A throwaway line I posted as a comment on Sarah Hoyt’s blog “The learned professors of formal logic are so far up the wrong trees they can’t even see the ground” interested a reader.

If you don’t understand the technical jargon, ask.

I had been working on three valued logic (a logic incorporating values for “True”, “False”, and “Maybe”) on and off since the early 1980s, and I couldn’t figure out why it didn’t work. When I consulted Ackerman’s work on “Three Valued Logic”, I found that some work had already been done, but none of the systems presented were wholly satisfactory and set it aside for a few years. Later, when I took a course on formal logic, I came back to it. I got as far as recreating modal functions for “definitely true” and “Possibly true”, and was able to recreate some of the classical laws of modality; for instance: “Not possible” equals “certainly not’, “not necessarily” equals “possibly not” I returned to the reference and I was disappointed to learn that everything I had done so far had been anticipated by the Polish logician Jan Lukasiewicz in the 1920s. I started looking up the work of C.I, Lewis, who is noted for his work in modal logic using an axiomatic approach. Lewis and Langford discussed the work of Lukasiewize and noted that it had a severe deficiency, in that the standard methods of modus ponens and transitive conditionals were not valid, which crippled its usefuless as a system of logic. On the other hand, the Lewis systems have the major disadvantage of not being able to be expressed using truth tables. I wondered technically why the two systems were incompatible, and set myself to the task of figuring it out. I searched more deeply in the literature of both multivalued logic and modal logic, looking for interconnections, but beyond a proof that the more useful Lewis systems could not be reduced to a finite number of values and a demonstrated inadequacy of the Lukasiewicz logic in providing standard methods of logical proof, the subject was scarcely mentioned. I also explored the axioms of intuitionism; this has a similarity to the Lewis Systems in that it is developed axiomatically and cannot be reduced to a finite number of truth values, but there was no unified approach. There was a throwaway line that “despite a promising beginning, the connections of three valued logic and modal logic didn’t work”. I wanted to know _why_ it didn’t work.

In the early 1990s I found a clue. I was looking again through a discussion of multi-valued logics, using updated works from Malinowski, and Bolc and Borowitz, and noticed that of the three valued logics which discussed logical equivalence, their equivalences were generally not equivalence relations; that it, they were not reflexive, symmetric, and transitive, which were the criteria laid down for equivalence relations in studies of finite math and the foundations of mathematical structure. I thought “why not add one to Lukasiewicz logic”. These results were much more satisfactory, and I could establish several of the axioms of the Lewis System S5 as valid three valued tautologies. But not all of them. Next, I noticed that I could express the table for my equivalency by applying the “definitely” operator to the Lukasiewiz biconditional. Then I thought “What’s good for the biconditional must be good for the conditional as well. I tried this, and in a dazzling, blinding flash of hindsight, I realized “of course it must be so”.

The Lukasiewicz conditional as defined allows conditionals to have the third logical value. In order to have valid reasoning, you need logical laws that do not introduce error. Modus ponens and the transitive law of the conditional when applied to the unmodified Ludasiewiz conditional allow false conclusions to be reached from true premise, and the truth tables show how. As a remedy, it is necessary and sufficient to modify these methods to require “strict” or “Definite” conditionals and forbid doubtful ones. This is reinforced by noticing that the “Strict” version of the Lukasiewicz conditional is an ordering relation; reflexive, antisymmetric, and transitive, which assures “if p then q” is definitely true, then q is not less true than p”. The common material conditional is such an ordering relation and assures this for classical two valued logic, which is why it works; most of the analogues proposed in three valued logics do not, which is why they don’t.

This is not quite the same as the strict conditional proposed by Lewis. Lewis held strictly to the law of the excluded middle, and incorporated it into his logical systems; Lukasiewicz denied it as a general law. In my view, this introduces a subtle inconsistency which breaks Lewis’ systems and makes them non-truth-functional.

I tried publishing these results about 2000; my paper was rejected by the one journal I attempted, in part in the grounds that they symbolism did not exceed the high school level. Excuse me? The ideas are simple enough for high school students to understand; they don’t need elaborate symbolism. I tried rewriting them and submitting them to selected experts individually, and met with profound lack of interest. Although the ideas behind multi-valued logic, modal logic, intuitionism, and fuzzy logic are closely related, in my observation the formal systems used to express them have become their own distinct and rather ossified bodies of research. Hence my comments about learned professors of logic. It’s intellectually painful to see how close the founders of these various fields came to a simple, robust theory that spans several such branches, without quite getting it right.

# The learned professors

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