plural, so must the variables be (and contra-positively).
xorxes: Can you explain why it cannot be done? I haven’t run into any problems so far.
Clifford: The basic reason it won’t work is simply that every constant entails a particular generalization and every universal entails an instance to every constant. A plural constant would not entail an individual generalization and even more so, a singular universal would not entail a plural instance.

He was right — it doesn’t work. And here’s why:

Two basic rules of predicate logic are that

  1. if c is a constant in the domain of the variable x and [∀x] P(x) is true, then it can be inferred that P(c) is true as well, and
  2. if P(c) is true, then it can be inferred that [∃x] P(x) is true.

Neither of those inferences work when there are plural constants and singular variables.

If c is a plural constant and x is a singular variable, and if [∀x] P(x) is true, P(c) may or may not be true, because there are things that hold for one thing but not for multiple things together. “is an individual” for example, or “weighs 60kg”.

If c is a plural constant x is a singular variable, and if P(c) is true, it does not follow that [∃x]P(x), because if, say, the students (c) are surrounding a building (P), it does not follow that there exists a student who is surrounding the building. lo tadni cu sruri lo dinju does not entail su’o da sruri lo dinju.

This failure on the part of mixed singularplural logic to comply with basic rules of predicate logic is not just a theoretical problem, but also a very real practical problem.

Say we want to state that people are gathering, that is, we want to make the existential statement that there are people X such that X are gathering. How do we do this with singular variables? Let’s try

su’o prenu cu jmaji
[∃x : prenu(x)] jmaji(x)
“Some people are gathering.” “There exists a person that is gathering.”

This doesn’t cut it, again because of the singular variable. No singular person is gathering. How, then, do we express it?

The simple truth is that we can’t. Not in singular logic. We need plural quantification for this sort of thing. But plural quantification is not part of Lojban, outside of experimental cmavo! That’s right: A seemingly basic statement like this is not expressable in Lojban without the use of cmavo which were proposed much later and which are still considered experimental (read: weird, situational, hypothetical, not part of the core).

Another example that exposes this giant gap in the language: You cannot say that nobody is gathering. No matter what you try, you cannot do it. The obvious attempt

no prenu cu jmaji
“No person is gathering.”

fails horribly. Since no = na ku su’o, the sentence expands to

na ku su’o prenu cu jmaji
¬[∃x : prenu(x)] jmaji(x)
“It’s false that there exists a person that is gathering.”

What is actually being negated here is the claim that some singular person is gathering, nothing more. It doesn’t say anything about multiple people gathering or not gathering! Again there is no way to fix it. Don’t think that this only concerns “gathering”; this problem applies to absolutely any predicate that can possibly be satisfied non- distributively.

I’ve shown two examples of things we plain cannot express in Lojban, because when xorlo happened we got plural constants but no plural variables.

Now that we know that plural variables are absolutely necessary, let’s get into the workings of plural logic.

Plural Logic

Plural logic, unlike singular logic, allows its variables to take on plural values. This means that not only single individuals like Susan, my house, or the number π can be substituted for a variable, but also both Susan and David, the yellow houses in my street, or all irrational numbers.

Because singular quantification only iterates over single individuals, everything is automatically distributive. In plural quantification the situation is a bit different, and we actually get two different kinds of predicate satisfaction, because while it can distribute over single individuals, it doesn’t have to. More on these two types later. Let’s first cover the basics.

The two basic quantifiers ∃ and ∀

Just as with singular logic, two quantifiers form the basis of the logic:

  • the existential quantifier ∃ (expressed by the experimental su’oi) and
  • the universal quantifier ∀ (expressed by the experimental ro’oi).

(su’oi and ro’oi were proposed by xorxes)

These quantifiers are plural quantifiers, meaning that the quantified statement can be satisfied by more than one thing at a time. Variables preceded by these quantifiers are plural variables, and I will write them as doubled letters in this post (e.g. xx, yy, zz).

Plural quantifiers can achieve things that singular quantifiers can’t. Let’s revisit an earlier example involving the verb “to gather”:

“Some people are gathering at the well.”
[∃xx : prenu(xx)] jmaji(xx, lo jinto)

440px-rebecca_murillo

This statement would not work with a singular existential quantifier, as no single person can be gathering at the well (by now this should sound familiar). With a plural quantifier, however, there is no problem because the variable can have multiple people as a value at once, and those can gather.

