Piero Scaruffi(Copyright © 2013 Piero Scaruffi | Legal restrictions )
These are excerpts and elaborations from my book "The Nature of Consciousness"
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. Gazdar’s system was fundamentally a revision of Katz’s system from predicate logic to intensional logic.
The US mathematician Richard Montague developed 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.
Montague’s 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, Montague provides 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.
Montague assigns 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.
Montague’s 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).
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