I haven't given up on the other aspects of self-directed education (in case anyone is actually reading this), but in the last few weeks, I've found some logic-oriented blogs on the net. Starting with Logblog I'll start referring to those once I get done with my "confessions" here.

My attitude toward professional logicians is, If you can't join em, fight em. No one competent to understand what I am talking about has shown any interest. So, since I read that one of the components of a successful blog is to be controversial, and since I am clearly suffering delusions of enlightnment, I'm going to turn guerrilla and snipe at the Establishment.

At the point in my studies I had reached, I was not aware that I had almost independently reconstructed the 3-valued logic of Lukasiewicz, and I didn't understand the reasoning or philosophy behind the Lewis systems S1-S5 beyond what I could determine from the axioms.

I ventured onto Compuserve (The internet was just then beginning to grow), and asked there whether anyone had any comment. I was referred to Joe Celko, who was described as working on a 3-valued logic that dealt with missing values in Data bases, who referred me in turn to a discussion going on in the pages of Data Base Programming and Design. I read the articles with great interest, and found myself sympathizing with both sides on the debate. The 3-valued approach the proponents were using was similar to what I was doing, but I agreed that a sound theory was lacking. I didn't have the answers, either, but again, there were unanswered questions.

I went back to school to try to get my BS in Mathematics, and took a course in classical logic. The approach used in that course was natural deduction, and it basically covered propositional logic without going into predicate logic, but I took the opportunity to study that on my own. I took note of the fact that theorems of logic corressponded to truth-functionally true statements (that is, a statement that evaluated as true on every assignment of truth values), their negations were truth-functionally false, and others were contingent, and I wondered whether the middle value I was using could be used to describe these.

I transferred to ASU for a semester, and later lived next to the University for a year, and took the opportunity to examine the literature a little closer. I was dismayed to discover that my discoveries had indeed been anticipated, and I almost gave up. However, there were still unsettled questions. One of the comments I encountered was that "In spite of the promising combination of trivalence and modality, modal logic on this basis was never fully developed." I wanted to know why, and there was no further discussion, no references, no reasons why it didn't work. The other was the objection to interpretation. Lukasiewicz intended his truth value to represent the uncertainty of the future contingent, but an objector (no reference given) pointed to the "law of the excluded middle" and, apparently, there was no answer. When I worked on this, I decided that there were two different kinds of uncertainty involved. Using constants instead of variables or tables !P expresses the idea "True or false, but it's not certain which (!T=T, !U=F, !F=T) while ?P expresses the doubtfulness associated with the middle truth value (?T=F, ?U=T, ?U=T).

It's trivial to show that ~!P = ?P and ~?P = ~P, but !~P=P and ?~P=?P. This meant, to me, that "uncertainty" is an ambiguous concept, with two formally similar but contradictory interpretations. I didn't fully work out the details of how these were associated with the "and" and "or" at this point.

After this, I moved back to Utah, close enough to BYU that I could consult the literature there, and ventured onto the internet, this time at the newsgroup math.logic. One person noted that according to my tables, [](P v Q) = []P v []Q, and <>P & <>Q = <>(P & Q), which aren't accepted in traditional (e.g. Lewis-type) modal logic, while someone else referred me to Bolc & Borowic's work on multi-valued logic. I labored over these for some time, trying to figure how I could get [](P v Q) & ~[]([]P v []Q); and (<>P & <>Q) & ~[](<>(P & Q), but no matter how I transformed and tortured these statements, I got contradictions. Eventually, I decided that they were genuinely contradictions. To simplify the problem, supposing that P and Q are mutually exclusive, so that Q = ~P, and then applying the various transformation rules, these boil down to trying to assert the excluded middle on one side and deny it on the other. No wonder there's a contradiction!

## Sunday, May 22, 2005

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