(The first 2 paragraphs here are set-up; those in the know can start at the third paragraph.) I teach a class called "Critical Thinking and Reasoning" most semesters. It covers argument recognition, reconstruction, and evaluation. At the very end of the semester, we talk about how one can show a particular argument form is valid. For propositional/ sentential logic, we use truth tables: if there is no row of the truth table where all the premises are true and the conclusion false, then the argument form is valid. For categorical logic, we use Venn diagrams. If you are not familiar with how this works, it's very straightforward and simple; here is a quick introduction for the unacquainted. The basic idea is that you diagram all the premises on a single diagram, and then check whether you have 'already' diagrammed the conclusion.
One interesting thing about presenting both of these to an intro class is their difference over ex falso quodlibet: in classical propositional logic, any argument with inconsistent premises is valid, whereas in categorical logic there are invalid arguments with inconsistent premises. This is reflected in the semantics for the two logics, truth-table or Venn diagram, respectively. That is, if you diagram 'All A are B' and 'Some A are not B', you have not already diagrammed 'All C are D' (or whatever -- I mean, or quodlibet). Whereas, in the truth table case, if one of your premises is p and another is 'not-p', it is of course impossible for there to be a row where both all the premises are true and the conclusion is false -- since there is no row in which all the premises are true.
Everything so far is completely uncontroversial. Now comes the point I've been wondering about for the last couple of days. What if we set up a validity test for a propositional language that looked more like the Venn-diagram test for categorical logic? That is, instead of thinking of validity in the usual way as absence of counterexamples (in the propositional case, no row of the truth table has all true premises and a false conclusion), we demand that the diagram of all the premises be a diagram of the conclusion. In categorical logic, each of the points in the Venn diagram represents an individual object, and the circles represent sets of objects; in a propositional Venn diagram, we let each point be a case/circumstance/state of affairs/whatever, and let the circles be sets of cases etc. The resulting propositional logic would NOT be classical, since ex falso quodlibet would not hold.
Why care about setting up such a Venn-diagram validity test for propositional logic? Here's why: when relevance logicians (and anyone else who doesn't like EFQ) accept the 'no counter-example' account of validity, they are forced to say some pretty counterintuitive things, first and foremost that there is a case (or whatever) in which both p and 'not-p' are true (and some other q false). If they don't say this, then they can't say EFQ is invalid. But if this Venn-diagram validity test for a propositional language is viable, we can reject EFQ without accepting true contradictions. All that's required is tweaking the no-counterexample notion of consequence -- a tweak that is already used in critical thinking textbooks.
Finally, this idea seems so obvious that somebody must have already explored it. Any pointers to the relevant literature?