## Quantification in English

The rules

Now that we have an idea of what game semantics looks like, let’s see how one could apply it to English. The natural rules that correspond to our rules for propositional connectives ($\vee, \wedge, \neg, \rightarrow$) can be adapted by simply replacing the formal symbol with the corresponding English conjunctive construction. This presupposes that we know how to get rid of pronominal reference from one conjunct to the other (or that we limit ourselves to some fragment of English where this can be done easily enough, for example we require that pronominal antecedents be proper names) and how to transform a negative sentence into a positive one (we’re going from the outside in, not generating something from the inside out). As for universal and existential quantification, the corresponding English words which Hintikka sets up analogical rules for are some / a(n) and every / any / each.

For existential quantification it is:

(G.some) When the game has reached a sentence of the form

X – some Y who Z – W,

a person may be chosen by myself. Let the proper name of that person be ‘b‘. Then the game is continued with respect to the sentence

X – b – W, b is a Y, and b Z.

An analogical rule (G.a(n)) can be devised for a(n).For universal quantification:

(G.every) When the game has reached a sentence of the form

X – every Y who Z – W,

a person may be chosen by Nature. Let the proper name of that person be ‘d‘. Then the game is continued with respect to the sentence

X – d – W if d is a Y and if d Z.

The rule (G.any) is analogical. Here we assume that Y and Z are singular and that the ‘who’ is the subject of ‘who Z’. The rules could be modified to deal with more general situations, but Hintikka does not go into that here.

The dependencies

So, we have defined some transformations which allow us to change an English sentence into a simpler English sentence. The natural question is, which order to we apply them in, and, since we are dealing with a game, what is their “informational status” (i.e. dependence, in the sense I talked about last time). I will discuss the first question in the next post (where we’ll finally get to the relevance of all this to linguistics itself), let us now concentrate on the latter. Hintikka argues that there are English sentences which do not exhibit a linear ordering of quantification. He provides an example:

Some novel by every novelist is mentioned in some survey by every critic.

This he interprets as being true only if the novel depends only on the novelist and the survey only on the critic. I.e. it is true in a state of affairs where

The bestselling book by every author is referred to in the longest essay by every critic

but not in a state of affairs where

The bestselling book by every author is referred to in the obituary essay on him by every critic.

I don’t know how about you, but I’d probably be quite willing to assent to the original statement even in such a situation. I would probably interpret it is “for every author and every critic, there is a book by the former which is mentioned in an essay by the latter” (but don’t mind my opinions). Against this, Hintikka gives an argument which I’m not sure I like, namely he appeals to the principle of charity and claims that even though the “literal meaning” of the sentence is as he says, people tend to interpret it the other way, because that’s the way “which is most likely to make the intended meaning of [the] utterance true” – just like the natural interpretation of “someone is hit by a car every week on this street” is simply an instance of the principle of charity, rather than an exception to the left-to-right ordering principle of English quantifiers.

Consider the sentence

John has not shown any of his paintings to some of his friends.

The natural interpretation is clearly that there are friends of John’s to whom he has shown none of paintings. But this seems to violate the left-to-right ordering principle (we would expect the scope of “any” to be greater than the scope of “some”). The principle of charity line of argument doesn’t seem to be applicable here, as it would not be anything out of ordinary if there were there were no paintings of John’s which he has shown to all of his friends. Partially-ordered quantification comes to the rescue, as we can simply interpret the two quantifiers to be independent of each other, which however turns out to be equivalent to the natural interpretation. Hintikka makes it clear that this is not due to some special behaviour of the “not”, as a similar analysis can be given for “John has shown all his paintings to some of his friends” (think it through yourself).

The other argument given is that “some novel by John is mentioned in some survey by every critic” clearly follows from the original sentence, but the “linear” interpretation (“there is one novel by John which is mentioned in some survey by every author”) is, according to Hintikka, “patently counter-intuitive”. Similarly with “some novel by every author is mentioned by every critic”. It is also argued that this nesting be arbitrarily complicated, that is that every sentence with branching quantifiers corresponds to some grammatical English sentence. Myself, I wouldn’t go as far as calling them patently counter-intuitive but I admit that in this case Hintikka’s interpretation seems acceptable. What I think Hintikka neglects, however, is that his sentences are a bit too abstract – I would say that their interpretation can also be expected to depend on stress (and possibly other prosodic features, too). I don’t think it’s possible to interpret the sentence “some novel by John is mentioned in some survey by every critic” (italics denoting stress) the way Hintikka does.

The implications

But however you feel about his particular examples, it seems undeniable that there are English sentences which can be naturally interpreted his way (for me, it would probably be sentences like “a particular book of every author is mentioned in a particular essay of every critic”). At the very least, even if you are unimpressed by any of his examples, they show that there is no a priori reason to assume that classical first-order predicate calculus can in general adequately capture quantification in natural language.

The importance of these findings (if true) is summarized at the end of Hintikka’s 1973 paper Quantifiers vs. Quantification Theory (according to the editor the first paper to discuss game semantics for natural language):

“Since our results suggest that the whole of f.p.o. quantification theory is built into the semantics of English quantifiers, it follows that the semantics of a relatively small fragment of English, viz. of English quantifiers plus a few supporting constructions, is a much subtler and much more complicated subject than anyone seems to have suspected. In the eyes of a logician, it seems to be powerful beyond the wildest dreams of linguists and of philosophers of language.”

Next time, I’ll try to wrap up this little introduction to Hintikka with his thesis concerning the interplay of syntax and semantics.

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