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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 nonmonotonic 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 resourcesensitive 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 kind or other. Nevertheless, even most of the latter systems have later been extended to firstorder formalisms capable of supporting non-trivial mathematical theories. So, besides intuitionistic and constructive mathematics, which are without doubt the most advanced areas of non-classical mathematics, by the beginning of the 21st century there have been developed dozens of mathematical theories axiomatized in some non-classical logic, and so effectively using some non-classical mode of reasoning for the derivation of their theorems. To give an incomplete list of examples, there are books and papers on intuitionistic and constructive mathematics, inconsistent infinitesimal calculi and naïve set theories, relevant arithmetic, fuzzy arithmetic and fuzzy set theory, etc. with scores of researchers active in these areas.