In this chapter we give the basics of hybrid logic. The chapter is structured as follows. In the first section of the chapter we give an informal motivation of hybrid logic. In the second section we give the formal syntax and semantics and we give translations forwards and backwards between hybrid logic and first-order logic. In the third section we discuss the work of Arthur Prior and describe how...
In this chapter we introduce the proof-theory of propositional hybrid logic. The chapter is structured as follows. In the first section of the chapter we sketch the basics of natural deduction systems and in the second section we introduce a natural deduction system for hybrid logic. In the third section we sketch the basics of Gentzen systems and in the fourth section we introduce a Gentzen system...
Based on tableau systems, we in this chapter prove decidability results for hybrid logic using tableau systems. The chapter is structured as follows. In the first section of the chapter we sketch the basics of tableau systems. In the second section we give a tableau-based decision procedure for a very expressive hybrid logic including the universal modality. In the third section we show how the decision...
In this chapter we compare and contrast the natural deduction system given in Section 2.2 to a modified version of a hybrid-logical natural deduction system given by Jerry Seligman. The chapter is structured as follows. In the first section of the chapter we describe the natural deduction systems under consideration, in particular, we define our version of Seligman’s system. In the second and third...
In this chapter we prove a functional completeness result for the hybrid logic of the universal modality. The chapter is structured as follows. In the first section of the chapter we describe the natural deduction system under consideration, in the second section we give an introduction to the notion of functional completeness, and in the third section we give general rule schemas for natural deduction...
In this chapter we introduce first-order hybrid logic and its proof-theory. The chapter is structured as follows. In the first section of the chapter we introduce first-order hybrid logic. In the second section we introduce a natural deduction system for first-order hybrid logic (taken from Braüner (2005b)) and in the third section we introduce an axiom system for first-order hybrid logic (also taken...
In this chapter we introduce intensional first-order hybrid logic and its proof-theory. This chapter is structured as follows. In the first section of the chapter we introduce intensional first-order hybrid logic, including two different kinds of models—standard models and generalized models. In the second section we introduce a natural deduction system which is complete with respect to generalized...
In this chapter we introduce intuitionistic hybrid logic and its proof-theory. Intuitionistic hybrid logic is hybrid modal logic over an intuitionistic logic basis instead of a classical logical basis. The chapter is structured as follows. In the first section of the chapter we introduce intuitionistic hybrid logic (this is taken from Braüner and de Paiva (2006)). In the second section we introduce...
In this chapter we compare the hybrid-logical natural deduction system given in Section 2.2 to a labelled natural deduction system for modal logic. The chapter is structured as follows. In the first section of the chapter we describe the labelled natural deduction system under consideration and in the second section we define a translation from this system to the hybrid-logical natural deduction system...
The material in this chapter is primarily of a conceptual nature, the goal of the chapter being to put into perspective hybrid logic and the proof-theory of hybrid logic. The chapter is structured as follows. In the first section we explicate what we mean when we say that a proof-system is well-behaved. In the second section we shall try to give an answer to the following question: Why does the proof-theory...