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These are excerpts and elaborations from my book "The Nature of Consciousness"
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The Math of Grammars First-order predicate logic,
the most commonly used logic, may be too limited to handle the subtleties of
language. The "generalized
phrase-structure grammar" pioneered by the British linguist Gerald Gazdar, for example, makes use of Intensional Logic, which is a variant of
the lambda calculus. Gazdar abandoned
the transformational component and the deep structure of Chomsky's model and focused on rules that analyze syntactic trees rather than
generate them. The rules translate
natural language sentences in an intensional-logic format. This way the
semantic interpretation of a sentence can be derived directly from its
syntactic representation. Gazdar
defined 43 rules of grammar, each one providing a phrase-structure rule and a
semantic-translation rule that show how to build an intensional-logic
expression from the intensional-logic expressions of the constituents of the
phrase-structure rule. Gazdars system
was fundamentally a revision of Katzs system from predicate logic to intensional logic. The US mathematician Richard
Montaguedeveloped the most sophisticated of intensional-logic approaches
to language. His intensional-logic
system employed all sorts of logical tools: type hierarchy, higher-order quantification,
lambda abstraction for all types, tenses and modal operators; and its model
theory was based on coordinate semantics. In this version of
intensional logic the sense of an expression determines its reference. The intensional-logic formula makes explicit
the mechanism by which this can happen.
Reality consists of two
truth values, a set of entities, a set of possible worlds and a set of points
in time. A function space is constructed inductively from these elementary
objects. Montagues logic determines the possible sorts of functions from possible
indices (sets of worlds, times, speakers, etc.) to their denotations (or
extensions). These functions represent the sense of the expression. In other words, sentences denote extensions
in the real world. A name denotes the
infinite set of properties of its reference. Common nouns, adjectives and
intransitive verbs denote sets of individual concepts, and their intensions are
the properties necessarily shared by all those individuals. Through a rigorously
mechanical process, a sentence of natural language can be translated into an
expression of intensional logic. The model-theoretic interpretation of this
expression serves as the interpretation of the sentence. Rather than proving a
semantic interpretation directly on syntactic structures, Montagueprovides the semantic
interpretation of a sentence by showing how to translate it into formulas of
Intensional Logic and how to interpret semantically all formulas of that logic. Montagueassigns a set of basic expressions
to each category and then defines 17 syntactic rules to combine them to form
complex phrases. The translation from
natural language to intensional logic is then performed by employing a set of
17 translation rules that correspond to the syntactic rules. Syntactic
structure determines semantic interpretation. Montagues work was based on the idea of categorial grammars pioneered by the
German mathematician Yehoshua Bar-Hillel(A Quasi-arithmetical Notation
for Syntactic Description, 1953), the man who had organised the first
conference on machine translation in 1952. Categorial grammar is built up from
two primitive categories: noun phrase and verb phrase. A sentence is composed
of a noun phrase (Piero, the red apple, the president of the US) and a verb
phrase (wrote this book, is rotting,
has canceled his trip). They can be related in an arithmetic way by using the
same rules of fractions: a verb phrase VP is the sentence S divided by the noun
phrase NP. Categorial grammars provide a unity of syntactic and semantic
analyses. Montague's semantics is truth-conditional (to know the meaning of a sentence is
to know what the world must be for the sentence to be true, or the meaning of a
sentence is the set of its truth conditions), model-theoretic and uses possible
worlds (the meaning of a sentence depends not just on the world as it is but on
the world as it might be, i.e. on other possible worlds). Back to the beginning of the chapter "Language: Minds Speak" | Back to the index of all chapters |
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