Representing model theory in a type-theoretical logical framework

  • Authors:

  • Fulya Horozal;Florian Rabe

  • Affiliations:

  • -;-

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2011

Quantified Score

Hi-index 5.23

Visualization

Abstract

In a broad sense, logic is the field of formal languages for knowledge and truth that have a formal semantics. It tends to be difficult to give a narrower definition because very different kinds of logics exist. One of the most fundamental contrasts is between the different methods of assigning semantics. Here two classes can be distinguished: model theoretical semantics based on a foundation of mathematics such as set theory, and proof theoretical semantics based on an inference system possibly formulated within a type theory. Logical frameworks have been developed to cope with the variety of available logics unifying the underlying ontological notions and providing a meta-theory to reason abstractly about logics. While these have been very successful, they have so far focused on either model or proof theoretical semantics. We contribute to a unified framework by showing how the type/proof theoretical Edinburgh Logical Framework (LF) can be applied to the representation of model theoretical logics. We give a comprehensive formal representation of first-order logic, covering both its proof and its model theoretical semantics as well as its soundness in LF. For the model theory, we have to represent the mathematical foundation itself in LF, and we provide two solutions for that. Firstly, we give a meta-language that is strong enough to represent the model theory while being simple enough to be treated as a fragment of untyped set theory. Secondly, we represent Zermelo-Fraenkel set theory and show how it subsumes our meta-language. Specific models are represented as LF morphisms. All representations are given in and mechanically verified by the Twelf implementation of LF. Moreover, we use the Twelf module system to treat all connectives and quantifiers independently. Thus, individual connectives are available for reuse when representing other logics, and we obtain the first version of a feature library from which logics can be pieced together. Our results and methods are not restricted to first-order logic and scale to a wide variety of logical systems, thus demonstrating the feasibility of comprehensively formalizing large scale representation theorems in a logical framework.