This is a continuation of a series of posts on three valued logic that began with “The Learned Professors”.

Logic doesn’t get very far without the ability to combined sentences and expressions. As with classical two valued logic, the principal means of combining are two binary operations; the logical “and” and the logical “or”. If the truth values T, U, and F, are ordered from “Most true” to “least true”, the value of “P and Q”, P & Q, is the least true of the two expressions, while “P or Q”, p \/ Q is the most true.

There is nothing new here; most of the three valued logics that have been created take this approach, and it seems entirely reasonable.
It becomes more interesting when negation and the modal operators becomes involved. De Morgan’s laws apply so that ~ (P & Q) = ~P \/ !Q and ~ (P \/ Q) = ~P & ~Q. The modal operators [] and <> are distributive, so that [](P & Q) = []P & []Q; <>(P \/ Q) = <>P \/ <>Q. [](P \/ Q) = []P \/ []Q; <>(P & Q) = <>P & <>Q.

This is a significant difference from the modal logic of C. I. Lewis, who argued that if P and Q are mutually exclusive so that Q = ~P, <>P & <>Q should be true, while <>(P & Q) should be false. However intuitively appealing this might be, it doesn’t work here. Once we are committed to using the truth value U for P, we are also committed to accepting that <> (P & ~P), for instance. “It is possible that the cat is black and the cat is white”. Fortunately, we are not required to accept such contradictions as true; [] (P & ~P) “It is definitely the case that P and ~P” is false. It is sufficient to refrain from denying that they are ever possible. Sometimes, there are intermediate states for which a such description makes sense. Even if there are not, it is inconsistent to relax the law of the excluded middle on one side, and then assert it strongly on the other.
It is a basic assertion that <>P & <> ~P “It is possible for both P and ~P” signifies that P is an equivocal statement and has the truth value U. If, on the other hand, you assert ~<> (P & ~P), “It is not possible for both P and not P”, this a commitment to dichotomy. If it holds for all statements involved, the three valued logic simply reduces to the two valued case. If P and Q really are mutually exclusive, <>(P & Q) can be considered a vacuous possibility that vanishes as soon as either P or Q is known.

As it works out, the similar behavior of dichotomous uncertainty and equivocal uncertainty means that you can get away with using U for either type, as long as you observe decorous restraint and refrain from asserting that the principle of the excluded middle is a law that cannot be violated. The laws of the logic are perfectly general and apply equally and indifferently to both cases.

This is a continuation of a series of blog posts on a three valued logic, and employs concepts and notations previously described,
beginning with “The Learned Professors”

Classical logic has one form of negation; a three valued logic has at least three.

~P signifies “It is not the case that P”, and takes a value of F where P has a value of T and T where P has a value of F. The question of what to do when P has a value of U naturally arises. The standard negation gives this a value of U. This sounds reasonable, since when “the cat is black” is uncertain or unknown, so is “the cat is not black”.

However, it is also possible to define a strong negation, in which statements with the value of U are assigned a value of F. This sense of negation is used in intuitionistic logic. I do not use a separate symbol for it, since this can be represented by ~<>P.
Similarly, a weak negation can be defined, in which statements with the value of U are assigned a value of T. Likewise, this can be represented by ~[]P. In evaluating expressions, these are evaluated in order right to left so that one closest to the valuable is evaluated first.

It may be observed that ~<>P has the same table as []~P, and ~[]P as <>~P. This is one of the oldest and earliest expressions of modal logic and dates to Aristotle. “Not possible” is equivalent to “necessarily (or certainly) not”, and “Not necessarily” is equivalent to “possibly not”.

It gets a little trickier when the other modes are considered. !P = !~P; The negation of a dichotomously uncertain statement is also dichotomously uncertain, and ?P = ?~P, the negation of an equivocally uncertain statement is also equivocally uncertain. However, the negation of claim that a statement is equivocally uncertain is equivalent to a claim that the statement is dichotomously uncertain, ~!P = ?P, and the negation of a claim that a statement is dichotomously uncertain is equivalent to a claim that the statement is equivocally uncertain. Since ordinary language does not distinguish equivocal and dichotomous uncertainty, it is easy to shift definitions and thus arrive at contradiction and confusion. This is a cases where the symbolism helps clarify the discussion, and aids in the distinction between use and mention of a claim. Also, !!P and !?P, but ~?!P and ~??P, or in words, the claim that a statement is dichotomous (either true or false) is itself dichotomous, and the claim a statement is equivocal (possibly true and possibly false) is also dichotomous. The claim that a statement is dichotomously uncertain may not equivocal, and the claim that a statement is equivocally uncertain may not itself be equivocal. Natural language does not make these distinctions or enforce these rules, but in order to avoid contradiction and confusion, the three valued logic must. This is more or less equivalent to a law of the excluded fourth. Three valued logic, like two-valued logic, is a tool with its own uses and limitations, not a cure-all for all philosophical or logical conundrums.

