# Lojban Wave Lessons/More on connectives and quantifiers

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## Lesson 25: Logical connectives

If you ask a Lojbanist: Do you want milk or sugar in your coffee? she'll answer: Correct.

Witty as this joke might be, it illustrates a weird property of the English way of asking this question. It is phrased as a true/false question, but it really isn't. In Lojban, we can't have this kind of inconsistency, and so we must find another way of asking this kind of question. If you think about it, it's pretty hard to find a good and easy way, and it seems Lojban have picked a good way instead of an easy way.

To explain it, let us take two separate bridi: Bridi 1: I like milk in my coffee and bridi 2: I like sugar in my coffee. Both of these bridi can have the state true or false. This yields four combinations of which bridi is/are true:

A ) 1 and 2 B ) 1 but not 2

C ) 2 but not 1 D )neither 1 nor 2

I, in actuality, like milk in my coffee, and I'm indifferent as to whether there is sugar in it or not. Therefore, my preference can be written A ) true B ) true C ) false D ) false, since both A and B yields true for me, but neither C nor D does. A more compact way of writing my coffee preferences would be TTFF for true, true, false, false. Similarly, a person liking his coffee black and unsweetened would have a coffee preference of FFFT. This combitation of "true" and "false" is called a truth function, in this case for the two statements I like milk in my coffee and I like sugar in my coffee. Note that the order of the statements matters.

In Lojban, we operate with 4 truth functions, which we consider fundamental:

A: TTTF (and/or)

O: TFFT (if and only iff)

U: TTFF (whether or not)

E: TFFF (and)

In this example, they would translate to something like: A:Just not black coffee, O: Either both milk or sugar, or nothing for me, please, U: Milk, and I don't care about if there's sugar or not and E: Milk and sugar, please..

In Lojban, you place the word for the truth function between the two bridi, selbri or sumti in question. That word is called a logical connective. The words for truth functions between sumti (and just for sumti!) are .a .o .u and .e. How nice. For instance: I am friends with an American and a German would be lo merko .e lo dotco cu pendo mi.

How would you say: I speak to you and no one else?

Answer: mi tavla do .e no drata Note that this actually states that it's true that "I speak to you".

One more: I like cheese whether or not I like coffee

ckafi = x1 is a quantity/contains coffee from source/of grain x2

Answer: mi nelci lo'e cirla .u lo'e ckafi

You can perhaps deduce that there are sixteen possible truth functions, so we need to learn twelve more in order to know them all. Eight more can be obtained by negating either the first sentence or the second. The first is negated by prefixing the truth function word with na, the second is negated by placing nai after the word. For example, since .e represents TFFF, .e nai must be both 1 is true and 2 is false, which is FTFF. Similarly, na .a is Just not: 1 is true and 2 is false, which is TTFT. Doing this type of conversion in the head real-time is very, very hard, so perhaps one should focus on learning how logical connectives work in general, and then learn the logical connectives themselves by heart.

Four functions cannot be made this way: TTTT, TFTF, FTFT and FFFF. The first and the last cannot be made using logical connectives at all, but they are kind of useless anyway. Using a hypothetical logical connective in the sentence I like milk FFFF sugar in my coffee is equivalent to saying I don't like coffee, just more complicated. The last two, TFTF and FTFT, can be made by prefixing .u with good ol' se, which just reverts the two statements. se .u , for instance is B whether or not A, which is TFTF. The final list of all logical connectives can be seen below.

TTTF: .a

TTFT: .a nai

TTFF: .u OR .u nai

TFTT: na .a

TFTF: se .u

TFFT: .o OR na .o nai

TFFF: .e

FTTT: na .a nai

FTTF: na .o OR .o nai

FTFT: se .u nai

FTFF: .e nai

FFTT: na .u OR na .u nai

FFTF: na .e

FFFT: na .e nai

Logically, saying a sentence with a logical connective, like for instance mi nelci lo'e cirla .e nai lo'e ckafi is equivalent to saying two bridi, which are connected with the same logical connective: mi nelci lo'e cirla .i {E NAI} mi nelci lo'e ckafi. This is how the function of logical connectives is defined. We will get to how to apply logical connectives to bridi in a moment.

