In Rogerson and Restall’s, the “class of implication formulas known to trivialize ” is recorded. The aim of this paper is to show how to invalidate any member in this class by using “weak relevant model structures”. Weak relevant model structures verify deep relevant logics only.

We collect together some misgivings about the logic R of relevant inplication, and then give support to a weak entailment logic $DJ^{d}$ . The misgivings centre on some recent negative results concerning R, the conceptual vacuousness of relevant implication, and the treatment of classical logic. We then rectify this situation by introducing an entailment logic based on meaning containment, rather than meaning connection, which has a better relationship with classical logic. Soundness and completeness results are proved for $DJ^{d}$ with respect (...) to a content semantics, which embraces the concept of meaning containment. (shrink)

Normal 0 21 false false false PT-BR X-NONE X-NONE MicrosoftInternetExplorer4 Normal 0 21 false false false PT-BR X-NONE X-NONE MicrosoftInternetExplorer4 Normative fragment of natural language make up sentences that express acts and describe norms. In this fragment there are criteria of logic thuth and relation of consequence between sentences which constitute a natural deontic logic. This paper adopts at ranslation function from the set of sentences of the normative fragment of natural language in to the set of formulae in the (...) formal language and claims that such function translates logically true sentences of the natural language into provable formulae of the formal calculus. With Von Wright's deontic calculus (1951), it does not fit and generates paradoxes, which are known as Prior's paradoxes. Cruz's paraconsistent deontic propositional calculus, D 1 (1993) avoids some paradoxes, except that generated by the formula OB-O (A- B). One builds a relevant deontic propositional calculus that aims to avoid these paradoxes and keeps intact all other fundamental features of deontic operators, since the formula B - (A - B) is improable in some relavant calculi.  . (shrink)

The rules for Core Logic are stated, and various important results about the system are summarized. We describe its relationship to other systems, such as Classical Logic, Intuitionistic Logic, Minimal Logic, and the Anderson–Belnap relevance logicR. A precise, positive explication is offered of what it is for the premises of a proof to connect relevantly with its conclusion. This characterization exploits the notion of positive and negative occurrences of atoms in sentences. It is shown that all Core proofs are relevant (...) in this precisely defined sense. We survey extant results about variable-sharing in rival systems of relevance logic, and find that the variable-sharing conditions established for them are weaker than the one established here for Core Logic (and for its classical extension). Proponents of other systems of relevance logic (such asRand its subsystems) are challenged to formulate a stronger variable-sharing condition, and to prove thatRor any of its subsystems satisfies it, but that Core Logic does not. We give reasons for pessimism about the prospects for meeting this challenge. (shrink)

“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.