While a singular existential quantifier reads something like

“There is some x such that it (x) …”,

a plural existential quantifier reads like

There are some xx such that they (xx) …”.

 Here’s an example:

su’oi jbopre cu jmaji la .nuĭòk.
[∃xx : jbopre(xx)] jmaji(xx, la .nuĭòk.)
“Some Lojbanists are gathering in New York.”

While the singular universal quantifier reads something like

“Any x is such that it (x) …”

a plural existential quantifier reads like

“Any [number of] xx are such that they (xx) …”.

For example:

ro’oi jbopre poi simxu lo ka tavla cu co’a simxu lo ka pendo
[∀xx : jbopre(xx) ^ simxu(xx, lo ka tavla)] simxu(xx, lo ka pendo)
“Any [number of] Lojbanists that talk to each other become friends.”

In plural logic, the universal quantifier ∀ is still the dual of the existential quantifier ∃, just like in singular logic, which means that

∀ ≡ ¬∃¬

∃ ≡ ¬¬

The following two sentences are equivalent in meaning:

ro’oi jbopre cu remna
[∀xx : jbopre(xx)] remna(xx)
“Any [number of] Lojbanists are humans.”

na ku su’oi jbopre na ku remna
¬[xx : jbopre(xx)] ¬remna(xx)
“It’s not the case that there exist some Lojbanists such that they are not humans.”

As are these two sentences:

su’oi toldi cu xunre
[xx : toldi(xx)] xunre(xx)
“There exist some butterflies which are red.”

na ku ro’oi toldi na ku xunre
¬[xx : toldi(xx)] ¬xunre(xx)
“It’s not the case that any number of butterflies are not red.”

Numerical Statements

A numerical statement is one that contains a bare numerical quantifier, like 3 Lojbanists”, or 5 cows”.

Let’s first revisit the singular example

ci da nenri lo tanxe
[3x] nenri(x, lo tanxe)
“Three things are inside the box.”

We said above that this claims that if we substitute everything in the domain for da (x), we get a true bridi exactly three times. In other words, the sentence really means “There are exactly three things inside the box”. Not just three, no. Exactly three. This means that

re da nenri lo tanxe
[2x] nenri(x, lo tanxe)
“Two things are inside the box.”

is automatically false.

This phenomenon is described by saying that Lojban’s quantifiers are exact. When one number is true, all other numbers are false.

This rule makes it easy to be precise about specific cases in very few words. An equivalent of “exactly” or “only” is nowhere to be found in the Lojban sentence; this extra bit of meaning comes for free whenever you use a (numerical) quantifier.

Imagine what it would be like if saying mi zvati lo zarci automatically meant that I’m the only one at the market. I would never be able to claim that I’m at the market with others. In other words, the sentence would claim more than desired, and, in a way, more than I actually said. This kind of thing goes against Lojban’s philosophy of making shorter sentences less specific than longer ones and requiring more words for more details.

This general principle is known as facultative precision.

Exact quantifiers break this principle.

To summarize, then, there are two practical problems with singular numerical quantifiers:

  1. They are automatically distributive (like any singular quantifier).
  2. They are automatically exact.

This comes pre-baked into the quantifiers. The problem with pre-baked stuff is that you can’t get rid of it. You can’t substract the distributivity nor the exactness, no matter how many words you add.

Fortunately, plural logic offers a very simple way to solve both problems at once. Instead of doing what the previous example did, we could do this:

ci da nenri lo tanxe
[xx : 3mei(xx)] nenri(xx, lo tanxe)
“Three things are inside the box.” (lit. “There exist some things that are three in number which are inside the box.”)

What’s the difference? The numerical quantifier is combined with a plural existential, which has two important effects:

  1. The “things” can satisfy the predicate collectively/non-distributively.
  2. It does not exclude the possibility of other things (or other broda in the case of PA broda) inside the box.

In other words, both of the pre-baked bits of singular quantifier semantics were replaced by more general, more flexible, plural semantics.

With this in place, the law of facultative precision is respected fully, because now when we wish to add certain details, like, for instance, the fact that there are no other things in the box, or that the things we’re talking about are satisfying the predicate distributively, we can indicate that by adding more words. More words for more details and fewer words for fewer details.

