A Rich Probabilistic Type Theory for the Semantics of Natural Language

Shalom Lappin
In a classical semantic theory meaning is defined in terms of truth conditions. The meaning of a sentence is built up compositionally through a sequence of functions from the semantic values of constituent expressions to the value of   the expression formed from these syntactic elements (Montague, 1974). In  this framework the type system is categorical. A type T identifies a set of  possible denotations for expressions from the values of its constituents.  There are at least two problems with this framework. First, it cannot represent the gradience of semantic properties that is pervasive in speakers' judgements concerning truth, predication, and meaning relations. Second, it offers no account of semantic learning. It is not clear how a reasonable account of semantic learning could be constructed on the basis of the categorical type systems that a classical semantic theory assumes. Such a system does not appear to be efficiently learnable from the primary linguistic data (with weak learning biases), nor is there much psychological data to suggest that it expresses biologically determined constraints on semantic learning.  A semantic theory that assigns probability rather than truth conditions to sentences is  in a better position to deal with both of these issues. Gradience is intrinsic to the theory by virtue of the fact that speakers assign values to declarative sentences in the continuum of real numbers [0,1], rather than Boolean values in {0,1}. Moreover, a probabilistic account of semantic learning is facilitated if the target of learning is a probabilistic representation of meaning.  We consider two strategies for constructing a probabilistic semantics. One is a top-down approach where one sustains classical (categorical) type and model theories, and then specifies a function that assigns probability values to the possible worlds that the model provides. The probability value of a sentence relative to a model M is the sum of the probabilities of the worlds in which it is true. The other is a bottom-up approach where one defines a probabilistic type theory and characterizes the probability value of an Austinian proposition relative to a set of situation types (Cooper (2012)). This proposition is the output of the function that applies to the probabilistic semantic type judgements associated with the syntactic constituents of the proposition.
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