## Media and Meaning: A Schematic Approach to Representational Semantics and its Applications

Media and Meaning: A Schematic Approach to Representational Semantics and its Applications is a Ph.D. dissertation written by Alan Bush.

Principal Adviser: Professor John Etchemendy. Advisers: Professor Jon Barwise, Professor John Perry, Professor Johan van Benthem

Informally, this dissertation is an exploration of why logic works and how to create new approaches to logical semantics. It explains Tarski's model-theoretic semantics which were created to model the meaning of sentences. It further defines and describes new approaches to logical semantics which have certain advantages for the modeling the meaning of object-oriented data.

(The following is an excerpt from the Introduction. To download a copy of the dissertation, click here).

“‘What is logic?’ Any language, regardless of its expressive devices, gives rise to a consequence relation, a relation that supports inferences from sentences in the language to other sentences in the language. The study of this relation is the study of the logic of that language” (Etchemendy 1999, 21).

A central aim of logic is the development of theoretical perspectives on the phenomenon of consequence. Carrying out this project includes proposing an answer to the question of what logical consequence is, and giving methods for constructing models of logical consequence for specific languages. The methodology of model-theoretic semantics developed by Alfred Tarski is the generally accepted technique for constructing mathematical models of logical consequence. As argued by John Etchemendy, Tarski interpreted his construction of model-theoretic semantics as giving an analysis of the logical consequence relation, reducing the concept of logical consequence to the simpler concepts of generalization and satisfaction (Etchemendy 1990; Tarski 1956). Etchemendy emphatically praises the power, importance and value of model-theoretic semantics. But at the same time, he demonstrates that Tarski’s reductive interpretation of model-theoretic semantics is seriously flawed, failing both conceptually and extensionally.

Etchemendy points us in the direction of a different way of thinking about the relationship of model-theoretic semantics to the concept of logical consequence and the task of modelling consequence for specific languages. He shows us how to see model-theoretic semantics as a technique for illuminating the consequence relation for specific languages of assertion; provided that we begin by making certain basic assumptions about what consequence is. On Etchemendy’s view, model-theoretic semantics does not reduce logical consequence to more basic concepts. Instead, the model-theoretic approach relies on a prior understanding of logical consequence in general as a way of enabling the construction of models of logical consequence for particular languages. This insight and general approach are at the core of the work in this dissertation.

Etchemendy uses the term “representational semantics” to identify the approach to logical consequence he advocates. We are going to take Etchemendy’s account of how the technique of model-theoretic semantics implements a representational semantics as the starting point of our investigations. We will abstract away from the particulars of that account, and construct a general conceptual framework we call the “representational schema.” The representational schema gives a general form for techniques used to construct theories of logical consequence implementing representational semantics. We use the term “schema” in the sense of a patterned arrangement of constituents within a specified system. In the course of this dissertation, we will use this schema to help us understand, apply, compare, and create a variety of techniques for constructing representational semantics.

There are two natural tests for any proposed general schema. First, does it capture additional instances beyond the one from which it was abstracted? In this case, additional instances would be other techniques for constructing representational theories of consequence beyond the specific model-theoretic approach described by Etchemendy. Secondly, is it productive? That is to say, does the schema increase our understanding of existing instances, and does it enable the development of new ones?

In the pages below, we will show how the representational schema is capable of subsuming not just model-theoretic semantics, but also a class of techniques for constructing theories of logical consequence whose central concept can be abstracted from a corollary to Lindenbaum’s Lemma; a class we call order-consistency semantics. We will use the schema to describe a general methodology for applying representational techniques to construct theories of logical consequence for arbitrary (partially or fully) interpreted languages. That methodology will be used to apply (or outline the application of) techniques subsumed by the representational schema to a number of interpreted languages, including propositional logic, feature logics (sentential languages with feature structures as models), and languages in which feature structures are considered as assertions in their own right. We will show how the schema helps us to compare and contrast differing techniques of representational semantics across a number of important dimensions: including the mode by which they explain the consequence relation; the range of interpreted languages to which they are applicable; their degree of epistemological commitment; and the ease with which they can be used in particular applications. We will further see how the representational schema serves as a base point for various vectors of extension, and use the schema to construct several new techniques of representational semantics. One will reduce the epistemological commitments of the specific model-theoretic technique described by Etchemendy, another will give a model-theoretic technique capable of handling partial models, and a third will be an order-consistency technique which does not require the assumption of Lindenbaum’s Lemma (and thus does not require the assumption of maximal extensions). Particular applications and potential benefits of these extensions will be identified and described.

Works cited above:

Etchemendy, John. 1990. The concept of logical consequence. Cambridge, MA: Harvard University Press; reprint CSLI Publications and Cambridge University Press, 1999.

———. 1999. Reflections on consequence. Unpublished.

Tarski, Alfred. 1956. On the concept of logical consequence. In Logic, semantics, metamathematics. Oxford: Oxford University Press. (A translation from Tarski, Alfred, 1936. O pojciu wynikania logicznego. Prezglad Filozoficzny 39: 97-112.)

Copyright © 1999. Alan Bush. All rights reserved.