The other thing we’ve (happily) given up is automatic distributivity. We will see in the next section how to manipulate distributivity according to our needs.

But there is another perk: By treating numerical statements as proposed above we also automatically simplify scope.

Yes, scope becomes simpler! In official Lojban, a sentence of the form PA broda cu brodi PA brode is frequently not an appropriate way to describe a situation, because it means “There exist exactly PA broda which are each such that there exist exactly PA brode that each brodi. We also cannot swap PA broda with PA brode and retain the original meaning.

ci gerku cu batci re remna
[3x : gerku(x) 2y : remna(y)] batci(x,y)
“There exist exactly three dogs that are such that there exist exactly two humans that they bite.”

has a substantially different meaning from

re remna cu se batci ci gerku
[2x : remna(y) 3x : gerku(x)] batci(x,y)
“There exist exactly two humans that are such that there exist exactly three dogs that bite them.”

With plural numerical quantifiers, however,

ci gerku cu batci re remna
[∃xx : 3mei(xx) ∧ gerku(xx) [∃yy : 2mei(yy) ∧ remna(yy)]] batci(xx,yy)
“There exist some dogs that are three in number for which there exist some humans that are two in number such that the dogs bite the humans.”

and

re remna cu se batci ci gerku
[∃yy : 2mei(yy) ∧ remna(yy) [∃xx : 3mei(xx) ∧ remna(xx)]] batci(xx,yy)
 “There exist some humans that are two in number for which there exist some dogs that are three in number such that the dogs bite the humans.”

are equivalent, because we are dealing with two existential quantifiers, and the order of quantifiers in a statement does not matter when they are all of the same type (all universal or all existential). (But the order does matter when the quantifiers are of two different types!)

With plural numerical quantifiers these sentences become less committal, and the order of terms is no longer important, making it much easier to use for the common general cases (while reserving the more complex mechanisms for more complex situations and meanings).

We can say pa ractu cu zvati lo kumfa without having to worry that we’re overlooking a rabbit — “A rabbit is in the room.”

So essentially, pa goes from meaning “there exists exactly one thing which …” to just “a”, re goes from“there exist exactly two things each of which …” to just “two”, and so on. A simpler quantifier logic indeed.

Quantifier exactness — a short detour

You might be wondering: If we give up singular logic‘s quantifier exactness, how can we get it back when we need it? This is what we will look at in this little box.

First of all, what does quantifier exactness even mean in the context of plural quantification? There are actually several ways a sentence like “Exactly five people are gathering” can be read:

  1. There are some people who are gathering and who are exactly five in number. There may be other people gathering indepently from our five, but the ones we’re talking about are absolutely certainly exactly five in number.
  2. There are some people who are gathering and who are exactly five in number, and there are no people who are more than five in number who are gathering.
  3. There are some people who are gathering and who are exactly five in number, and there are no people who are more than five who are gathering and no people who are less than five who are gathering.

Reading 1) is the default of the plural logic outlined in this article, and I believe it is a good default. There is nothing inexact about this. “Taking away quantifier exactness” does not mean making quantifiers less exact, in that sense. It only removes the unwanted exclusion of other claims. The quantifier (“five”, in this case) still means “five” and not “approximately five”.

As for reading 2), this is another natural way to interpret the sentence. It corresponds to ([5xx : prenu(xx)] jmaji(xx)) ∧ (¬[>5xx : ∧ prenu(xx)] jmaji(xx)), which is not the default of our plural logic, but which is easily expressed in plural Lojban as mu je nai za’u mu prenu cu jmaji (this expands to ge mu prenu cu jmaji gi na ku za’u mu prenu cu jmaji).

Reading 3) is not very natural. It also happens to not be compatible with distributive predicates. When “exactly three people smile”, we do not mean to say that “there exist no people that are two in number that smile”. Reading 3) will therefore be discarded. (but you can express it analogously to reading 2, if you want to)

We have now covered the possible readings of the sentence in question and know how to handle each one in Lojban. There are, however, still other ways to deal with exactness, for example those that involve po’o:

ci da po’o nenri lo tanxe
“There exist some three things such that they are the only things inside the box.”
“There are only three things in the box.”