By putting a j in front of the core word of a logical connective, it connects two selbri. An example is mi ninmu na jo nanmu I am a man or a woman, but not both

ninmu = x1 is a woman

This is tanru-internal, meaning that it loosely binds selbri together, even when they form a tanru: lo dotco ja merko prenu means a German or American man, and is parsed lo (dotco ja merko) prenu. This binding is slightly stronger that normal tanru-grouping (still weaker than specific grouping-words), and as such, lo dotco ja merko ninmu ja nanmu is parsed lo (dotco ja merko) (ninmu ja nanmu). The selbri logical connectives can also be attached to .i in order to connect two sentences together: la .kim. cmene mi .i ju mi nanmu I'm called Kim, whether or not I'm a man. The combination .i je states that both sentences are true, much like we would assume had no logical connective been present.

Unfairly hard question: Using logical connectives, how would you translate If you're called Bob, you're a man.?

Answer: zo .bab. cmene do .i na ja do nanmu or Either you're not named Bob and a man, or you're not named Bob and not a man, or you're named Bob and a man. But you can't be named Bob and not be a man. The only combination not permitted is: You're called Bob, but not a man. This must mean that, if it's true that you're called Bob, you must be a man.

If we try to translate the sad, sad event of I cried and gave away my dog, we run into a problem.

Attempting to say the sentence with a je between the selbri gave and cried, would mean the same word for word, but unfortunately mean that I cried the dog and gave away the dog, cf. the definition of logical connectives. One can cry tears or even blood, but crying dogs is just silly.

However, we can get around by using bridi-tail logical connectives. What they do is that any previous sumtcita and sumti attaches to both selbri bound by the bridi-tail logical connective, but any following sumti or sumtcita only applies to the last mentioned: The bridi splits up from one head to two tails, to speak metaphorically.

The form of a bridi-tail logical connective is gi'V, with the V for the vowel of the truth function.

How could you correctly translate the English sentence to Lojban?

Answer: mi pu klaku gi'e dunda le mi gerku

What does ro remna cu palci gi'o zukte lo palci mean?

palci = x1 is evil by standard x2

Answer: People are evil if and only if they do evil things.

Furthermore, there is a forethought all-but tanru internal group of connecters made by prefixing g in front of the truth function vowel. Forethought in this context means that they need to go in front of the things they connect, and thus you need to think about the grammatical structure of the sentence before saying it. All-but tanru internal means that it serves both as a connective between sumti, bridi, selbri and bridi-tails, but not between two selbri of one tanru. Let me show you how it works, rewriting the Lojban sentence above:

go lo remna cu palci gi lo remna cu zukte lo palci

The first logical connective in these kinds of constructs are what carries the vowel which signal what truth function is being used. The second logical connective is always gi, and like .i, it has no truth function. It simply serves to separate the two terms being connected. If you want to negate the first or second sentence, a nai is suffixed to either the first (for the first sentence) or second (for the second sentence) logical connective.

Provided that the constructs are terminated properly, it has remarkable flexibility, as the following few examples demonstrate:

mi go klama gi cadzu vau le mi zdani I go, if and only if walk, to my home or I can only go to my home by walking. Notice that the vau is needed to make le mi zdani apply to both cadzu and klama.

se gu do gi nai mi bajra le do ckule Whether or not you, then not I, run to your school or I won't run to your school no matter if you do or not

