There are a number of other logical systems competently specified in the six books in question. Some of these (e.g., the sorted logic featured in Alan Frisch's ``The Substitutional Framework for Sorted Deduction'', in Brachman et al. which marries classical logic to semantic networks) serve narrow but important purposes. Others are systems that anyone seriously interested in LAI (and related mathematical, linguistic, and philosophical matters) ought to be familiar with. One family of such ``required reading'' systems are those formed in the attempt to genuinely come to grips with the expressive power of natural language. Invariably, members of this family build upon the logical systems already described in this paper; the perfect example is Montague semantics, and perhaps the perfect introduction to Richard Montague's work (and related matters) comes by way of Thayse 1989 and Thayse 1991.
Montague, in a series of three seminal papers (all of which are conveniently reprinted in Thomason 1974), initiated a program intended to formulate a syntax and semantics for natural language having the same rigor and precision as the syntax and semantics for formal languages. The introduction to this program found in Thayse 1989 and Thayse 1991 is first rate. The Thayse 1989 part of this treatment starts with an extensional, type-theoretic logic; proceeds to consider Montague's intensional logic; pauses to engagingly discuss the historical, philosophical, and linguistic context of Montague's program; then presents for converting natural language to formal language; and, finally, briefly discusses some alternative approaches to reaching Montague's dream. Thayse 1991 is an articulate discussion of some of the limitations of Montague's approach (e.g., anaphora isn't accommodated) and some of the proposed solutions - solutions that show LAI making genuine progress.Anaphoric constructions are handled by infusing Montague's approach with Hans Kamp's discourse representation theory (1984). Another problem afflicting Montague's original approach is that it can't deal with contexts that are not referentially opaque. Thayse 1991 finds a solution in the situation semantics of Barwise & Perry 1983. We recommend that interested readers begin with the initial portion of the advanced material in Thayse 1991, which constitutes a quick introduction to Montague's logic; from there the reader can move back to the coverage in Thayse 1989, and then back to the advanced topics treated in Thayse 1991. We attempt now to give a sense of Montague's work.
Montague's approach borrows numerous elements from the logical systems visited above. For example, his logic is ``multimodal'', in that the logic combines operators for possibility and necessity with temporal operators:
Belief and knowledge, which we formalized above using the two basic operators from modal logic, are formalized by Montague with a two-place predicate for each concept - e.g., believe(a,e) - the first place occupied by a symbol referring to the agent, the second to an event. There is insufficient space here for even a compressed presentation of how the step-by-step process of Montague's translation scheme, when applied to natural language, yields a formal counterpart. But central aspects of the process are as follows: At the heart of the translation is an isomorphism between the analysis of an expression in natural language and its associated analysis in the intensional logic. This isomorphism comes by way of a type-theoretic syntax for the logic and a translation function that maps each syntactic category of natural language to a type of the intensional logic. Each natural-language expression A of syntactic category is assigned a logical translation A' in the logic; and f is the logical type of category . The approach assumes that most sentences of natural language can be analyzed according to two different moods: de re and de dicto.The distinction here will be familiar to many philosophers, but others may be seeing it for the first time. Consider the sentence `John believes that a woman talks'. The de dicto reading is that this sentence means that `John believes that there exists at least one woman who talks', where this belief doesn't refer to a particular woman. On the other hand, a de re reading would be that the sentence means that `John believes that there exists a well-determined woman who is talking'. Two additional operators handle the de re and de dicto moods; these are the intension and extension operators - and , respectively the first indicating de re, the second de dicto. The final tool used in Montague's translation is the lambda operator. Our readers are doubtless familiar with such ``set builder'' statements as
that is, the set of all people able to beat Deep Blue, a famous chess-playing computer program. The -operator allows us to say similar things in the logic itself. For example, if the two-place relation Wxy is true if and only if x has written y, and if u is a constant denoting Umberto Eco, then intuitively speaking the following represents the set of works that Eco has written.
In order to perhaps whet your appetite for Montague's program, here is what all of this machinery produces in the logic for the English sentences ``John examines and solves a problem'' and ``John will not talk'', respectively:
Those interested not only in Montague's work, but in the full range of proposals for how to genuinely deal with the formidable power of natural language (especially natural language's ability to allow reference to nonexistent objects), are encouraged to read Graeme Hirst's ``Existence Assumptions in Knowledge Representation'', in Brachman et al. (We strongly recommend that readers searching for answers to the problems lurking in this area also investigate SNePS (Shapiro & Rapaport 1987).)