A constructive approach to state description semantics

Dedication: To Ed Keenan, a champion of model-theoretic semantics, on the occasion of his retirement, with best wishes for a long continual of scientific work.

Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

2021

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to that formalization purpose, of some modal schemes, such as S4 and S5. It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon.

CILC09-24 Convegno Italiano di Logica …, 2009

Following the approaches and motivations given in recent works about constructive interpretation of description logics, we introduce the constructive description logic KALC. This logic is based on a Kripke-style semantics inspired by the Kripke semantics for Intuitionistic first order logic. In the paper we present the main features of our semantics and we study its relations with other approaches. Moreover, we present a tableau calculus which turns out to be sound and complete for KALC.

Electronic Notes in Theoretical Computer Science, 2011

Constructive modal logics come in several different flavours and constructive description logics, not surprisingly, do the same. We introduce an intuitionistic description logic, which we call iALC (for intuitionistic ALC, since ALC is the name of the canonical description logic system) and provide axioms, a Natural Deduction formulation and a sequent calculus for it. The system iALC is related to Simpsonʼs constructive modal logic IK the same way Mendler and Scheeleʼs cALC is related to constructive CK and in the same ...

Electronic Notes in Theoretical Computer Science, 2009

We give a comprehensive formal representation of first-order logic using the recently developed module system for the Twelf implementation of the Edinburgh Logical Framework LF. The module system places strong emphasis on signature morphisms as the main primitive concept, which makes it particularly useful to reason about structural translations, which occur frequently in proof and model theory. Syntax and proof theory are encoded in the usual way using LF's higher order abstract syntax and judgments-as-types paradigm, but using the module system to treat all connectives and quantifiers independently. The difficulty is to reason about the model theory, for which the mathematical foundation in which the models are expressed must be encoded itself. We choose a variant of Martin-Löf's type theory as this foundation and use it to axiomatize first-order model theoretic semantics. Then we can encode the soundness proof as a signature morphism from the proof theory to the model theory. We extend our results to models given in terms of set theory using an encoding of Zermelo-Fraenkel set theory in LF and giving a signature morphism from Martin-Löf type theory into it. These encodings can be checked mechanically by Twelf. Our results demonstrate the feasibility of comprehensively formalizing large scale representation theorems and thus promise significant future applications.

2014

Formal semantics based on Modern Type Theories (MTTs) provides us with not only a viable alternative to Montague Grammar, but potentially an attractive full-blown semantic tool with advantages in many respects. We shall introduce the MTT-based semantics and then study several issues such as adjectival modification, co-predication and coordination. Key comparisons to Montague Grammar are done all along, discussing the advantages of MTTs over simple type theory. For example, subtyping is crucially needed for proper semantic treatments of some linguistic features but has proven difficult in a Montagovian setting; coercive subtyping is adequate for MTTs and has become a key foundation for the MTTbased semantics. MTTs have been implemented in proof assistants, which can be used to implement MTT-based semantics and hence conduct computer-assisted reasoning in natural language. We shall consider NL inference implemented in Coq to show that such activities can be supported effectively.

2020

In this paper, we argue that formal semantics based on modern type theories (MTT-semantics) is both model-theoretic and proof-theoretic, and hence has unique advantages as a semantic framework. Being model-theoretic, it provides a wide coverage of various linguistic features partly because the rich type structure in MTTs can be used effectively to represent various collections, playing a role as sets do in Montague’s model-theoretic semantics. Being proof-theoretic, its foundational languages have a proof-theoretic meaning theory and provide a solid foundation for natural language reasoning using proof assistants. After presenting the basic arguments, we shall then focus on further development of the first, and arguably less understood, aspect: MTTsemantics is model-theoretic. We shall develop a notion of signature to allow new forms of subtyping and definitional entries and show that such formal contextual tools support useful ways of representing incomplete possible worlds in sema...

2014

The definition of identity in terms of other logical symbols is a recurrent issue in logic. In particular, in First-Order Logic (FOL) there is no way of defining the global relation of identity, while in standard Second-Order Logic (SOL) this definition is not only possible, but widely used. In this paper, the reverse question is posed and affirmatively answered: Can we define with only equality and abstraction the remaining logical symbols? Our present work is developed in the context of an equational hybrid logic (i.e. a modal logic with equations as propositional atoms enlarged with the hybrid expressions: nominals and the @ operator). Our logical base is propositional type theory. We take the propositional equality, λ abstraction, nominals, ♦ and @ operators as primitive symbols and we demonstrate that all of the remaining logical symbols can be defined,ion the remaining logical symbols? Our present work is developed in the context of an equational hybrid logic (i.e. a modal log...