The mathematician G.F.C. Griss is known for his program of negationless intuitionistic mathematics. Although Griss’s rejection of negation is regarded as characteristic of his philosophy, this is a consequence of an executability requirement that mental constructions presuppose agents’ executing corresponding mental activity. Restoring Griss’s executability requirement to a central role permits a more subtle characterization of the rejection of negation, according to which D. Nelson’s strong constructible negation is compatible with Griss’s principles. This exposes a ‘holographic’ theory of negation in (...) negationless mathematics, in which a full theory of negation is ‘flattened’ in a putatively negationless setting. (shrink)

Although the work of G.F.C. Griss is commonly understood as a program of negationless mathematics, close examination of Griss’s work suggests a more fundamental feature is its executability, a requirement that mental constructions are possible only if corresponding mental activity can be actively carried out. Emphasizing executability reveals that Griss’s arguments against negation leave open several types of negation—including D. Nelson’s strong negation—as compatible with Griss’s intuitionism. Reinterpreting Griss’s program as one of executable mathematics, we iteratively develop a pair of (...) bilateral constructive logics and argue for their adequacy as accounts of the propositional basis of Griss’s work. We conclude by observing connexive features exhibited by the two bilateral logics and by investigating the difficulties connexive principles reveal for the development of executable mathematics. (shrink)

The Craig interpolation theorem is shown for an extended LJ with strong negation. A new simple proof of this theorem is obtained. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The paper extends a framework of information-based semantics for intuitionistic logic with a paraconsistent negation and consistency-sensitive epistemic modalities. In this framework information states represent information collected from various sources and as such they can be inconsistent because they receive contradictory information either from a single inconsistent source or from various mutually incompatible sources. The modalities reflect only those sources that are consistent and trusted. For the paraconsistent logic generated by this framework, we present a Hilbert style system, we prove (...) completeness, the disjunction property, the finite model property and various other results. (shrink)