On generalized Van Benthem-type characterizations

Abstract

Abstract. The paper continues the line of [6], [7], and [8]. This results in a model-theoretic characterization of expressive powers of arbitrary finite sets of guarded connectives of degree not exceeding 1 and regular connectives of degree 2 over the language of bounded lattices.Keywords. model theory, modal logic, intuitionistic logic, propositional logic, asimulation, bisimulation, Van Benthem's theorem. This paper is a further step in the line of our enquiries into the expressive powers of intuitionistic logic and its extensions. This line started in late 2011, when we began to think about possible modifications of bisimulation relation in order to obtain the full analogue of Van Benthem modal characterization theorem for intuitionistic propositional logic. For the resulting modification, which was published in [6], we came up with a term "asimulation", since one of the differences between asimulations and bisimulations was that asimulations were not symmetrical.Later we modified and extended asimulations in order to capture the expressive powers of first-order intuitionistic logic (in [7]) and some variants of basic modal intuitionistic logic (in [8]) viewed as fragments of classical first-order logic. Some other authors were also working in this direction; e.g. in [2] this line of research is extended to bi-intuitionistic propositional logic, although the author prefers directed bisimulations to asimulations.In this paper we publish a general algorithm allowing for an easy computation of asimulation-like notions for a class of fragments of classical first-order logic that can be naturally viewed as induced by some kind of intensional propositonal logic via the corresponding notion of standard translation. The group of appropriate intensional logics includes all of the above mentioned logics (except, for obvious reasons, the first-order intuitionistic logic) but also many other formalisms. It is worth noting that not all of these formalisms are actually extensions of intuitionistic logic, in fact, even the classical modal propositional logic which is the object of the original Van Benthem modal characterization theorem 1 , is also in this group. Thus the generalized asimulations defined in this paper have an equally good claim to be named generalized bisimulations, and if we still continue to call them asimulations, we do it mainly because for us these 1 For its formulation see, e.g. [3, Ch.1, Th. 13].

Craig Interpolation Theorem Fails in Bi-Intuitionistic Predicate Logic

“…Although the explanations there are given relative to the intuitionistic logic, all of them are also applicable to the bi-intuitionistic case. A more general and systematic inquiry into the dependence between the expressive power of a logic and a form of its characteristic simulation relation can be found in [16], where the first author basically shows, among other things, that, in the presence of classical and in the language, the symmetry of a characteristic simulation is equivalent to a presence of a connective expressing non-constant and non-monotone Boolean function.…”

Maximality of bi-intuitionistic propositional logic

“…. Moreover, preservation under bi-asimulations is known to semantically characterize BIL as a fragment of classical first-order logic, see [1,21] for proofs.…”

“…The resulting logic is known as Heyting-Brouwer or Bi-intuitionistic logic. In recent years, the study of this very natural logic has received some degree of attention from different scholars [1,2,13,14,15,21,23,24,25]. Our own work in this field has focused on the semantic study of bi-intuitionistic logic.…”

A Lindström Theorem for Intuitionistic Propositional Logic

“…The notion of an asimulation was introduced in [16] 2 and used to characterize the expressive power of intuitionistic languages as fragments of first order logic over any class of structures axiomatizable by a first order theory. Later the notion of asimulation was modified to capture also expressive powers of intuitionistic first-order logic [17], various systems of basic intuitionistic modal logic [18], and also some systems which are not connected with intuitionistic logic at all [19].…”