About this time I transferred from Phoenix College to ASU, and while I was there, I took the opportunity to do some extracurricular library digging. I found a bibliography of the history of symbolic logic in the Journal “Symbolic Logic” and troubled myself to look up references. One of them that I found was an early attempt at a logic of indecision. This didn’t work because it was inconsistent: a calculation done one way and a calculation done another way came up with different results. I believe I know why, now, but It would take some digging (on the order of an archaeological research with a grant and a team of undergraduate laborers) to find the references again. I also looked up the work of C. I. Lewis as published in Symbolic Logic, by Lewis and Langford, 1932, and found the axioms for his systems. I also found his definition for the “Strict implication” he used in his axiomatic systems, and that the had examined the same Lukasiewicz three valued logic I had been working with, and found it deficient because a a couple of important logical laws were missing. I will return to this subject later.

I also took a course on mathematical structure, which considered how mathematical object can be defined, the relations among them and the properties of those relations, and the mathematical operations that can be performed on them. I didn’t think to apply this to mathematical logic at the time, that came later, but it appears that it has not occurred to many logicians or philosophers, either. But this subject lay fallow for a while.

I believe it was after I left ASU that I began seriously flogging the algebra to discover why it was yielding some seriously paradoxical results. Let us take, for example “Charlie is a boy” versus “Charlie is a girl”, and what happens if we give this a truth value of U, for undecided. The algebra gives “Charlie is a boy and Charlie is a girl” a truth value of undecided. But what if these are mutually exclusive condition? Should not the expression be false? This point was raised by Lewis and it was his decision to incorporate the premise that they would be into his logic. As it turns out, the reason is that the “law” of the excluded middle is so easily and often taken for granted that we try to sneak it in anyway, even when the logic explicitly rejects it as a law.

As it turns out, the concept “The fribbles are thurgish” is undecided (to use an absurdity for illustrative purposes) can be mean two different things. It could mean that the thribbles are thurish is either true or false, but we don’t know which, or it could mean that there is an intermediate state between thurgish and not thurgish. Another example? “The door is open”. We could mean that the door is either definitely open or it is definitely closed, but we don’t know which is the case, or it could mean that the door is ajar, an intermediate state between fully open and fully closed. Another one. “it is raining”. We may not know which is the case, or there may be states of the weather that make it difficult to determine which is this the case.

The logic makes it clear that these are two different cases. One of them, which I call the “Dichotomous” sense, can be symbolized, but does not have a single truth value. The other sense, which I call the equivocal sense, can be symbolized and does have the single truth value U. What makes it tricky is that the negation behaves the same in both sases. The negation of a dichotomous statement “It is raining” and evaluated as true or false, would be naturally be “it is not raining”, and evaluated as false or true. If “The door is open” is considered equivocal because the door is ajar, “the door is not open” is also equivocal for the same reason. Without the symbolism to make the distinction clear, these two different kinds of uncertainty can be easily confused. “if it is possible that some proposition is true and it is possible that that proposition is not true then that proposition is equivocal. ” This is a theorem of the logic I use and I can prove it.

For practical purposes, the algebra works the same for either interpretation. What you cannot do, however, is used “undecided” as a logical value on one hand, and then turn around and try to invoke the law of the excluded middle on the other. BLAATTT! REEE!. Violation! If you’re going to say “Charlie is a girl” is undecided, you must also admit Charlie is a girl and Charlie is not a girl (i.e. a boy)” to also be undecided. You don’t have to accept such a contradiction as true. You just have to refrain from claiming it to be impossible. There is no problem with setting up a condition “Charlie is a girl” if and only if “Charlie is not a boy”. Once you know one, you know the other, and the case where they are equivocal slides through because they have the same truth value, even though they mean different things. But if you do insist ahead of time that it is impossible to say both “Charlie is a girl” and “Charlie is not a girl” then you have no business using three valued logic at all and the ordinary two valued case is sufficient.

The logic admits no compromise. Either you accept the law of the excluded middle as a general law, or do not. No, rejection of the law of the excluded middle doesn’t mean that it is never true. It only means that you can’t count on for all kinds of statement. In this particular case and a few others, it bites down hard.

“Mother may I go out to swim? Yes, my darling daughter. Hang your clothes on a hickory limb, but don’t go near the water. “

Your intuition may lead you astray until you have properly trained it. The algebra is consistent.

At about this stage in my research, I became interested in the connections with modal logic, the logic of necessity and possibility. Here, my primary source was the symbolic version developed by C. I. Lewis. I adopted his box-and-diamond notation, but I also noted that his versions were developed on an axiomatic basis and did not use truth tables. I took a look at his axioms and tried to fit them with theorems or tautologies of the three valued logic, and they didn’t quit fit. They were close, but didn’t quite match. There was even a mathematical proof that the Lewis systems could not be reduced to a finite number of truth values. There was enough similarity, and I had produced enough theorems to match most of the laws of Boolean Algebra, that I wondered what the difference was. I also took a course in logic at Phoenix College, using “The Logic Book” by Bergmann, Moor, and Nelson, which covered the natural deduction method of logical proof. We only covered propositional logic, but it gave me a better insight into the subject. I also made contact with people who were discussing the possibilities of using three-valued logic in Database programming, particularly as it was implemented in the language SQL. I didn’t quite have the full satisfactory theory yet, but I notice that there was a fierce debate over “Practically Useful” versus “Theoretically Sound” going on in articles and letters of Data Base Programming and Design Journal. The editors called a moratorium and the journal went into electronic publication only, so I lost track of the debate, and the journal has now ceased publication. It did raise some theoretical issues, which bedeviled me for a time.