This is a simple way to exclude the possibility of anything else being in the box. (you can also move po’o right after the quantifier to achieve a different, unusual effect. I’ll leave it as an exercise to the reader to figure out what that effect is)

We can see, then, that plural logic has no difficulty expressing any of this. This means that in effect we do not lose anything — we let it go, because we do not need it.

Let us now move on to a treatment of—

Distributivity and non-distributivity

What is distributivity? When is a predicate (or an argument place of a predicate) distributive and when is it not?

A predicate is distributive if, whenever some things satisfy the predicate, then each one of them satisfies the predicate. For example, whenever some things are cats, then each one of them is a cat. Therefore, “to be a cat” is a distributive predicate, or distributive in its first argument place.

lo pendo pu cisma
[lo xx : pendo(xx)] cisma(xx)
“The friends smiled.”

is distributive, and therefore has the same meaning as

ro’oi da poi pa mei gi’e menre lo pendo pu cisma
[∀xx : 1mei(xx) [lo yy : pendo(yy)] xx  yy] cisma(xx)
“Any number of things that are one in number and among the friends smiled.”
“Each one of the friends smiled.”

If some things satisfy a predicate but not each one of them satisfies the predicate, the predicate is called nondistributive. For example, when some things (or people) are gathering at a table, not each one of them is gathering at the table. Therefore, “to gather” is a non-distributive predicate, or non-distributive in its first argument place.

lo since ca’o sruri mi
[lo xx : since(xx)] sruri(xx, mi)
“The snakes are surrounding me.”

is not distributive, and therefore has a different meaning from

ro’oi da poi pa mei gi’e menre lo since ca’o sruri mi
[xx : 1mei(xx) ∧ [lo yy : since(yy)] xx yy] caho.sruri(xx, mi)
“Each one of the snakes is surrounding me.”

This describes a different situation. In many cases, however, a distributive reading of a non-distributive predicate is simply nonsensical:

lo .inglico tcadu cu so’i mei
[lo xx : inglico.tcadu(xx)] sohi.mei(xx)
“The English cities are numerous.”

makes no sense when predicated distributively: “Each one of the English cities is numerous” is self-contradictory.

We see, then, that a predicate can be satisfied either distributively or non-distributively (also described as collectively).

su’oi tadni poi ci mei cu sruri lo dinju gi’e dasni lo mapku
[∃xx : tadni(xx) ∧ 3mei(xx)] sruri(xx, lo dinju) ∧ dasni(xx, lo mapku)
“Some students that are three in number surround the building and wear hats.”

Here, we can observe how a single plural term satisfies different predicates in the same sentence either distributively or nondistributively (or collectively). cimei is satisfied collectively, tadni is satisfied distributively, sruri is again satisfied collectively, and finally dasni is satisfied distributively. How an argument place is to be satisfied is part of the meaning (i.e., definition) of a predicate.

A predicate can also be distributive in its first argument place and non-distributive in its second (and vice versa). One such predicate is me (or menre). The following image is taken from guskant’s gadri: an unofficial commentary from a logical point of view:

688px-display7-svg

(Make sure to read her article if you haven’t already. It is one of not many logical treatments of Lojban our there.)

The definition of me (menre) or “among” (symbolized here by “≤”) is:

                           ko’a me ko’e ≡ ro’oi da zo’u ga nai da me ko’a gi da me ko’e

A ≤ B [∀xx] (xx A → xx ≤ B)

This relation turns out to be one of the two primitives of our logic (the other being ∃). It is used to define other important relations such as sameness (symbolized by “≈”; A ≈ B iff A B ∧ B A), but is also central in defining outer quantifiers in Lojban:

PA PA da poi ke’a me

We can (and have to) use me in combination with the condition “x1 is an individual” to force distributivity on a sumti when

  • a predicate is nondistributive but a distributive reading is desired
  • we want to emphasize individual satisfaction (e.g. in contrastive contexts)

lo rokci cu ki’ogra li 20
[lo xx : rokci(xx)] kihogra(xx, 20)
“The rocks weigh 20kg.”

The rocks weigh 20kg together (let’s assume that ki’ogra is non-distributive). How do we say that each single rock weighs 20kg?

ro’oi da poi pavmei je me lo rokci cu ki’ogra li 20
[∀xx [lo yy : rokci(yy)] pavmei(xx) ∧ xx  yy] kihogra(xx, 20)
“Any number of things that are an individual and among the rocks weigh 20kg.”
“Any single one of the rocks weighs 20kg.”

