3VL - doubtful inference

The impression I have gathered is that Lukasiewicz 3-valued logic hasn't had a great deal of respect in the logical community. This is entirely understandable. As it has been presented so far, there are chronic difficulties with interpretation, and you can't do the same kinds of things with it that you can with classical logic. Most of the serious work that has been done with it has been using Lukasiewicz' "Polish notation", which is unfamiliar to most people who work with logic, and in Europe rather than the United States.

But the small step of defining a strict conditional for it makes an incredible difference. It's like giving it the power pill that turns lowly shoeshine boy into Underdog; like turning Bruce Banner into the Incredible Hulk; like turning a lightning bug into lightning; moonlight into sunlight, climbing over a mountain peak and seeing the Pacific Ocean on the other side. The difference in effectiveness is so huge, that it's amazing no one has seen it before. But if it has been seen, I haven't found it in the literature.

It was like a dazzling flash of hindsight. Revelation followed revelation so swiftly, and in such interconnected fashion, that I no longer recall their exact sequence of events. But I can describe some of them.

One of the early ones is that I realized the reason why Lukasiewicz logic hadn't been workable before.
I was already aware that, using the original Lukasiewicz conditional, Modus Ponens fails as a tautology. But it fails in only one case, Namely, when P is doubtful and Q is false. The truth table labels this as doubtful.
But of course!! The Lukasiewicz conditional allows the expression of doubtful conditionals, and if it were true without restriction, it would be possible to start with a doubtful premise and a doubtful conditional, and advance to a false conclusion. But by forbidding dubious conditionals and assuring that it is definitely the case that if P, then Q we repair the deficiency. The original truth table quite correctly labels a case where modus ponens can and should fail.
A few more examples, such as transitivity, yielded similar results, and the basis for a whole theory of doubtful inference falls out, naturally and easily.
Of course this must be so!! One of the purposes of logic, after all, is to assure that our rules of reasoning are correct and that we do not start from true premises and reason to false conclusions. And the strict conditional has just the kind of ordering properties, on three values, that the ordinary material conditional has for two values; when P =>Q is true, The conclusion Q is at least as true as the premises P.