One of the things things that logical arithmetic is good for is in proving logical formulas. An arbitrary expression in Boolean alebra, is usually conditional. It depends on the truth values of the variables it uses. However, for a certain class of formulas, the overall expression can be evaluated as true in every case. These are called truth-functionally true, or tautologies, and are the theorems, laws, or rules of boolean algebra. For instance, (A AND B) is equivalent to (B AND A). This is always true, regardless of whether A and B are true or false.

There are other ways of proving logical formulas. One of them is the axiomatic approach, in which one starts with a few axioms and a few rules of inference, and derives all the others. But wait, you may ask. Isn’t this a bit circular? Well, yes, it is, when applied to logic itself, and I much prefer the truth table approach when it can be used. For classical two-valued propositional logic, these give the same laws.

There are variants of standard logic which are examined for experimental purposes, in which some of the standard rules do not apply, and which do not or cannot use truth tables. These commonly use an axiomatic approach. It isn’t to my personal taste, but that’s just me.

Another topic that needs examination is the concept of a valid argument. Traditionally, rhetorical logic from the Greeks on was concerned with arguments that were logically valid. The goal is to start from true assumptions and reach true conclusions without introducing error in the reasoning itself. There are some forms of argument, commonly known as fallacies, which are considered invalid because they may introduce an error. By the time you boil things down to propositional logic, there is only one criterion that matters: Does the material conditional hold? If it does, your argument or proof is valid. If it doesn’t, then it isn’t. It’s that simple: Easy peasy. If you want to talk about classes of things, or talk about subjects where statements can be vague or uncertain, or fiddle with the laws of logic, it gets increasingly more complicated.

The other thing that needs to be considered is that from the time of Aristotle onward, logic was concerned with statements that were either true or false. No others were to be considered. This is easiest to do in mathematics, where we can control the definitions and decree certain things by convention. It also means that logic works less well in the real world. Take, for example “It is raining”. Aside from being dependent on time and place, there are certain states of the weather that make it hard to give that a definite true or false answer. If you are careful with definitions, you can control or limit the uncertainty…but then you have the problem of getting other people to agree with your definitions. Again, this is easiest to do in mathematics, unless perhaps you have engineers getting into two-fisted fights over Bessel functions or Laplace transforms.

As it turns out, uncertainty is a very slippery concept to grasp if all you have is words, and it’s easy to get mired in contradiction and confusion.

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.

I promised that I would continue with my discoveries, but as I sketched out the post ahead of time, I realized that it would work better if I backtrack and do a little history. In 1847, the British mathematician George Boole published a pamphlet on the Mathematical Analysis of Logic, and in 1854, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. At a later point in my independent study, I went back and read them, and found, as with many pioneering tracts, hardly anybody reads them anymore, and for good reason. Boole had the bright and revolutionary idea of using algebraic variables for whole classes of objects instead of just numbers, using the number 0 to represent falsehood and 1 to represent truth, and the addition operation to mean “or” and multiplication to mean “and”. His original methods were quite cumbersome and have been vastly improved and simplified, but modern symbolic formal logic can be traced to his work. Symbolic, because it uses symbols instead of words, and formal, because it is more concerned with form than content.

Boole dealt with statements about classes of objects, such as “All men are mortal”, “Some roses are red”, and “No king is a fink”. The terms “all”, “some”, and “none” are nowadays referred to as quantifiers, and the rest of the sentence is the predicate, so this branch of the subject is now referred to as predicate logic. Mathematics employs this branch of the subject.

A simpler version, which deals with particular statements, such as “Socrates is mortal, “This rose is red”, or “The King of Id is a fink”, is called propositional logic. This was developed somewhat after Boole’s work.

It was discovered fairly early after Boole, that it vastly simplified the calculation if “or” were taken in the inclusive sense, so that she “She wore jeans or she wore a dress” would be true if she wore both (jeans under the dress, for instance), instead of the exclusive sense. (She wore either jeans or she wore a dress, but not both.) The inclusive sense has become conventional in logic, while English uses the exclusive one. We won’t talk about the logically literate having fun with using the inclusive sense when the exclusive is intended, by answering such questions as “Do you want cake or ice cream” with”Yes”.

So, in propositional logic, every statement is given a truth value, which is either true or false. Other possible values are simply not discussed, which is one of the limitations of logic.

The “arithmetic” of logic is quite simple. For “or”, you have “True or true equals true”. “True or false equals true”. “False or true equals true.” False or false equals false.”

For “and”, you have “true and true equals true”. “True and false equals false.” “False and true equals false.” false and false equals false.

There is also negation “Not true equals false”, and “not false equals true”.

These are often summarized in truth tables. There is nothing mysterious about truth table: they just like addition and multiplication tables, only much simpler.

On the other hand, the algebra of logic, Boolean algebra, it is called, after Boole, is a bit more complicated. There, as in ordinary algebra, you use letters to stand for statements, or more exactly, the truth values of statements, and there are rules for the combination and simplification of variables and expressions. The goal is not so much to “solve for the unknown as it to “make the expression as simple as possible.

Since this isn’t an algebra lesson, (aren’t you thankful)? that’s as far as I will go right now. Unless you want be a computer scientist, mathematician, or logician, you probably won’t need or use it. For those few who someday might, I will assign homework. No, I won’t collect or grade it. Like writing a textbook, you’d have to pay me for that. But mathematical logic is not a spectator sport and if you don’t flog your brain through the exercises, you won’t learn it.