Lojban Wave Lessons/27

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Lojban Wave Lessons: Foreword | ← Lesson 26 | Lesson 27 | Lesson 28 →

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of Lojban is for the most part experimental, new or complex, so you should expect a lot of changing definitions, outdated definitions, disagreements and misunderstandings on the part of the author of this text.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

Lojban Wave Lessons: Foreword | ← Lesson 26 | Lesson 27 | Lesson 28 →