From the paper at reference 1 by Haspelmath, already noticed at reference 2, I have lighted on the paper about quantifiers at reference 3, by Croft. A chap from the US who has spent quality time in the UK, is now in New Mexico and is keen on redwoods, probably including what we know as the Wellingtonia with search key ‘wgc’. So he must be a good chap.
He starts off by observing that the sentence 'every man loves some woman' is well known to be ambiguous: more particularly, the scope and precise meaning of the quantifiers ‘every’ and ‘some’ are not tied down. And while, as he explains, various subtle markers in such sentences in English may help, the basic ambiguity lurks. Other languages can do these things in different ways, but he does not go into that very much.
One of the problems is that we tend to analyse such sentences with the help of brackets and explicit quantifiers. So, for example, we might analyse this sentence into ‘(for every man, (there is a women such that (this man loves that woman)))’, otherwise (for all x (there is a y such that (L(x,y))). Which while unambiguous, is rather clumsy. So what are we to do instead?
Some preliminaries
Specifics
In the beginning, I said things about particular things, for example: ‘I am in Epsom and this is a green letter box’. Suggesting that I am in direct contact with a green letter box. And even in ‘Peter said that he was in Epsom and saw a green letter box’, I am allowing that contact by hearsay.
But if I say: ‘there is a green letter box in Epsom’, I am asserting a property about the set of green letter boxes, without asserting direct contact with a green letter box with myself or anyone else. Grammarians might say that in the first case I was being specific, in the second non-specific.
Cocaine provides another example: if I test the waste water of Epsom and find a threshold level of cocaine, I can truthfully assert that it is very likely that someone in Epsom is using cocaine without having any knowledge of who that someone might be. In which I put aside confusing factors like the people of Epsom have very bad teeth and getting through a lot of dental anaesthetics.
Sentences
We suppose that we are talking about well defined sentences (‘Peter jumped over the wall’) and clauses (the bit after the ‘that’ in ‘Mary said that Peter jumped over the wall’). For brevity, we call them all sentences in what follows.
Quantifiers
Sentences can include one or more quantifiers, with ‘Peter jumped over all the walls’ and ‘Peter jumped over a wall’ containing the universal quantifier ‘all’ and the existential quantifier ‘[there is] a’ respectively, for the moment assuming that the second wall is non-specific.
These quantifiers are what we call in what follows in-text, attached to the relevant noun. This is convenient in natural, spoken languages.
While in logic, it is convenient to have what we call pre-text quantifiers, coming immediately before the sentence to which they apply.
In logic: ‘E x (Prop(x))’ : there is some x such that the proposition Prop(x) is true.
Where we use ‘E’ in place of the turned upper case E usually used for these purposes, as in the snap from Wikipedia above. And we will use ‘U’ in place of the turned upper case A usually used for the universal quantifier. At least ‘U’, unlike ‘A’ is unlikely to appear in natural text. All this because the turned letters do not themselves appear in regular fonts in Microsoft Word.
In natural, if rather forced, language: ‘there is a wall x such that Peter jumped over x’.
In the case that there is more than one such quantifier, the order may be important, order which is brought out more clearly in the pre-text format.
More on the problem
Suppose we have the simple sentence: ‘the man loves a woman’. We know that we are talking about a particular man, but we do not know for sure, unless the context is helpful or there are more clues, whether we are talking about a particular woman (specific) or that we are just asserting the existence of such a woman (non-specific). So, from the point of view of analysing this sentence with a computer, it would be helpful if there were some marker to make clear which reading is intended.
Suppose now we graduate to two quantifiers. Let us suppose we have a simple sentence: ‘Q man loves Q woman’, where the ‘Q’s stand for in-text quantifiers. The sentence might be a bit more complicated than this, but we do have exactly two quantified nouns.
We suppose that both quantifiers have the same scope, in this case the whole sentence. So the issue is whether they are universal or existential and their order. I make it six permutations, exhibited in the next section.
Let X be the set of relevant men, Y be the set of relevant women. We might, for example, rather unrealistically restrict ourselves to adults presently living in Epsom. Of more fancifully to the divine inhabitants of Mount Olympus.
Let Prop(x,y) be the proposition that for some x from X and some y from Y it is the case that x loves y.
It might be argued that ‘love’ is not quite like this and X and Y coincide. But I don’t think that this needs to be the case for the examples below to work and there are certainly verbs which are not symmetric in this way. So if x hems y, then x is a person and y is a garment or a piece of cloth, not the other way around. If x throws y, then the class of things that throw is not the same as the class of things that can be thrown.
And even when X and Y coincide, with most verbs, we do not have it that Prop(m,n) if and only if Prop(n,m); that it can happen, for example, that I love you and you don’t love me. Indeed, I am having trouble finding a common verb for which this rule would be true. But this does not matter for present purposes.
The examples
UxUyProp(x,y) : for all x in X, for all y in Y, Prop(x,y) is true: all the men love all the women. Which is more or less the same as ‘all the women are loved by all the men’. Reversing the order of the quantifiers results in differences of stress and attention, rather than of structure.
UxEyProp(x,y) : for all x in X, there is a y in Y such that Prop(x,y) is true: every man has a loved one.
