If φ and φ→ψ are NCL-valid, then so is ψ.The proof didn't turn out to be hard, but it does involve some rather curious lemmas that hardly resemble anything like what I've seen before in my study of mathematical logic. It really feels like exploring a whole new world. Sara discovered a crucial feature of valid NCL-implications:
If φ→ψ is NCL-valid, then eitherI confess that I was skeptical when Sara shared this conjecture with me; it felt too ham-fisted and alien. But the conjecture in fact holds, and it paves the way toward a positive solution of the composition problem. (The other principal lemma in the proof is a characterization of NCL-valid negations.) Just today we submitted a joint paper about all this (“A curious dialogical logic and its composition problem”) to the upcoming Indian Conference on Logic and its Applications, to be held in Delhi in January, 2011. We expect to learn in about a month whether it will be accepted. (In the paper we call NCL simply “N”. The name NCL comes from a more benighted time when we thought, based on a priori reasoning about rulesets, that since NCL is only slightly different from a ruleset that characterizes classical logic, it would correspondingly give rise to a logic that might be different from classical logic but “close” to it.) We were restricted to 12 pages, but we have lots more to say about NCL and couldn't pack it all in. We expect to produce a fuller account in a journal article.
- φ is an atom,
- φ is a negation, or
- ψ is already NCL-valid.