The tanru-internal equivalent of gV is gu'V. These are exactly the same, except that they are exclusively tanru-internal, and that they bind a selbri to the gi tighter than normal tanru-grouping, but weaker than explicit binding-sumti:

la xanz.krt. gu'e merko gi dotco nanmu is equivalent to

la xanz.krt. merko je dotco nanmu

And so you've read page up and page down just to get the necessary knowledge in order to be able to learn how to ask Would you like milk or sugar in your coffee? in Lojban. Simply place a question logical connective instead of another logical connective, and like ma, it asks the listener to fill in a correct response. Unfortunately, these question-logical connectives don't always match the morphological pattern of the logical connectives they ask for:

ji = Logical connective question: Asks for a sumti logical connective (A)
je'i = Logical connective question: Asks for a tanru-internal selbri logical connective (JA)
gi'i = Logical connective question: Asks for a bridi-tail logical connective (GIhA)
ge'i = Logical connective question: Asks for a forethought all-but tanru internal logical connective (GA)
gu'i = Logical connective question: Asks for a forethought only tanru internal logical connective (GUhA)

So... how would you ask if the persons wants milk or sugar in her coffee?

ladru = x1 is/contains milk from source x2
sakta = x1 is/contains sugar from source x2 of composition x3

Possible answer: sakta je'i ladru le do ckafi though I guess something more English and less elegant could also suffice like do djica lenu lo sakta ji lo ladru cu nenri le do ckafi

## Lesson 26: Non-logical connectives

The word "logical" in "logical connective" refers to the association a logical connective has with a truth function. Not all useful connectives can be defined through a truth function, however, and so there are other connectives beside the logical ones.

The meaning of a logical connective is defined the same as two different bridi connected with that logical connective. For instance, mi nitcu do .a la .djan. is defined to be equivalent to mi nitcu do .i ja mi nitcu la .djan.. This definition is useful to bear in mind, because it implies that sometimes, sumti cannot be connected with logical connectives without chaning the meaning. Consider the sentence: "Jack and Joe wrote this play." One attempt at a translation would be: ti draci fi la .djak. e la .djous.

draci = x1 is a drama/play about x2 by writer/dramatist x3 for audience x4 with actors x5

The problem with this translation is that it means ti draci la .djak. ije ti draci la .djous., which is not really true. Neither Jack nor Joe wrote it, they did so together. What we want here is of course a mass, and some way to join Jack and Joe in one mass. This has little to do with a truth function so we must use a non-logical connective, which are of selma'o JOI. We'll return to this Jack and Joe-problem in a little - first: Four of the known JOI:

The Lojban connective joins sumti and forms a:

The functions of these words are simple: lo'i remna jo'u lo'i gerku considers both the set of humans and the set of dogs distributively (as individuals). Remember from lesson twenty-two (quantifiers) that "distributivity" means that what is true for the group is also true for each of the individuals alone. Similarly loi ro gismu ce'o loi ro lujvo ce'o loi ro fu'ivla is a sequence consisting of the mass of all gismu, followed by the mass of all lujvo, followed by the mass of all fu'ivla. As with all of the JOI which has an inherent order, se may be put before ce'o to inverse the order: "A ce'o B" is the same as "B se ce'o A". How can you correctly translate "Jack and Joe wrote this play"? Answer: ti draci fi la .djak. joi la .djous. The cmavo of JOI are very flexible: They can act both as sumti connectives and tanru-internal connectives, so they can be used to connect sumti, selbri and bridi. This flexibility means that one must be careful to use famyma'o correctly when using a JOI. What is wrong with the bridi lo dotco jo'u mi cu klama la dotco gugde? Answer: jo'u is put after a selbri, so it expects a selbri after it to connect to, but none is found. Had a ku been present before the connective, it would have been grammatical If several JOI are used, bo and/or ke may be used to override the usual left-grouping:
 mi joi do ce'o la .djak. joi bo la .djous. cu pu'o ci'erkei damba lei xunreMe and you, and then Jack and Joe are about the play against the reds

Contrast it with

 mi joi do ce'o la .djak. joi la .djous. cu pu'o ci'erkei damba lei xunreFirst me and you, then Jack will together with Joe play against the reds.