EyUxProp(x,y) : there is a y in Y such that for all x in X Prop(x,y) is true: there is a woman whom every man loves.
ExUyProp(x,y) : there is an x in X such that for all y in Y, Prop(x,y) is true: there is a man who loves all the women.
UyExProp(x,y) : for all y in Y, there is an x in X such that Prop(x,y) is true: every woman has a lover.
ExEyProp(x,y) : there is an x in X and a y in Y such that Prop(x,y) is true: there is a man who loves a woman. Which is more or less the same as ‘there is a woman who is loved by a man’. Love does exist. There is love somewhere. Reversing the order of the quantifiers here results in differences of stress and attention, rather than of structure.
The present issue being how all this is sorted out in natural language, which likes to be short and to the point, qualities sometimes paid for with ambiguity. Not helped by there being no bracketed structures like ‘ExUyProp(x,y)’: one could write the brackets and one could possibly say them, but it would be terribly clumsy, and probably difficult, not to say impossible, to understand if there were to be one more than one layer of brackets, particularly, say, in the context of a conversation in a public house.
At reference 1, Croft goes to some pains to tease out the subtle markers that you do get in natural languages, more particularly in English, which go to reduce, if not eliminate this ambiguity.
A solution to the problem
The present argument being that, if one was starting over, this can all be done quite easily.
Let there be four quantifiers to take the place of ‘a’ in our simple sentence: ‘a man loves a woman’: OU, outer universal; OE outer existential; IU, inner universal; and, IE existential. There is a rule of grammar which says that in a sentence with two quantifiers, you have just one outer quantifier and one inner quantifier.
Then ‘ExUyProp(x,y)’ maps to something like ‘Prop(OEx,IUy)’ or ‘OE man loves IU woman’ and we have got rid of the bracketed structure without introducing ambiguity – at the expense of four new particles to learn.
The idea is that these particles are not used for anything else, are not co-opted to help with some other problem of expression. So there is no interaction between the business of quantification and anything else. Perhaps wasteful, but simple. Computer friendly.
Another solution to the problem
Let there be two quantifiers to take the place of ‘a’ in our simple sentence: ‘a man loves a woman’: U, universal; and, E existential. There is a rule of semantics which says that in a sentence with two quantifiers, the first is always presumed to be the outer quantifier and the second the inner.
So here, replacing all by ‘U’ and some by ‘E’, the otherwise ambiguous ‘some man loves all women’ would be interpreted as ExUyProp(x,y), that is to say ‘there is a man who loves all the women’ While ‘all women are loved by some man’ would be interpreted as UyExProp(x,y) that is to say ‘every woman has a lover’.
Where we pay for halving the number of particles by having, on occasion, to use the passive tense. Noting that when the two quantifiers are the same, the order is irrelevant. Noting that in this context, the use of plurals is redundant: the subject of a universal is (very nearly) always plural, that of an existential is always singular.
Interaction with both the business of passive and that of plurals. Which might cause complications elsewhere.
Yet another solution to the problem
Picking up on this last point, we might drop the quantifier altogether and say ‘women loved by man’ for ‘every woman has a lover’ and ‘man loves women’ for ‘there is a man who loves all the women’. But I imagine that such brevity would run into other problems. We need something to flag up the quantification.
Supplementary questions
So the first question is, why does English not do something like the first solution, something straightforward and simple. Why has English come to do quantifiers by nudges and winks?
Is the answer that particles have to be short words which can be said as well as written and a language only has space for so many of them? One can’t just invent a new particle for every new bit of function one comes across, that one needs. One has to do something complicated with those that one has already got. I think that Haspelmath makes a similar point, in a slightly different context, in reference 1. Put another way, evolution is lazy and will tend to make do and mend with what it has got, rather than attempt to start over: all vertebrates are organised in pretty much the same way, despite the huge variety in superficial appearances.
Perhaps a look at what the designers of Esperanto came up with would be helpful here. Maybe I should take a proper look at reference 4.
A second question is whether the scope of the quantifiers is always as simple as these examples would suggest, a sentence or a clearly demarcated clause, this last being something like the ‘that’ bit in ‘I said that some man loves all women’.
A third question is how all this interacts with the indefinite articles ‘a’ and ‘some’ when quantification is not involved.
A fourth question is that of probability. Is it good enough in natural language, if the reading suggested by syntax and context is right most of the time? That we can make do without the precision that computers are fond of.
Perhaps if I read Croft again I will find the right answers. Or perhaps the error of my ways.
References
Reference 1: Indefinite pronouns – Martin Haspelmath – 1997.
Reference 2: https://psmv4.blogspot.com/2021/06/implicature.html.
Reference 3: Quantifier scope ambiguity and definiteness – Croft, William – 1983.
Reference 4: https://en.wikipedia.org/wiki/Esperanto. More use than the fat Esperanto dictionary which sits on one of my bookshelves. Dictionary as in OED rather than an Esperanto-English dictionary.
Reference 5: Plena Ilustria Vortaro de Esperanto – Sennaciea Asocio Tutmondo – 1970. The fat dictionary.
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