Constructive predicate logic with strong negation and model theory.

Answer sets and nonmonotonic S4

“…In general, database rules like (1) are not interpreted as logical formulas and the answer sets of a database are characterised by a fixpoint definition, without reference to any underlying logical system. Even the notation is carefully chosen so as to minimise confusion with the customary logical connectives.…”

“…This idea was first carried out by Gelfond for general logic programs whose rules (1) contain neither disjunction or nor strong negation. His embedding [6] into autoepistemic logic made the plausible assumption that weak negation, as 'failureto-prove', can be regarded as a modal concept, so that the subformula 'notA' of a program rule (where A is an atom) is translated by ',,~ BA', where 'B' is a modal or doxastic belief operator.…”

“…For reasons of space we do not present here a formal system corresponding to N. The reader is referred instead to [l 1,5], where axiom systems are presented. These are obtained from the standard axioms for intuitionistic predicate logic by adding new axioms governing strong negation, '~'.…”

Nelson's paraconsistent logics

“…The reader is also referred to Akama [8] and Almukdad and Nelson [9] for sequential formulations of Nelson's logics. A Kripke semantics for strong negation was developed by Thomason [23]; also see Gurevich [13] and Akama [3,5]. A strong negation model for N − is of the form (W, ≤, w 0 , val , D), where W is a set of worlds with the distinguished world w 0 such that for all w ∈ W : w 0 ≤ w; ≤ is a reflexive and transitive relation on W × W , val is a three-valued valuation assigning 1 (true) or 0 (false), −1 (undefined) to the atomic formula p(t) at w ∈ W with parameter t ∈ D(w) satisfying:…”

“…Alternatively, we can use a four-valued valuation in a strong negation model for N − . A completeness proof for N may be found in Akama [3,5]. Thomason [23] proved that N with the constant domain axiom (CD): ∀x(A(x)∨B) → (∀xA(x)∨B), where x is not free in B, has a Kripke semantics with constant domains.…”