In general, the solution involves a predicate that expresses individualness. One such predicate is pavmei.

There are two places, syntactically speaking, where the individualness restriction can be inserted into a sentence, depending on which part is supposed to be marked as distributive:

  • In the prenex, as part of a restrictive relative clause: “Some xx, each of which …, broda”
  • As part of the main claim, often with poi’i: “Some xx are such that each one among them brodas”

For example:

lo rokci cu poi’i ro’oi pavmei je me ke’a cu ki’ogra li 20
[lo xx : rokci(xx) [∀yy : pavmei(yy) ∧  yy  xx]] kihogra(yy, 20)
“The rocks are such that each one among them weighs 20kg.”

or

ci jbopre poi ro’oi pavmei je me ke’a cu xagji cu casnu zo lo
[3xx : jbopre(xx)   [∀yy : pavmei(yy) ∧  yy  xx] xagji(yy)] casnu(xx, ‘lo’)
“Three Lojbanists each on of which is hungry discuss the word lo.”

ro’oi pavmei je me ke’a is short for ro’oi da poi ke’a pavmei gi’e menre ke’a xi re, but an even shorter option is available: ro’oi pa ke’a.

What all of this shows is that singular predication is just a special case of plural predication. With a full plural logic in place, a separate singular logic becomes unnecessary.

Note that the same pattern we used in order to force distributivity in the previous rokci example can also be used to perform singularization from within a plural, to express sentences like this one:

su’oi ratni va poi’i lo gunma be ke’a cu mlatu
[∃xx : ratni(xx) [lo yy : gunma(yy, xx)]] va.mlatu(yy)
“Some atoms over there are such that the mass composed of them is a cat.”
“There are some atoms that make up a cat together over there.”

Plural logic is flexible in all directions.

One last note about predicates not being defined clearly as distributive/non-distributive in xorlo; xorxes wanted lo to be absolutely non-committal with regards to distributivity, therefore, in his model, all predicates are left vague with regards to distributivity. In my proposed full plural logic, predicates are defined as distributive or non-distributive, so it is usually unnecessary to force distributivity via explicit universal quantification.

Any vs All

English sometimes uses “any” and “all” interchangeably, or alternates between them in a seemingly random fashion, thereby obscuring the fact that they needn’t be synonymous, even in those cases where “any” does not mean “some”.

As we will discover now, we are going to need two different kinds of universal quantifiers, one of which we have already met above in the form of ∀ (ro’oi). This quantifier works for a lot of things without problems, like for instance the following sentence:

ro’oi jbopre cu poi’i ga nai do retsku fi ke’a gi ke’a cusku mo’e lo se memkai be ke’a be’o su’i pa danfu
Any number of Lojbanists is such that if you ask them a question, you’ll get that many answers plus 1.”

And what about the following sentence?

ro’oi jbopre cu fricysi’u lo ka ma kau smuni zo le ce’u
[∀xx : jbopre(xx)] fricysihu(xx)
Any number of Lojbanists differ on what le means to them.”

This almost works, but not quite, because the claim includes individual Lojbanists, and no single thing can satisfy simxu. A small addition can fix a sentence like this:

ro’oi jbopre je re mei cu fricysi’u lo ka ma kau smuni zo le ce’u
[∀xx : jbopre(xx) ^ 2mei(xx)] fricysi’u(xx)
“Any two Lojbanists differ on what le means to them.”

(ro’oi broda je PA mei could be abbreviated ro’oi PA broda)

There are, however, sentences that require a different quantifier altogether, despite using the word all”:

“All [of the] people who can speak Lojban are five in number.”

“All [of] my friends are meeting together to plan a surprise.”

“All [of the] students are surrounding the building.”

Using ro’oi here would not work. Not any number of Lojban speakers are five in number (some are one, some are two, etc.). Not any number of my friends are meeting together. Not any number of students are surrounding the building.

Therefore, a different quantifier is needed.

