On the Negative Disjunction Property

In the field of intermediate logics, the concept of the disjunction property (DP)  plays an important part. Lloyd Humberstone has drawn my attention to an analogious principle  called the Negative Disjunction Property ( NDP) which applies when the disjuncts involved are negated. The author investigates the NDP in the case of intermediate propositional logics. Key words: intermediate logic, disjunction property, negative disjunction property, Heyting algebra, Janko

On the Negative Disjunction Property

In the field of intermediate logics, the concept of the disjunction property (DP)  plays an important part. Lloyd Humberstone has drawn my attention to an analogious principle  called the Negative Disjunction Property ( NDP) which applies when the disjuncts involved are negated. The author investigates the NDP in the case of intermediate propositional logics. Key words: intermediate logic, disjunction property, negative disjunction property, Heyting algebra, Janko

Sub-classical Boolean Bunched Logics and the Meaning of Par

We investigate intermediate logics between the bunched logics Boolean BI and Classical BI, obtained by combining classical propositional logic with various flavours of Hyland and De Paiva\u27s full intuitionistic linear logic. Thus, in addition to the usual multiplicative conjunction (with its adjoint implication and unit), our logics also feature a multiplicative disjunction (with its adjoint co-implication and unit). The multiplicatives behave "sub-classically", in that disjunction and conjunction are related by a weak distribution principle, rather than by De Morgan equivalence. We formulate a Kripke semantics, covering all our sub-classical bunched logics, in which the multiplicatives are naturally read in terms of resource operations. Our main theoretical result is that validity according to this semantics coincides with provability in a corresponding Hilbert-style proof system. Our logical investigation sheds considerable new light on how one can understand the multiplicative disjunction, better known as linear logic\u27s "par", in terms of resource operations. In particular, and in contrast to the earlier Classical BI, the models of our logics include the heap-like memory models of separation logic, in which disjunction can be interpreted as a property of intersection operations over heaps

Structural completeness in propositional logics of dependence

In this paper we prove that three of the main propositional logics of dependence (including propositional dependence logic and inquisitive logic), none of which is structural, are structurally complete with respect to a class of substitutions under which the logics are closed. We obtain an analogues result with respect to stable substitutions, for the negative variants of some well-known intermediate logics, which are intermediate theories that are closely related to inquisitive logic

Multiple Conclusion Rules in Logics with the Disjunction Property

We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics

Uniform Definability in Propositional Dependence Logic

Both propositional dependence logic and inquisitive logic are expressively complete. As a consequence, every formula with intuitionistic disjunction or intuitionistic implication can be translated equivalently into a formula in the language of propositional dependence logic without these two connectives. We show that although such a (non-compositional) translation exists, neither intuitionistic disjunction nor intuitionistic implication is uniformly definable in propositional dependence logic