# Monthly Archives: April 2015

## Formal semantic analysis of natural language quantifiers

Natural language quantifiers are an interesting subset of words in that it is possible to define them formally using set theory, by taking them to be binary relations between sets. For example, here are the formal definitions of some English quantifiers.

• “every” is the binary relation $\forall$ between sets such that for every pair of sets $A$ and $B$, $\forall(A, B)$ if and only if $A \subseteq B$. For example, the sentence “every man is in the room” is true if and only if the set of all men is a subset of the set of everything in the room.
• “some” (in the sense of “at least one”) is the binary relation $\exists$ between sets such that for every pair of sets $A$ and $B$, $\exists(A, B)$ if and only if $A \cap B$ is non-empty. For example, the sentence “some (at least one) man is in the room” is true if and only if the set of all men and the set of everything in the room have at least one member in common.
• More generally, each natural number $n$ (as an English word, in the sense of “at least $n$”) is the binary relation $\exists n$ between sets such that for every pair of sets $A$ and $B$, $\exists n(A, B)$ if and only if $|A \cap B| \ge n$. (We are assuming here that the number is not interpreted to be exhaustive, so that the statement “two men are in the room” would still be seen as true in the case where three men are in the room.) For example, the sentence “two men are in the room” is true if and only if the set of all men and the set of everything in the room have at least two members in common.
• “most” (in the sense of “more often than not”) is the binary relation $M$ between sets such that for every pair of sets $A$ and $B$, $M(A, B)$ if and only if $|A \cap B| \ge |A \setminus B|$.

Admittedly, some natural language quantifiers, like “few” and “many”, cannot be satisfactorily defined in this way. But quite a lot of them can be, and I’m going to just focus on those that can be in the rest of the post. From now on you can take the term “natural language quantifiers” to refer specifically to those natural language quantifiers that can be given a formal definition as a binary relation between sets.

Now, once we have taken this approach to natural language quantifiers, an interesting question arises: which binary relations between sets correspond to natural language quantifiers? Clearly, no individual language could have natural language quantifiers corresponding to every single binary relation between sets, because there are infinitely many such binary relations, and only finitely many words in a given language. In fact, we can be quite sure that the vast majority of binary relations between sets will never correspond to natural language quantifiers in any language, because most of them are simply too obscure. Consider, for example, the binary relation $R$ between sets such that for every pair of sets $A$ and $B$, $R(A, B)$ if and only if $A$ is the set of all men and $B$ is the set of all women. If this corresponded to an English quantifier, which might be pronounced, say, “blort”, then the sentence “blort men are women” would be true, and every other sentence of the form “blort Xs are Ys” would be false. I don’t know about you, but I can’t think of any circumstances under which such a word would be of any use in communication whatsoever.

Another problem with our supposed quantifier “blort” is that it can’t reasonably be called a quantifier, because its definition has absolutely nothing to do with quantities! You probably know what I mean here, but it’s worth trying to spell out exactly what it is (after all, the whole point of formal analysis of any subject is that trying to spell out exactly what you mean often leads to interesting new insights). It seems that the problem is to do with the objects and properties that are referred to in the definition of “blort”. Our definition of “blort” refers to the identities of the two arguments $A$ and $B$—it includes the phrases “if $A$ is the set of all men” and “if $B$ is the set of all women”. But the definition of a quantifier should refer to quantities only, not identities. From the point of view of set theory, “quantity” is just another word for “cardinality”, which means the number of members a given set contains. So perhaps we should say that the definition of a natural language quantifier can only refer to the cardinalities of the arguments $A$ and $B$. This is still not a proper formal definition, because we have not been specific about what it actually means for a definition to “refer” to the cardinalities of the arguments only. If we take the statement very literally, we could take it to mean that the definition of a natural language quantifier should be a string consisting only of the substrings “$|A|$” and “$|B|$” (with $A$ and $B$ replaced by whichever symbols you want to use to refer to the two arguments), interpreted in first-order logic. But that’s ridiculous, and not just because such a string would evaluate to a natural number rather than a truth value. In order to find out what the proper constraints on the string should be, let’s have a look again at the definitions we gave above.

• For “every”, we have that $\forall(A, B)$ if and only if $A \subseteq B$, or, equivalently, $|A \setminus B| = 0$.
• For each natural number $n$, we have that $\exists(A, B)$ if and only if $|A \cap B| \ge n$.
• For “most”, we have that $M(A, B)$ if and only if $|A \cap B| \ge |A \setminus B|$.