This quantifier needs to select all the things together. Let Λ be this quantifier, then we can define it as follows:

[Λxx] F(xx)   ≡   [∃xx ∧ [∀yy] yy  xx ] F(xx)

[Λxx : F(xx)] G(xx)   ≡   [∃xx : F(xx) ∧ [∀yy : F(yy)] yy ≤ xx ] G(xx)

The first reads: “‘All xx F‘ is equivalent to ‘There exist some things xx for which any [number of] yy are among xx, such that xx F.’“.

The second reads: “‘All [of the] F are G’ is equivalent to ‘There exist some xx that F for which any [number of] yy that F are among xx, such that xx G'”.

This new quantifier would of course need to be spelled differently from ro.

I propose ru’o, which is a sound mix between ro and mu[ln]o. Let’s see ru’o (Λ) in action.

ru’o banka’e be lo jbobau cu mu mei
[Λxx : bankahe(xx, lo jbobau)] 5mei(xx)
“All [of the] people who can speak Lojban are five in number.”

ru’o pendo be mi cu pensi’u
[Λxx : pendo(xx, mi)] pensihu(xx)
“All [of] my friends are meeting together.”

ru’o tadni cu sruri lo dinju
[Λxx : tadni(xx)] sruri(xx, lo dinju)
“All [of the] students are surrounding the building.”

Let us also define another useful shorthand:

ru’o PA da brodasu’oi da poi ge ro’oi de zo’u de menre ke’a gi ke’a PA mei cu broda

Then we can use ru’o PA like CLL ro PA:

ru’o mu stizu ca zvati lo purdi
[Λ5xx : stizu(xx)] zvati(xx, lo purdi)
“All five chairs are now in the garden.”

ru’o ze torcrida cu sruri lo jubme
[Λ7xx : torcrida(xx)] sruri(xx, lo jubme)
“All seven dwarfs surround the table.”

We should now have covered everything essential.

It’s time for a —

Summary

By making all quantifiers plural by default, the basic inferences

[x] P(x) P(c)

and

P(c) → [∃x] P(x)

are made valid again.

Sentences that couldn’t be expressed before, like “Some people are gathering” and “Nobody is gathering” can finally be expressed (and quite easily, too).

The principle of facultative precision, or zipf’s law of least effort, spreads to quantifiers so that more basic or more general claims require fewer words than those that are very specific.

Many kinds of sentences that were intuitive but wrong in singular Lojban become correct, thus making quantifiers more intuitive on top of making them more powerful.

Therefore, I propose that all quantifiers become plural quantifiers and that da become a plural variable. Instead of having experimental su’oi and ro’oi, make them part of the core by changing their spelling to su’o and ro respectively. Finally, add one new quantifier for “all”: ru’o.

In order to get an idea of what a proposal like the above would result in, it’s always good to be offered —

A Fair Comparison — Singular versus Plural

Below you will find a table with a systematic comparison of how pure singular logic, xorlo and full plural logic handle all the things I have talked about. Where a system is incapable of producing a satisfying translation (due to conflicts with its own philosophy) the tag NOT POSSIBLE will indicate this.

Let brodi be any distributive predicate and brodo any collective/non-distributive predicate, then the equivalences in the following table hold. Keep in mind that

  • in the singular column, it is assumed that all non-distributive predicates (brodo) can only (meaningfully) be satisfied by sets. Also, due to its reliance on sets singular logic makes ontologically different claims than the pure plural version, but I will not tag it with NOT POSSIBLE because it is nevertheless a self-consistent system, unlike the singular variables + plural constants system
  • in the singular variables + plural constants column, all quantifiers are distributive, but predicates can be satisfied both distributively and non-distributively by plural constants.
  • in the plural column, all quantifiers are plural quantifiers, which means that su’o corresponds to su’oi, PA to su’oi PA, and ro to ro’oi). Remember that a subscript D marks a place as being satisfied distributively.
Singular Logic
(CLL)
Singular Variables
+ Plural Constants
(xorlo)

