This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs (...) of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

Ontology is the philosophical discipline which aims to understand how things in the world are divided into categories and how these categories are related together. This is exactly what information scientists aim for in creating structured, automated representations, called 'ontologies,' for managing information in fields such as science, government, industry, and healthcare. Currently, these systems are designed in a variety of different ways, so they cannot share data with one another. They are often idiosyncratically structured, accessible only to those who (...) created them, and unable to serve as inputs for automated reasoning. This volume shows, in a nontechnical way and using examples from medicine and biology, how the rigorous application of theories and insights from philosophical ontology can improve the ontologies upon which information management depends. (shrink)

The Foundational Model of Anatomy (FMA) symbolically represents the structural organization of the human body from the macromolecular to the macroscopic levels, with the goal of providing a robust and consistent scheme for classifying anatomical entities that is designed to serve as a reference ontology in biomedical informatics. Here we articulate the need for formally clarifying the is-a and part-of relations in the FMA and similar ontology and terminology systems. We diagnose certain characteristic errors in the treatment of these relations (...) and show how these errors can be avoided through adoption of the formalism we describe. We then illustrate how a consistently applied formal treatment of taxonomy and partonomy can support the alignment of ontologies. (shrink)

This paper provides an axiomatic formalization of a theory of foundational relations between three categories of entities: individuals, universals, and collections. We deal with a variety of relations between entities in these categories, including the is-a relation among universals and the part-of relation among individuals as well as cross-category relations such as instance-of, member-of, and partition-of. We show that an adequate understanding of the formal properties of such relations – in particular their behavior with respect to time – is critical (...) for formal ontology. We provide examples to support this thesis from the domain of biomedicine. (shrink)

Reviewed Works:Gaisi Takeuti, Proof Theory.Georg Kreisel, Proof Theory: Some Personal Recollections.Wolfram Pohlers, Contributions of the Schutte School in Munich to Proof Theory.Stephen G. Simpson, Subsystems of $\mathbf{Z}_2$ and Reverse Mathematics.Solomon Feferman, Proof Theory: A Personal Report.

This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.

This paper embeds the core part of Discourse Representation Theory in the classical theory of types plus a few simple axioms that allow the theory to express key facts about variables and assignments on the object level of the logic. It is shown how the embedding can be used to combine core analyses of natural language phenomena in Discourse Representation Theory with analyses that can be obtained in Montague Semantics.

Many types of part-whole relations have been proposed in the literature to aid the conceptual modeller to choose the most appropriate type, but many of those relations lack a formal specification to give clear and unambiguous semantics to them. To remedy this, a formal taxonomy of types of mereological and meronymic part-whole relations is presented that distinguishes between transitive and intransitive relations and the kind of entity types that are related. The demand to use it effectively brings afore new requirements (...) for automated reasoning over a hierarchy of relations. To ensure logically and ontologically correct inferencing over both the class and role hierarchy, the new reasoning service RBox compatibility for Description Logics reasoners is introduced. The proposed combination of formal semantics and the new reasoning service will improve the representation of the application domain when using part-whole relations in conceptual models and ontologies. (shrink)

This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will (...) be essential reading for all those working in logic & computer science. (shrink)

The Description Logic Handbook covers all aspects of the research in the field of knowledge representation. Written by some of the most prominent researchers in the field, and covering the basic technical material and implementational aspects, it is both a unique reference and a self-study guide.

Here, the author, develops a type theory, studies its properties, and explains its uses in applications to computer science. In particular, type theory is shown to offer a powerful and uniform language for programming, program specification and development, and logical reasoning.

To enhance the treatment of relations in biomedical ontologies we advance a methodology for providing consistent and unambiguous formal definitions of the relational expressions used in such ontologies in a way designed to assist developers and users in avoiding errors in coding and annotation. The resulting Relation Ontology can promote interoperability of ontologies and support new types of automated reasoning about the spatial and temporal dimensions of biological and medical phenomena.

John Searle began to discuss his recently published book `The Construction of Social Reality' with Anthony Freeman, and they ended up talking about God. The book itself and part of their conversation are introduced and briefly reflected upon by Anthony Freeman. Many familiar social facts -- like money and marriage and monarchy -- are only facts by human agreement. They exist only because we believe them to exist. That is the thesis, at once startling yet obvious, that philosopher John Searle (...) explores in The Construction of Social Reality. (shrink)

I will explore possibilities for formulating linguistic semantics in terms of records and record types of the kind used in recent developments of Martin-L¨of type theory (Betarte, 1998, Betarte and Tasistro, 1998, Coquand, Pollock and Takeyama, 2003, Tasistro, 1997). I will suggest that they give us the tools to develop a theory which includes aspects of Montague semantics, using the lambda calculus1, Discourse Representation Theory (DRT)2, situation semantics3 and Head-Driven Phrase Structure Grammar (HPSG)4 in a single theory. I will also (...) argue that formulating these theories in terms of record types may provide us not only with a unified approach but also with certain improvements over the individual theories. (shrink)