In order for these to count as quantifiers, our definition must allow us to compare the cardinalities as well as refer to them. We also need to refer to the cardinalities of combinations of the two arguments of $A$ and $B$, such as $A \cap B$ and $A \setminus B$, as well as $|A|$ and $|B|$. And, although none of the definitions above involve the logical connectives $\wedge$ (AND) and $\vee$ (OR), we will need them for more complex quantifiers that are formed as phrases, such as “most but not all”.

The question of exactly which combinations of sets we need to refer to is quite an interesting one. Given our two arguments $A$ and $B$, we can see all the possible combinations by drawing a Venn diagram:

There are four disjoint regions in this Venn diagram, corresponding to the sets $A \cap B$, $A \setminus B$, $B \setminus A$ and (not labelled, but we mustn’t forget it) $U \setminus (A \cup B)$ (where $U$ is the universal set). We also might need to refer to regions that are composed of two or more of these disjoint regions, but such regions can be referred to by using $\cup$ to refer to the union of the disjoint regions.

But do we need to be able to refer to each of these disjoint regions? Note that in the definitions above, we only needed to refer to $|A \setminus B|$ and $|A \cap B|$, not to $|B \setminus A|$ and $|U \setminus (A \cup B)|$. In fact, it is thought that these are the only two disjoint regions that definitions of natural language quantifiers ever need to refer to. Quantifiers which can be defined without reference to $|U \setminus (A \cup B)|$ are called extensional quantifiers, and quantifiers which can be defined without reference to $|B \setminus A|$ are called conservative quantifiers. So now, if all this seems like pointless formalism to you, you might be relieved to see that we can make an actual falsifiable hypothesis:

Hypothesis 1. All natural language quantifiers are conservative and extensional.

To give you a better sense of exactly what it means for a quantifier to be conservative or extensional, let’s give some examples of quantifiers which are not conservative, and not extensional.

• Let $NE$ be the binary relation between sets such that for every pair of sets $A$ and $B$, $NE(A, B)$ if and only if $|U \setminus (A \cup B)| = \emptyset$. For example, if we suppose $NE$ corresponds to an English quantifier “scrong”, the sentence “scrong men are in the room” is true if and only if everything which is not a man is in the room (it’s therefore identical in meaning to “every non-man is in the room”). “scrong” is conservative, but not extensional.
• Let $NC$ be the binary relation between sets such that for every pair of sets $A$ and $B$, $NC(A, B)$ if and only if $|B \setminus A| = \emptyset$. For example, if we suppose $NC$ corresponds to an English quantifier “gewer”, the sentence “gewer men are in the room” is true if and only if there is nothing in the room which is not a man. “gewer” is extensional, but not conservative.

Wait a minute, though! I don’t know if you noticed, but “gewer” as defined above has exactly the same meaning as a real English word: “only”. The sentence “only men are in the room” means exactly the same thing as “gewer men are in the room”. (It’s true that we can say “only men are in the room” might just mean that there are no women in the room, not that there is nothing in the room that is not a man—there could be furniture, a table, etc. But “only” still has the same meaning there—it’s just that the universal set is taken to be the set of all people, not the set of all objects. In semantics, the universal set is understood to be a set containing every entity relevant to the current discourse context, not the set that contains absolutely everything.)

Does that falsify Hypothesis 1? Well… I said “only” was a word, but I didn’t say it was a quantifier. In fact, the people who propose Hypothesis 1 would analyse “only” as an adverb, rather than a quantifier. I guess this makes sense considering “only” has the “-ly” suffix. But that’s not proper evidence. Some people have argued that “only” cannot be a determiner (and hence cannot be a quantifier) based on syntactic evidence: “only” does not pattern like other determiners. The example I was given at university was the following sentence:

The girls only danced a tango.

Here, “only” occurs in front of the VP, rather than the NP, hence it must be a determiner.

I’m sure my lecturer could have given better evidence, but he was just pressed for time. But the obvious problem with this argument is that there is a well-known group of determiners which can appear in front of the VP, rather than the NP: the “floating” quan tifiers, such as “all”:

The girls all danced a tango.

Anyway, I remain not totally convinced that “only” is not a quantifier, and, taking a very brief look at the literature, it seems like a far-from-uncontroversial topic, with, for example, de Mey (1991) arguing that “only” is a determiner, after all (although I don’t really understand its argument, having not read the paper very carefully). Payne (2010) mentions that “only” should be seen as a kind of adverb-quantifier hybrid, which I guess is probably the best way to think about it, although it is kind of inconvenient if you’re trying to analyse these words in a formal semantic approach.

