Maximality of bi-intuitionistic propositional logic

Abstract

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 the previous work done by the authors for intuitionistic logic (Badia and Olkhovikov, 2020, Notre Dame Journal of Formal Logic, 61, 11–30).

Nature of Knowledge in Philosophy

“…These requirements include objectivity, subjectivity, and evidence. The objective requirement is the 'truth condition' which implies that for anyone to claim knowledge of any proposition, such a proposition must be initially true [23,24]. It is epistemically inappropriate to claim knowledge of something which is false.…”

“…The second, to that, is knowledge by reasoning which stresses the act of inferring new knowledge from what has already been known. This type of knowledge cannot be established using physical evidence but by reasoning or logic [23,38]. For example, given the fact that there is a teacher implies that there must be a learner.…”

Craig Interpolation Theorem Fails in Bi-Intuitionistic Predicate Logic

“…In the 1970s, Rauszer started an intense study of various technical aspects of both propositional and predicate bi-intuitionistic logic in a series of interesting articles spanning over a decade [18–23]. This work has been picked up in recent years by a number of scholars [1–4, 6, 7, 9, 13, 17, 26, 27] and results on all sorts of proof-theoretic and model-theoretic properties of these systems have been produced.…”

“…

In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication φ → with no interpolant.

…”

“…sadly neglected in the history of the subject and it contains valuable techniques such as a version of the unraveling construction used in [13] (see [8,Definition 4.1]). Let us note, however, that the central result in the present paper does not actually require any appeal to completeness.…”