Full
Plural Logic
(Proposed)
su’o da brodi
[∃x] brodi(x)
“There exists something which brodi.”
su’o da brodi
[∃x] brodi(x)
“There exists something which brodi.”
su’o da brodi
[∃xx] brodi(xxD)
“There exists something which brodi.”
(same in all systems)
su’o da brodo
[∃x] brodo(x)
“There exists a set which brodo.”
(involves a set)
NOT POSSIBLE
Closest Approximation:
su’o da zo’u lo cmima be da cu brodo
[∃x [lo y : y ∈ x]] brodo(y)
“There exists something whose members collectively brodo.”
(has to rely on lo to get plural reference)
su’o da brodo
[∃xx] brodo(xx)
There exist some things that collectively brodo.”
(straightforward)
su’o da brodo gi’e ckaji lo ka ro cmima be ce’u cu brodi
[∃x] brodo(x) ^ [∀y : y ∈ x] brodi(y)
“There exists a set which brodo and is which is such that each of its members brodi.”
(clumsy)
NOT POSSIBLE
Closest Approximation:
su’o da zo’u lo cmima be da cu brodo gi’e ckaji lo ka ro menre be ce’u cu brodi
[∃x [lo y : y ∈ x]] brodo(y) ^ [∀z : z  y] brodi(z)
“There exists something whose members collectively brodo and who are such that each one among them brodi.”
(just absolutely painful)
su’o da brodo gi’e brodi
[∃xx] brodo(xx) ^ brodi(xxD)
“There exist some things that collectively brodo and distributively brodi.”
(straightforward)
ro da brodi
[∀x] brodi(x)
“Anything brodi.”
ro da brodi
[∀x] brodi(x)
“Anything brodi.”
ro da brodi
[∀xx] brodi(xxD)
“Anything brodi.”
(same in all systems)
ro da brodo
[∀x] brodo(x)
“Any set brodo.”
(involves a set)
NOT POSSIBLE
Closest Approximation:
ro da zo’u lo cmima be da cu brodo
[∀x [lo y : y ∈ x]] brodo(y)
“Any set is such that its members collectively brodo.”
(has to rely on lo to get plural reference)
ro da brodo
[∀xx] brodo(xx)
“Any number of things brodo.”
(straightforward)
ro da poi PA de cmima ke’a cu brodo
[∀x : [PAy : y ∈ x]] brodo(x)
“Any set with PA members brodo.”
NOT POSSIBLE
Closest Approximation:
ro da poi PA de cmima ke’a zo’u lo cmima be da cu brodo
[∀x [lo y : y ∈ x]] brodo(y)
“Any set that has PA members is such that its members collectively brodo.”
(has to rely on lo to get plural reference. painful.)
ro da poi ke’a PA mei cu brodo
[∀xx : PAmei(xx)] brodo(xx)
“Any PA things brodo.”
(can be shortened to ro PA da brodo)
ro da brodo gi’e ckaji lo ka ro cmima be ce’u cu brodi
[∀x] brodo(x) ∧ [∀y : y ∈ x] brodi(y)
“Any set brodo and is such that each of its members brodi.”
(clumsy)
NOT POSSIBLE
Closest Approximation:
ro da zo’u lo cmima be da cu brodo gi’e ckaji lo ka ro menre be ce’u cu brodi
[∀x [lo y : y ∈ x]] brodo(y) ∧ [∀z : z  y] brodi(z)
“Anything is such that its members collectively brodo and are such that each one among them brodi.”
(more pain)
ro da brodo gi’e brodi
[∀xx] brodo(xx) ∧ brodi(xxD)
“Any number of things collectively brodo and distributively brodi.”
(straightforward)
lo se cmima be ro da cu brodo
[lo x : ∀y cmima(y,x)] brodo(x)
“The set of all things brodo.”
NOT POSSIBLE
Closest Approximation:
su’o da poi ro de cmima ke’a zo’u lo cmima be da cu brodo
[∃x : [∀y : y x] [lo z : z∈ x]] brodo(z)
“There exists something which has everything as its members such that its members collectively brodo.”
(has to rely on lo to get plural reference. a bit roundabout)
ru’o da brodo
[Λxx] brodo(xx)
“All things together brodo.”
(new quantifier)
su’o da zo’u ro cmima be da cu brodo
[∃x [∀y : y ∈ x]] brodo(y)

