Comments on the Logic of Constructible

Nelson has presented a constructive arithmetic with a negation operation (−) different from the ordinary intuitionistic one (¬). In [5] he presents a variant of Kleene's realization semantics for intuitionistic arithmetic, and proves that relative to this interpretation the arithmetic language with – has the same expressive power as the usual intuitionistic one, and fact certain theories of arithmetic incorporating his negation are equivalent to corresponding systems of intuitionistic arithmetic. A Fitch style natural deduction formulation (cf.[1])) of the pure first order logic that may be extracted, from Nelson's work may be obtained from that for intuitionistic first order logic by first dropping the intuitionistic rule of negation introduction and then adding rules guaranteeing the De Morgan equivalences, their quantifier analogues, and the equivalence of anegated implication with the conjunction of its antecedent and the negation of its consequent. Given the De Morgan equivalences, conjunction and the existential quantifier may he treated as defined operators. Thomason, in [8], develops a classical model theory for what he claims is the first order logic of [5]. It is similar to his and Kripke's model theory for intuitionistic predicate logic (cf. [4], [7]), with appropriate changes in the treatment of negation, but holding the domain of objects " existing " at the various " stages " constant. Now, Görnemann (in [2]) and others (cf. references in [2]) have shown that holding the domain constant in the model theory of intuitionistic logic characterizes the predicate logic obtained from intuitionistic logic by adding the intuitionistically invalid rule D: from ∀x(F x ∨ A), where x has no free occurrence in A, to infer (∀x(F x) ∨ A). It is thus no surprise that the logic of [8] is stronger than the one I have described, and validates D (it may in fact be obtained from the