I wonder if there are any words in natural languages which have ever been analysed as non-extensional quantifiers. Google Scholar doesn’t turn up anything on the subject.

In any case, perhaps the following weakened statement of Hypothesis 1 is more likely to be true.

Hypothesis 1. In every natural language, the words that can be analysed as non-conservative or non-extensional quantifiers will exhibit atypical behaviour compared to the conservative and extensional quantifiers, so that it may be better to analyse them as adverbs.

## Thoughts on “The Master and Margarita” by Mikhail Bulgakov

The Master and Margarita is a book by Mikhail Bulgakov, written from 1928 to 1940, but published only in 1967 due to Soviet censorship, long after the author’s death. It’s considered one of the best works of Soviet literature if not the best. I finished reading it last week, and, although I haven’t read any other works of Soviet literature, I can believe that it’s one of the best.

I’m more familiar with Russian novels of the 19th century: the works of Tolstoy, Dosteyevsky, etc. So it was interesting to see how a 20th-century Russian novel compared, although of course I don’t know how representative The Master and Margarita (let’s call it TMM, because I’m going to have to write it a lot) is of 20th-century Russian literature in general. There are some clear differences between TMM and Tolstoy and Dostoyevsky’s novels. For example, while Tolstoy and Dostoyevsky stuck to realism in their writing (even though the devil made an appearance as a character in The Brothers Karamazov, he could be explained as a hallucination), TMM is not very realistic at all. It’s arguably a fantasy novel. I mean, the premise of the story is that the devil, together with a band of demon sidekicks, has come to Moscow to cause trouble. One of the characters, who you might see on the cover if you buy the book, is a talking cat called Behemoth. At several points he threatens people wielding a Browning handgun. There are some memorably absurd lines involving, like the following:

“I challenge you to a duel!” screamed the cat, sailing over their heads on the swinging chandelier.

If this strikes you as a pretty amusing line, that’s probably intentional. Because another way in which TMM is quite different from your average Tolstoy or Dostoyevsky book is that it’s a comedy. A Russian kind of comedy, mind you; it can be categorised best as farce, although there are elements of black comedy too, to a lesser extent. Basically, if you find it funny when bizarre, horrible things happen to people, but they kinda deserve it, then you’ll probably enjoy the humour in TMM. Oh, and although a lot of it went over my head, since I wasn’t ever a Soviet citizen, there’s political satire in here too. That’s why Stalin wouldn’t allow it to be published, after all. There is at least one point where a more serious political message is conveyed, one more horrifying than amusing: I’m thinking of the chapter called “Nikanor Ivanovich’s Dream”. I’ll confess that I completely missed the point of this chapter when I first read it; I only realised its significance when I was reading about the book online and came across people saying that it was an ‘obvious’ allusion to the secret police’s interrogation methods. But, even if you take the chapter just as what it is on the surface, a description of a strange dream, it’s an unsettling read. In fact, I believe that this is one of the chapters that was still heavily censored for a time even after the book was published.

It was never my impression, though, that the political aspect of TMM was central to it. I’ve seen a lot of descriptions of the book along the lines of ‘satire of Soviet life’, and I guess you can interpret it that way, but that isn’t how I interpreted it. It seemed to me that the book was mostly about something else; it was more than just a comedy, and more than just a political satire or exposé along the lines of Solzhenitsyn’s The Gulag Archipelago. That said, I can’t really tell you what this extra something is. Like most good works of art, I don’t think it has or is supposed to have a straightforward message. So I won’t say anything more about what the deeper meaning of TMM is; that’s up to you to reckon for yourself if you read the book. But I can give you a brief overview of the plot (which I guess can be thought of as the meaning at the surface level).

The book has two parts. In Part 1 the reader is treated to a number of descriptions of the disastrous encounters various residents of Moscow have with Satan (going by the alias of “Professor Woland”) and his gang of demons. This is fairly entertaining, and it includes “Nikanor Ivanovich’s Dream” and no doubt other politically meaningful excerpts. But if this was all there was to the book, I’d be disappointed. Part 2 is the meat of the book, in my opinion. That’s where we are introduced to the titular character, Margarita. There are two titular characters, actually, since there’s the Master as well, who is Margarita’s boyfriend. But it’s Margarita who the narrative follows. We are briefly introduced to the Master in part 1, when he is living in an insane asylum (where a lot of the people who have had the misfortune to meet Professor Woland have ended up). He was a writer (he refuses to tell his name), who wrote a magnum opus about Pontius Pilate, the Roman governor of Judea in Jesus’s time, but the censors refused to publish it, he got into political trouble, and he ended up being sent to the madhouse (but not before he decided to burn his book’s entire manuscript). By the way, a writer, who wrote a great book that was censored—does that sound familiar? Yes, it’s not too hard to see that the Master is, at least on one level, a stand-in for Bulgakov himself. You might say that the book he was writing represents TMM itself, in which case I guess you could say that Bulgakov wrote a book about himself writing it1. The manuscript-burning episode even has a parallel in Bulgakov’s life: apparently, he set an early manuscript of TMM alight as well.

