[with Tony Roy] “On Permutation in Simplified Semantics,” in progress.
This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Journal of Philosophical Logic 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) and permutation (A → (B → C)) → (B → (A → C)). 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.
In this note, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose a correction.
Comments
Don’t follow the reasoning. Maybe I have misunderstood. The counterexample to an invalid argument doesn’t have to be in the base world g. It can be in any world. Otherwise, how are you going to find a counterexample to the invalid argument from A to B->B?
Have I missed the point?
Posted by: John Slaney at April 14, 2006 08:59 PM
I think our explanation in the beginning is not quite as clear as it should be. Validity defined as truth preservation at the base world isn’t the standard relevant validity. So, it’s true that the models defined in my ‘Simplified Semantics’ don’t validate disjunctive syllogism in the usual relevant sense. However, they are restricted to models in which base worlds are consistent, so they provide no models for inconsistent but non-trivial extensions of R.
Posted by: Greg Restall at April 15, 2006 09:43 PM
Don’t follow the reasoning. Maybe I have misunderstood. The counterexample to an invalid argument doesn’t have to be in the base world g. It can be in any world. Otherwise, how are you going to find a counterexample to the invalid argument from A to B->B? Have I missed the point?
Posted by: John Slaney at April 14, 2006 08:59 PM