Transreal Logical Space of All Propositions

Transreal numbers provide a total semantics containing classical truth values, dialetheaic, fuzzy and gap values. A paraconsistent Sheffer Stroke generalises all classical logics to a paraconsistent form. We introduce logical spaces of all possible worlds and all propositions. We operate on a proposition, in all possible worlds, at the same time. We define logical transformations, possibility and necessity relations, in proposition space, and give a criterion to determine whether a proposition is classical. We show that proofs, based on the conditional, infer gaps only from gaps and that negative and positive infinity operate as bottom and top values.

In: 

Haeng Kon Kim; Mayhar A. Amouzegar; Sio-Iong Ao. Transactions on Engineering Technologies, World Congress on Engineering and Computer Science. London: Springer, pp. 227-242, 2015.