Over at Greg’s blog, there was a post on which is prior, logical truth or consequence. They are, in many cases, interdefinable. For example, we might want something like: A |= B iff |= A→B. Of course, this will depend on having the appropriate expressive resources in the language. The left to right direction fails for any logic without a conditional, defined or primitive. An example of this would be the conjunction-disjunction fragment of classical propositional logic. This looks a little artificial though. A more natural example is the logical system of Aristotle’s Prior Analytics. It has no conditional locutions, definable or primitive. This depends on using the interpretation found in Smith’s introduction to the Prior Analytics, which i believe derives form Corcoran’s work, rather than the axiomatic interpretation given by Lukasiewicz. There might be another counterexample coming from connectives defined using Kleene’s strong matrix because the conditional defined on that (at least, the standard one) results in no tautologies. For example, p→p gets the value one-half when p is assigned the value one-half. I’m not sure about this because I’m not sure how consequence is defined for that system.

Does the schema fail in the right to left direction? I don’t know of one and I’m doubtful there are any.

Why would one think that direction would always hold? The conditional is an object language expression that is supposed to capture the consequence relation. Sometimes the consequence relation can outstrip it. Things would be amiss if the object language outstripped the metalanguage. Here’s an idea. There are fewer restrictions on the stuff that appears to the left of the turnstile, e.g. there can be infinitely many things on the left, they can be gathered using sets instead of conjunctions. If the right to left direction is going to fail, we’d need more restrictions on the stuff appearing on the left-hand side of the turnstile than the right. When put like that, it doesn’t seem obvious that the right to left direction cannot fail, but the context would have to be very unusual if there is one. As a parting thought, it isn’t clear that we could place restrictions like that on the left-hand side of the turnstile and still have something recognizable as a consequence relation, i.e. something satisfying Tarski’s conditions.