Branching Quantifier - Relation To Natural Languages

Relation To Natural Languages

Hintikka in a 1973 paper advanced the hypothesis that some sentences in natural languages are best understood in terms of branching quantifiers, for example: "some relative of each villager and some relative of each townsman hate each other" is supposed to be interpreted, according to Hintikka, as:

.

which is known to have no first-order logic equivalent.

The idea of branching is not necessarily restricted to using the classical quantifiers as leafs. In a 1979 paper, Jon Barwise proposed variations of Hintikka sentences (as the above is sometimes called) in which the inner quantifiers are themselves generalized quantifiers, for example: "Most villagers and most townsmen hate each other." Observing that is not closed under negation, Barwise also proposed a practical test to determine whether natural language sentences really involve branching quantifiers, namely to test whether their natural-language negation involves universal quantification over a set variable (a sentence).

Hintikka's proposal was met with skepticism by a number of logicians because some first-order sentences like the one below appear to capture well enough the natural language Hintikka sentence.

where

denotes

Although much purely theoretical debate followed, it wasn't until 2009 that some empirical tests with students trained in logic found that they are more likely to assign models matching the "bidirectional" first-order sentence rather than branching-quantifier sentence to several natural-language constructs derived from the Hintikka sentence. For instance students were shown undirected bipartite graphs—with squares and circles as vertices—and asked to say whether sentences like "more than 3 circles and more than 3 squares are connected by lines" were a correctly describing the diagrams.