Connecting bridi with JOI can make some interesting implications of the relationship between the bridi:

 la .djak. morsi ri'a lo nu ri dzusoi .i joi le jemja'a po ri cu bebnaJack is dead because he was a infantry soldier and his general was an idiot.

implying that these two bridi massed together was the physical cause of his death: Had he only been in an armored vehicle or with a competent commander, he might had survived.

dzusoi = x1 is an infantry soldier of army x2
jemja'a = x1 is a general of army x2 in function x3
bebna = x1 is foolish/idiotic in property/aspect x2

Non-logical connectives may also be negated with nai, indicating that some other connective is appropriate: lo djacu ce'o nai .e'o lo ladru cu cavyfle fi le mi tcati - "Please don't pour first water then milk in my tea". This, of course, says nothing about which connective is appropriate - one might guess se ce'o (first milk, then water), only to find out that .e nai (only water, no milk at all) was the correct one.

cavyfle = x1, consisting of x2, flows into x3 from x4

Just like a logical connective is a plausible negation of a non-logical connective, answers to questions of the type ji or je'i can be both logical and non-logical: A: ladru je'i sakta le do ckafi B: se ce'o ("Milk or sugar in you coffee?" "First the latter, then the former"). In this case ce would make no sense at all, since sets can't be contained in coffee, and joi (both mixed together) would mean the same as jo'u (both of them), unless the respondant preferred unmixed sugar in his coffee.

The fifth JOI I present here is a bit of an oddball:

fa'u = Non-logical connective: Unmixed ordered distribution (A and B, respectively)

When only one fa'u is placed within a bridi (or several bridi connected together with connectors), fa'u may be assumed to be identical to jo'u. When several fa'u is used within one bridi, however, the constructs before fa'u each apply to each other, and the constructs after fa'u each apply to each other. Let's have an example:

 mi fa'u do rusko fa'u kadnoYou and I are Russian and Canadian

implying that mi goes with rusko and do goes with kadno, and implying nothing about any other combination. Of course, in this example, it would be much easier to say mi rusko .i do kadno.

These last three JOI connects two sets to make new sets:

jo'e = A union B
ku'a = A intersection B
pi'u = Cross product of A and B

These are probably not very useful for the average Lojbanist, but I might as well include them here.

The first one, jo'e, contains all the members of set A and those of set B. If anything is a member of both sets, they are not counted twice.

A set made with ku'a makes a new set from two sets. This new set contains only those members which are in both sets.

pi'u is a little more complicated. A set "A pi'u B" contains all the possible combinations of "a ce'o b", where a is a member of A and b is a member of B. It is thus a set of sequences of members. If, for instance, set A contained the members p and q, and set B contained members f and g, then A pi'u B would be a set consisting of the four members p ce'o f, p ce'o g, q ce'o f and q ce'o g.

## Lesson 27: Lojban logicː da, bu'a, zo'u, and terms

Here we are starting to talk about advanced Lojban. The Lojban in this and following lessons is rarely relevant when speaking Lojban in normal contexts, but it pops up quite often when speaking about language and logic.

These corners of Lojban is for the most part experimental, new or complex, so you should expect a lot of changing definitions, outdated definitions, disagreements and misunderstandings on the part of the author of this text. Sorry about that.

The stated topic of this lesson needs some justification: This lesson is not really about how do to logic in Lojban, since firstly, logic is presumably the same in all languages, and secondly, actually teaching logic would be totally impractical in one single lesson. Rather, this lesson explains some constructs which resemble those which logicians use. It turns out they have a remarkable wide range of uses in Lojban.

Getting involved in the more obscure details of these logical constructs can be mind-warpingly difficult, and there will always be some disagreement in the corners of this part of the language.

