A Lindström Theorem for Intuitionistic Propositional Logic

Abstract

It is shown that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain topological property reminiscent of compactness, the Tarski union property and preservation under asimulations.

Maximality of bi-intuitionistic propositional logic

“…If Θ is a signature and M is a bi-intuitionistic Kripke Θ-model, 3 then W is a nonempty set of worlds, states, or nodes, ≺ is a partial order on W called M's accessibility relation, and V is the evaluation function for Θ in M, that is to say, a function V : Θ → 2 W such that for every p ∈ Θ and arbitrary s, t ∈ W , it is true that:…”

“…In the context of propositional intuitionistic logic, the authors of the present paper obtained such a theorem in [3] using the Tarski union property, a form of compactness and preservation under asimulations (inspired by work done for modal logic in [9]). Since bi-intuitionistic logic is such a natural companion to intuitionistic logic, the question immediately arose whether a similar theorem would hold for bi-intuitionistic logic replacing asimulations (introduced in [20]) by an appropriate bi-intuitionistic analogue (such as that in [3]).…”

“…In this article we will adapt the somewhat delicate arguments from [3] to a biintuitionistic context. One of the most substantial difficulties of this task is the backwards looking modality, making the unraveling construction, which plays a central role in our proof, more complex.…”

“…Hence, to use it the way we need it requires new special care. 1 Finally, the paper is self-contained and it will not require any familiarity with [3] on behalf of the reader (which, of course, means that we will repeat even the parts from [3] which are rather easily adaptable, and not just the non-trivial parts).…”

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In the style of Lindström's theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of previous work done by the authors for intuitionistic logic [3].

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