Logical Musings #3

I don’t remember why I starting playing with three valued logic. I began with extending the truth tables for “or” and “and”, with a new value I called “Undecided”. I thought it reasonable to begun with “true OR undecided (true or false, don’t know which) equals true”; “undecided OR true equals true”, “false OR undecided equals undecided”, “undecided OR false equals undecided”, and “undecided OR undecided is undecided”.

For the logical and, “True AND undecided is undecided”, “undecided AND true is undecided”, “False AND undecided is false”, “undecided AND false is False”, “undecided AND undecided is undecided.

The negation of “undecided is undecided.

As I came to discover later, this is not quite right, but it will do for a start. I should also mention here that “undecided” is a very tricky and slippery concept to work with. This is one of the advantages of using symbols: By using this symbol I meant exactly THIS with this meaning, and not THAT with that meaning. But I’m getting a little ahead of myself.

Another logical connective that is usually defined is called the material conditional, and it is taken to mean “if… then…. It is called the material conditional and it has a specific meaning and truth table. ” ‘if true then true’ equals true.” ” ‘if true then false’ equals false.” ” ‘if false then true’ equals true”. ” ‘if false then false’ equals true.

This one confuses a lot of beginning logic students: especially, the last two entries, but since it is the standard definition, I’ll take it as given for now. The reason it works is one of they keys to properly understanding logic, both two-valued and multivalued versions There are some astonishingly eminent logicians who don’t quite get it. I will explain later.

The extension I chose was ” ‘if true then undecided’ equals undecided. ” “‘if false then undecided’ equals true. ” ‘if undecided then true’ equals false”. “‘ ‘if undecided then false’ equals undecided. “‘ ‘if undecided then undecided’ equals undecided. “

These additional five entries in each truth table give me 3×3 or nine, which is appropriate.

I so was pleased with myself that I betook myself to the ASU Hayden Library to find out if anyone had discovered this before. Alas, and to my dismay, someone had. There was entire book, “An Introduction to Many Valued Logics” by Robert Ackerman. I found a useful discussion of the reasons for considering many valued logics in the introduction, but I had basically rediscovered Stephen Cole Kleene’s strong three valued logic….and it didn’t work. Oh, logicians can get some use out of it by ‘designating’ truth values, but for practical purposes it is useless, because you get no useful theorems or laws of logic out of this and can do no useful deduction. Even so simple a statement as “if the bus is on time then the bus is on time”, which ought to be true, is labeled undecided when ‘the bus is on time ” is undecided. That’s no good. There were at least half a dozen other systems presented. The most interesting was the three-valued logic of Jan Lukasiewicz. I took note of it and promptly forgot his name (because it’s Polish).

At this point I need to back up and talk about logical theorems and valid deduction.

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