Learning these logical constructs requires one to learn a bit about constructs which are not logical in nature. Let's begin with zo'u

zo'u = Separates prenex from bridi

Before any zo'u is the prenex, after zo'u is the bridi. Informally, a prenex is a place in front of the bridi, where you put a bunch of terms. A term is an English word given to some kinds of Lojban constructs: Sumti, sumtcita with or without sumti attached, na ku and an abomination called termsets, which I refuse to include in these lessons. The prenex is not part of the bridi, but any terms put inside it gives us information about the bridi. One can, for example, use it to state a topic as shown thus:

lo pampe'o je nai speni zo'u mi na zanru - "Concerning lovers who are not spouses: I do not approve". The benefits of kind of sentence structuring is questionable, but it's always good to have some variation to play with. Furthermore, constructing sentences this way resembles Mandarin (and other languages) closely, meaning it might seem more intuitive for speakers of that language.

pampe'o = x1 is a lover of x2
zanru = x1 approves of x2 (plan, event or action)

Of course, the relation between the terms in the prenex and the bridi is vague. One can imagine any sumti in the prenex bearing the same relevance to the bridi as if they were put in the bridi after a do'e sumtcita, and any sumtcita in the prenex doing pretty much the same as if they were in the bridi. It is quite possible to put terms in prenexes without any clear hints as to how the term may relate to the bridi:

le vi gerku zo'u mi to'e nelci lo cidjrpitsa - "Concerning this dog here: I dislike pizza." It leaves you guessing about the reason for mentioning the dog.

cidjrpitsa = x1 is pizza with topping/ingredients x2

If the prenex contains na ku, it's pretty straight forward: The entire bridi is negated, just as if the bridi itself began with na ku.

So how long does a prenex last? It lasts until the following bridi is terminated. If that is not desired, there are two ways to make it apply to several bridi: One is to put some kind of connective after the .i separating the bridi, and another method is to simply include all of the text in tu'e ... tu'u-brackets. These brackets work pretty much by gluing all the bridi together and makes all sorts of construct apply to several bridi.

Now that we covered zo'u, the first "logical" words we can use it with are these:

da = logically quantified existential sumka'i 1
de = logically quantified existential sumka'i 2
di = logically quantified existential sumka'i 3

These words are all the same, like the mathematical variables x, y and z. Once you have defined them, however, they keep refering to the same thing. These words are defined in the prenex of bridi, meaning that when the prenex stop applying, the definition of these three words are cancelled.

The words da, de and di can refer to literally any sumti, which makes them kind of useless unless restricted in some way. The first and foremost way to restrict them is to quantify them: They are not called "logically quantified existential sumka'i" for nothing. They are sumka'i, they are most useful when quantified, and they are existential. What does it mean, being "existential"? It means that by default, they are used to assert that something exists. An example:

The statement pa da zo'u da gerku has pa da in the prenex, which means "There exists one thing such that it:", and then da, now defined, is used in the bridi da gerku. Translated to English, this means: "There exists one thing which is a dog". This is obviously false, there are around 400,000,000 of them in the world. If da and its sisters are not quantified, the number su'o is the default. Thus da zo'u da gerku means "There exist at least one thing which is a dog", which is true. Notice here, that any quantification must be more or less exact in order to be true: Of course one dog exists, but in Lojban, pa da zo'u da gerku means not only that does one dog exists, but also that no more than one does.

There are a few specific rules to these existential sumka'i:

- If the quantifier ro is used before da, it instead refers to "all which exists".

- Importantly, the usage of an existential sumka'i only asserts that such a thing exists in the domain of truth where it's used. Thus, in the sentence so'e verba cu krici lo du'u su'o da crida, does not state da crida, since its "domain of truth" is only inside the du'u-abstraction. Generally speaking, abstractions contain their own domain of truth, so using da and friends inside an abstraction is usually safe.

- If the same variable is quantified several times, the first quantification is the one which sticks: Any later quantified instance of that variable can refer only to things which are also being referred to by the first instance of that variable, and any later non-quantified instance of that variable will gain the first quantifier. To use an example: ci da zo'u re da barda .ije da pelxu means "There exists three things such that two of them are big and all three are yellow". re da, being after ci da, can only refer to two of the already stated three things. When da appears without a quantifier, ci is assumed.

