Logical musings #7

Please forgive me. This has been dancing through my head for the past 15 years or so, and I don’t think I’m going to get past it until I get it written down. By this time, the early 1990s or so, I don’t remember exactly, I obsessed with finding exactly why there was a mismatch between the Lewis systems of modal logic, and the Lukasiewicz three-valued logic, because there were so many points of similarity, and because I was getting comprehensible and consistent results from the algebra. Yes, these are considered incompatible, but exactly? And I couldn’t find out. After the 1920s, the modal logic community followed Lewis like lemmings off a cliff (except that lemmings apparently don’t really do that, making them somewhat smarter than logicians) and the multivalued logic community went into higher and higher numbers of logical values. The communities stopped speaking to each other and nobody I could find was even asking the question of why these were incompatible. I went to the original work of Luksasiewicz. He had already done everything that I had, so I was covering old ground there.

Then one day after I had moved to Utah, I looked again into a newer work on multivalued logic, specifically the three valued case, and there were several competing systems, and I noticed something peculiar about all the logical equivalence relations they were defining. None of them, except one, met the conditions for a mathematical equivalence, and the Lukasiewicz logic wasn’t one of them. (Nor convenience and to avoid misspelling the name, I’ will just call it Polish logic from now on) I thought “Easy enough, I will just define a relation that is.” I was quite pleased that I could now express all the truth tables I had worked out as proper equivalences, just like I could do in classical logic. Then I noticed that I could get this same table by applying the “Definitley” or “Necessarily” true operator to the Polish equivalence. Then it occurred to me that what was good for the equivalence, or the biconditional might be good for the conditional as well. So I tried it.

SHAZAM!!! The familiar triangular form of a mathematical ordering relationship fell out. I was struck by the most blinding, dazzling flash of hindsight I have ever experienced. OF COURSE IT MUST BE SO!!! I am calling this the strict Polish conditional. It had all the nice properties of the material conditional that make it so useful in classical logic. While the important logical laws of modus ponens and the transitive property fail with the Polish conditional, they succeed with the strict Polish conditional, and a comparison of the truth tables for them revealed why. The ordinary Polish conditional allows for doubtful conditionals (equivocal or dichotomous, take your pick it works out the same either way) and of course you cannot expect valid reasoning with those. Take “I get to the bus station on time”. If the conditional is “If I get to the bus station on time, I will get to ride the bus” is doubtful, then you cannot safely reach the conclusion “I will get to ride the bus”. However, if you can guarantee that the conditional holds, your reasoning is safe. At one stroke, the failure of the laws that bothered Lewis turned into a powerful argument for Polish logic.

I now had four conditionals to work with, The ordinary material conditional in two valued logic, the strict conditional as defined by Lewis, the Polish conditional, and the strict Polish conditional. I already knew that the ordinary material conditional didn’t work for 3-valued logic, and the strict Lewis conditional didn’t either, for the same reason: Neither accounts for a middle truth value. The ordinary Polish conditional was better, but still not quite up to the job. But the strict Polish conditional…ah, now that was safe. I could safely guarantee valid conclusions with it. I could block introduction of error by preventing true assumptions from reaching doubtful conclusions, and doubtful assumptions from reaching false conclusions.

So, I tried casting the axioms of the Lewis system S5 into Polish logic, with the exception that I used the strict Polish conditional instead of the strict Lewis conditional. Success!! Theorems, not just axioms! Furthermore, if the strict Lewis conditional is is true, then so is the strict Polish conditional. (Not conversely). So, I had a system of modal logic that is truth functional and can be done with truth tables, not the cumbersome axiomatic approach. Anything you can do in S5 I can do better!

This turns the lightning bug into lightning. It turns Billy Batson into Captain Marvel. It turns humble, lovable shoeshine boy into the mighty Underdog.

At the risk of sounding like a commercial from Ronco; “But wait! There’s more!”

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