I forgot to mention, by the way, that there are actually some chapters in Part 1 which do not progress the main story, but instead give us a chapter of the Master’s lost manuscript. So there are actually two stories in parallel: the main story set in Moscow in the 1930s, and the Master’s story about Pontius Pilate, set in Jerusalem in 1 AD. The Master’s story is, as far as I know, a pretty faithful retelling of the same one readers of the gospels will be familiar with; although I couldn’t really say since I haven’t read the gospels myself. But it’s an interesting side-narrative. In my experience books that try this dual narrative technique often suffer from one of the narratives being less interesting than the other. I’m thinking of Ursula Le Guin’s The Dispossessed here; that’s a great book, but I was always more interested in what was happening on Urras than Annares. But TMM is not one of the books that suffers from this; I often found the Biblical sub-narrative more interesting than the main one. This sub-narrative continues into Part 2, as well.

Anyway, the Master had a girlfriend, Margarita, and he didn’t get a chance to tell her that he was being taken away from his home, nor did he opt to disappoint her with the news once he was free to tell her, so when Part 2 begins she has no idea where he is and fears that he has died. Then she encounters a mysterious man in a park, who is actually Azazello, one of Professor Woland’s goons. He’s been entrusted with the task of getting Margarita to carry out a certain errand, which Woland anticipates she will do in return for being reunited with her love, the Master. Azazello isn’t a very good salesman—his pitch isn’t very clear, and I don’t think calling someone a “stupid bitch” ever helps—but when he recite a passage from the Master’s lost book, Margarita realises what he has to offer, and agrees to do whatever he wants.

What follows is the most wonderful section of the book, in my opinion. Azazello gives Margarita a special cream which, when she rubs it on her body, restores her youthful beauty and grants her the powers of a witch. Then she hops on a broomstick (while still completely naked—witches don’t need clothes) and goes flying out of Moscow and far to the east (although not before stopping at the apartment of one of the critics who negatively reviewed the Master’s book and gleefully smashing everything in it). It’s the kind of section that you want to read as slowly as possible, savouring every new sentence since the prose is just such a pleasure to read. In the end she lands somewhere deep in Siberia, where there is no trace of human habitation, and bathes in a river. When she steps onto the bank she is greeted by a band of pipe-playing frogs, dancing water-sprites, curtsying fellow witches and a goat-legged man brings her champagne. Then after a short stay, she is returned to Moscow in a flying car driven by a crow.

In Moscow, Margarita is taken to Woland’s apartment and meets his demon crew. There, she learns what the errand is that she has agreed to doing. It turns out not to be anything especially terrible: all she has to do is host a ball on Walpurgisnacht for the denizens of Hell: murderers and poisoners, free-thinkers and adultresses, pimps and brothel-keepers, and the composer Johann Strauss. Although she finds it somewhat exhausting, she carries it out without trouble and Woland is satisfied. Again we are treated to some wonderful writing as Bulgakov tells us about the fantastic things at the ball, including a swimming pool filled with brandy and a troupe of accordion-playing polar bears. After this point, I probably shouldn’t go into too much detail about what happens, in case you don’t want the ending spoiled. But I will tell you that there is a sort of happy ending. Indeed, one of the interesting things about the book is that Satan is not portrayed as the ruthless trickster you might expect to him; he sticks to his word and gives Margarita what she wants. Even the various people who suffer as a result of his crew’s actions tend to have done something to deserve it first. And as for the heavenly counterparts of these demons, they are nowhere to be seen; well, Jesus appears in the Master’s story but he doesn’t really have much presence in the main narrative.

So, overall, what’s my opinion on the book? Well, it’s an entertaining, enjoyable story, for sure. Russians and Russia enthusiasts, especially, will find a lot that will appeal to them in the book, and perhaps will gain insight from reading it. The book can be appreciated in many different ways: at one time it’s comic, another time tragic, and other times intellectually stimulating. I’m not surprised that TMM is often considered the greatest Russian novel of the 20th century.