- If there are several terms in the prenex, the terms are always read left to right. Sometimes, this matters: ro da de zo'u da prami de means "Concerning all the things X that exists, concerning at least one thing Y: X loves Y". This is the same as "All things love at least one thing.", where the "thing(s)" can be anything, including the thing itself. Note here that de can refer to different things for each da - the thing which is referred to by de is dependent on the da, since it came before it in the prenex, therefore each thing might love something different. If we switched the places of da and de in the prenex, a different result would arise: de ro da zo'u da prami de = "Concerning at least one thing Y, concerning all X which exists: X loves Y", meaning "There exists at least one thing which everything loves".

Of course, both claims are completely false. There are many things which loves nothing - rocks, or abstract concepts, for example. Likewise, it's impossible to concieve of something which everything loves, since "everything" also encompasses non-sentient things. We need better ways to restrict what these variables can point to. One good way of doing it is to make them the subject of a relative clause:

ro di poi remna zo'u birka di = "Concerning all X that exists, which is human: X has one or more arms." or "All humans have arms", which is true, at least when speaking in a potential, timeless sense.

birka = x1 is an arm of x2

When restricting claims using this kind of logical "existential" variable, it's very important to remember that unless there is an explicit no or ro as a quantifier, these kind of statements always imply that there actually exists something which can be referred to by da. Therefore, any kind of non-negated statement where da points to something which does not exist is false.

ro da was also originally defined to imply existence, but is now usually taken to be equivalent to no da naku, "nothing exists that does not ...". Having this equivalence hold no matter what the formula evaluates to is seen as more important than having {ro} mirror natural language use exactly.

In fact, you don't really need the prenex to define the variables. You can use them directly as sumti in the bridi, and quantify them there. You only need to quantify them the first time they appear, though. Thus, the sentence about humans having arms could be turned into birka ro di poi remna. The order of the variables still matters though, and so the prenex can be used to avoid having to mess up your bridi to place the variables in the correct order. When having more variables, a prenex is usually a good idea.

The second kind of logical words are basically the same as the three we have already been though, but these are brika'i instead of sumka'i:

bu'a = logically quantified existential brika'i 1
bu'e = logically quantified existential brika'i 2
bu'i = logically quantified existential brika'i 3

These work pretty much the same way as the other three, but there are a few points which are important to mention:

Since only terms can go in the prenex, these brika'i need to have a quantifier in order to make them into sumti. When quantified in the prenex, however, the quantifier works very different from quantifiers with normal selbri: Instead of quantifying the amount of things which fits the x1 of the selbri variable, it directly quantifies the amount of selbri which applies. Again, the default quantifier is so'u. Thus, instead of re bu'a zo'u meaning "Concerning two things which is in relationship X:", it means "Concerning two relationships X:"

It's probably good to see an example of bu'a put to practice:

ro da bu'a la .bab. = "Considering all X which exists: X is in at least one relationship with Bob" = "Everything is related to Bob in at least one way.". Notice again the order matters: su'o bu'a ro da zo'u da bu'a la .bab. means: "There is at least one relationship such that everything that exists is in that relationship with Bob". The first statement is true - for any one thing, one can indeed make up some selbri which relates any guy called Bob and it. But I'm not sure the latter is true - that one can make a selbri which can relate anything, no matter what it is, and Bob.

Let's have an example which quantifies selbri:

ci'i bu'e zo'u mi bu'e do - "Concerning an infinite amount of relationships: I am in all those relationship with you." or "There exists an infinite amount of relationships between us"

You can't quantify the selbri variables in the bridi itself, though. Then it will act as a sumti: mi ci'i bu'a do is not a bridi. There are some situation where this will become problematic - lesson twenty-nine will teach how to overcome those problems.