Editorial: Special issue on non-classical mathematics

The 20th century witnessed not only incredible advances in the model theory and proof theory of classical logic, but also a corresponding advance of non-classical (or, as they are also called, non-standard) logics—i.e. the logics that either non-trivially extend, or actually compete with, classical logic of the Aristotelian, Stoic and Boolean tradition. Since the beginning of the 20th century, various shortcomings of classical logic motivated the development of numerous families of alternative logical systems. Perhaps the earliest manifestation of a non-classical account of logic was in L. E. J. Brouwer's intuitionistic approach to mathematics in the beginning of the 20th century, later formalized by A. Heyting in the form of intuitionistic logic and followed by several schools of constructive mathematics. The first formally defined non-classical logic was the three-valued logic of Lukasiewicz, followed by a host of many-valued logics of wildly differing properties, finding applications in epistemology, the philosophy of vagueness or computer science (to name just a few). A criticism of material implication gave birth to modal logics, first axiomatized by C. I. Lewis; later the modal apparatus was applied to various other modalities, such as deontic, doxastic or epistemic, and extended to multimodal systems capable of formalizing multi-agent and dynamic aspects of reasoning. The non-discriminating treatment of contradictions in classical logic inspired, in view of various logical paradoxes, the development of paraconsistent logics. Considerations about belief revision, default reasoning, abduction and similar reasoning methods motivated the study of non-monotonic consequence relations, or non-monotonic logics. Several families of logics of different origins have been recognized as branches of a broader family of substructural logics—among them, relevant logics as another reply to the criticism of material implication, linear logics as resource-sensitive calculi, fuzzy logics as the logics of gradual truth and variants of the Lambek calculus, originally formalizing categorial grammar. Furthermore, numerous possible distinctions regarding existence, denotation and quantification structured the field of non-standard logics of quantification, including partial and free logics, logics of generalized and branching quantifiers, as well as various intensional and higher-order formalisms. While some of the aforementioned logics—such as intuitionistic logic or some paraconsistent or higher-order systems—have from the outset been intended to serve as a basis for non-classical mathematical reasoning, most others were originally developed as primarily propositional logics, aimed at capturing only the most fundamental reasoning patterns regarding propositional combinations of basic propositions, perhaps illustrating some philosophical point, or satisfying algebraic constraints of some …