This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993 ) concerning the modelling conditions for the axioms of assertion A → (( A → B ) → B ) (there called c 6) and permutation ( A → ( B → C )) → ( B → ( A → C )) (there called c 7). We show that the modelling conditions for assertion and permutation proposed (...) in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001 ), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections. (shrink)
This document collects natural derivation systems for logics described in Priest, An Introduction to Non-Classical Logic . It provides an alternative or supplement to the semantic tableaux of his text. Except that some chapters are collapsed, there are sections for each chapter in Priest, with an additional, final section on quantified modal logic. In each case, (i) the language is briefly described and key semantic definitions stated, (ii) the derivation system is presented with a few examples given, and (iii) soundness (...) and completeness are proved. There should be enough detail to make the parts accessible to students would work through parallel sections of Priest. (shrink)
In this short paper, I introduce two central notions for argument evaluation. The presentation is completely informal. It is possible to develop formal methods for working with validity and souneness, but it is also possible to apply the informal notions directly to problems in philosophy and beyond. In either case, it is important to understand the basic notions, in order to understand what is accomplished in reasoning. Exercises are included, with answers to selected exercises at the end.
In this short paper, I discuss certain aspects of a “common-sense” approach to truth and falsity. It is my experience that many will object to what I have to say. As you read, if you have objections, try to formulate them carefully, and ask yourself whether I attempt a reply.
Involving as it does impossible worlds and the like, the Routley-Meyer worlds semantics for relevant logic has seemed unmotivated to some. I set a version of relevant semantics in a context to make sense of its different elements. Suppose a view which makes room for structured properties — or related entities which combine in arbitrary ways to form structured ones. Then it may seem natural to say entailment supervenes upon the structures, so that P entails Q just when part of (...) the condition for being p is being q. If P stands in this relation to Q, a result is that there is no possible world where P but not Q, so that P classically entails Q. But the conditions are not equivalent. For all possible worlds, but not all properties, are maximal and consistent. I suggest that relevant semantics is naturally seen as modeling entailment grounded in property structure and makes sense insofar as it reflects this fundamental and intuitive notion. (shrink)