Bi-Simulating in Bi-Intuitionistic Logic

Abstract

Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindström and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary chains.

On generalized Van Benthem-type characterizations

“…Similarly, the rest functions which are not F T F -functions we will call ∃-special. 2 Further, Boolean functions (not necessarily rest functions) which are not F T F -functions we will call weakly ∃-special, whereas Boolean functions (again, not necessarily rest functions) which are not T F T -functions we will call weakly ∀-special. Thus, every non-rest function is both weakly ∀-special and weakly ∃-special.…”

“…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.…”

Infinitary Propositional Relevant Languages With Absurdity

“…This notation should not be confused with the classical notation where the second subscript is used to bound the possible length of a string of quantifiers. 3 An example of a connective definable in L Ñ 8ω (but not in L Ñ ωω ) is ω Ý Ñ (iterated entailment), which was introduced by Humberstone (see [10], p. 36). The formula φ ω Ý Ñ ψ means that for some natural number n ą 1,…”

“…By considering restricted classes of Routley-Meyer structures where the relation R has certain properties and only some valuations are admitted, we can get classes of models corresponding to a number of formal systems of relevant logic like B, T or R. Next we will consider some famous examples from [26]. 3 It is opaque whether there is a connection here. For instance, φ Ñ pφ Ñ p. .…”

“…This notation should not be confused with the classical notation where the second subscript is used to bound the possible length of a string of quantifiers. 3 An example of a connective definable in…”