“There exists a set such that each of its members brodo.”
NOT POSSIBLE
Closest Approximation:
su’o da zo’u ro cmima be da cu brodo
[∃x [∀y : y ∈ x]] brodo(y)
“There exists something such that each of its members brodo.”
(stuck in singular logic)
su’o da zo’u ro pavmei be ke’a cu brodo
[∃xx] [∀yy : yy  xx]] brodo(yy)
“There exist some things such that each individual among them brodo.”
(clean. can be shortened to ro PA ke’a)
PA da brodi
[∃!PAx] brodi(x)
“Exactly PA things each brodi.”
PA da brodi
[∃!PAx] brodi(x)
“Exactly PA things each brodi.”
PA jenai za’u PA da brodi
[PAxx] brodi(xx) ∧ ¬[>PAyy] brodi(yy)
“Some PA things each brodi and no more than PA things each brodi.”
(see the detour about exactness for why the singular sense of exactness is a bit nonsensical in plural logic)
su’o da poi PA de cmima ke’a zo’u ro cmima be da cu brodi
[∃x [∃!PAy : y ∈ x][∀z : z ∈ x]] brodi(z)

“There exists a set with PA members such that each of its members brodi.”
(going through pains to circumvent exactness)
su’o da poi PA de cmima ke’a zo’u ro cmima be da cu brodi
[∃x [∃!PAy : y ∈ x][∀z : z ∈ x]] brodi(z)
“There exists a set with PA members such that each of its members brodi.”
(going through pains to circumvent exactness)
PA da brodi
[PAxx] brodo(xxD)
“Some PA things distributively brodi.”
su’o da poi PA de cmima ke’a cu brodo
[∃x : [∃!PAy : y ∈ x]] brodo(x)
“There exists a set with PA members such that it brodo.”
NOT POSSIBLE
Closest Approximation:
su’o da poi PA de cmima ke’a zo’u lo cmima be da cu brodo
[∃x : [∃!PAy : y ∈ x] [lo y : y ∈ x]] brodo(y)
“There exists a set with PA members such that its members collectively brodo.”
(has to rely on lo to get plural reference. very verbose)
PA da brodo
[PAxx] brodo(xx)
“Some PA things collectively brodo.”
(clean)

That’s it for this article. If you would like to read more of this sort of thing, there are another two articles in the Simpler series that you might enjoy: A Simpler Connective System and A Simpler Morphology [coming soon].

If you have comments or questions leave them below.

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14 thoughts on “A Simpler Quantifier Logic

  1. At some point (the Any vs All section) I think you switched from “please read {ro’oi} and {su’oi} as the plural quantifiers” to “just use {ro} and {su’o} as plural in the examples” but you didn’t state that until way later in the summary. Or maybe I missed it because I read rather fast.

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      1. That’s what I’m saying, there is no “where exactly” 🙂

        The last example using {ro’oi} is {ro’oi da poi pavmei je me lo rokci cu ki’ogra li 20} and it says that {ro’oi} means ∀xx [lo yy : rokci(yy)]. The next example is {lo rokci cu poi’i ro pavmei je me ke’a cu ki’ogra li 20} and it says that {ro} means ∀yy : pavmei(yy). Those seem the same to me, unless I missed something, and there’s no intervening text saying “now consider that we replace {ro} with {ro’oi}”.

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  2. Great article! This is a very valuable addition to the Lojban documentation.

    I would emphasise that it is the quantification that is singular, rather than the variables.

    lu ro pa ke’a li’u iu

    I don’t get [Ex xx [A yy] yy <= xx]] F(xx)
    You never talk about yy except saying <= xx. Isn’t this always trivially satisfied?

    Nits:
    ci gerku, but 2x gerku(x)

    pendo(xx) cisma(xx) in the each of the snakes example

    su’o ratni … but the formula is about the rocks and the kilograms

    straighforward -> straightforward

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      1. [Λxx] Fxx ≡ [∃xx [∀yy] yy ≤ xx ]] F(xx)

        I don’t get the quantification over yy here. You don’t seem to say anything in particular about the yy.

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  3. banli .i’o karnysle

    However it seems you forgot to tell how to express the plural equivalent of the singular {PA da broda} (“The number of things that broda is exactly PA”) in the proposed system. I think this should at least be part of the “A Fair Comparison” table, and preferably be mentioned in the “Numerical Statements” section as well.

    mi’e la .ilmen. mu’o

    Like

    1. I had a few paragraphs about exactness in plural logic in the article but removed them shortly before publication.

      Thank you for taking the time to compose a comment about this instead of complaining elsewhere. 🙂

      The thing is that exactness makes a lot less sense in plural logic than it does in singular logic. I could re-add a short digression into the article.

      Like

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