It needs to be recognized that 3-valued logic is a simplified approach to modal logic, and does not attempt to cover all the profundities and complexities that philosophers have explored. For instance, all the modes are two-valued. There is no attempt to explore whether sentences that are definitely true are necessarily so; or if something is possible, whether it is necessarily possible or only contingently so. Rather than focus on what it cannot do, I prefer to draw attention to what it can do.

The simple statements “P is true” and “P is false” are somewhat ambiguous in 3 valued logic. It is useful to ask “is it definitely true or false, or possibly true or false?.” It makes a difference.

This is noteworthy because in computer science, the data base language SQL contains a partial implementation of three valued logic. For purposes of computing, a decision often needs to be made about whether an item should be included in a search or merger, or excluded. Here, the modal functions would be beneficial. The difficulty is that to my understanding, these are often performed in an implicit, automatic, and hidden fashion, and even worse, differently for different functions. This makes SQL more difficult than it ought to be. It may sometimes give baffling, incomprehensible, or contradictory results. It may require elaborate, careful circumlocution to get desired results.

The notions of “necessity” and “possibility” in traditional modal logic do not correspond exactly to the usage enforced by this logic. I am not an expert in philosophy and have not attempted to delve into these matters. “Alethic modalities”, “Epistemic modalities”, “deontic modalities”, and others are recognized. This logic doesn’t attempt to distinguish among them. While the results are mathematically consistent, they may or may not correspond to accepted understanding and conventions.

There is also difficulty with interpretation of the third logical value as “contingent”. This logic makes no distinction between contingently true and contingently false, and cannot reliably distinguish between “necessarily true” and “contingently true”. This would require a logic of at least four values. It is my contention that the three-valued logic is not yet sufficiently understood to successfully and reliably develop the four valued one.

One of the more insidious traps is with shifting concepts of “uncertain”. Uncertain may mean “possibly true”, or “not definitely false”, in which case it could correspond to <>P or ~[]~P. It could be used to mean the third logical value and correspond to ?P. I termed this sense “equivocal”. It could also be used to signify “Definitely true or definitely false, but we don’t know which”. This would correspond to !p. I have termed this sense “dichotomous”. It could be used in a broad sense to mean that the truth value of a statement is undetermined yet and could be T, U, or F. It is noteworthy that equivocal uncertainty and dichotomous uncertainty are very much alike and behave quite similarly, although in this logic, they are strictly contradictory. This has historically made discussion of “uncertain” slippery and difficult.

This is closely tied to the principle of the excluded middle. In general, I call it a principle, not a law, because it is a contingent statement applying to some statements, but not others. This, for instance, came up when Lukasiewicz proposed using his third logical value to signify future events such as “I will be in Warsaw next December 25.” One logician objected that this had to be either true or false, and so the third logical value could not be used to apply to it. Tweet! Foul! Insisting that a statement that you have used the third value for must be either true or false is already a contradiction. If you’re going to reassert the law of the excluded middle or one of its equivalents, there is no point in using the third value at all, and you can stick with classical two-valued logic. It has nothing to do with whether three valued logic is consistent or whether it works.

In classical two-valued logic, there are only two truth values and one mode. Every statement is judged as either true or false. While other statements may exist, classical logic excludes them from discourse, doesn’t talk about them, and doesn’t even admit they exist.

While most of us can agree on what “true” or “false” means, there is no single word that clearly expresses the third logical value. Uncertain, undecided, unknown, debatable, and contingent all have meanings or connotations that may be misleading. The clearest understanding of what it means is developed by the behavior of the various logical expressions.

Statements such as “Roses are red”, or “Wanda is a fish” or “1+1 = 2” or “Frodo Lives” are assigned letters, such as P, Q, R, S, and truth values, T, U, and F.

So far, so good. That’s the easy part. Unlike classical two valued logic, the three valued logic allows us to make statements about statements. These are referred to as “modes”, and there are six possibilities:
P is definitely true: Symbolized []P; true when P has the value T; false if it has the values U or F.
P is possibly true: Symbolized <>P; true when P has the value T or U; false if it has the value F.
P is not possibly true: Symbolized ~<>P; true when P has the value F, false if it has the value T or U.
P is not necessarily true: Symbolized ~[]P; true when P has the value F or U, false if it has the value F.
P is equivocal: Symbolized ?P: True when P has the value U, false if it has the value T or F.
P is dichotomous: Symbolized !P: True when P has the value T or F, false if it has the value U.

I have chosen to use the “box” and “diamond” notation of the Lewis systems of modal logic, although there is a difference from these systems, to be discussed later. In the system defined by Lukasiewicz and extended by Tarski, []P is written Lp and <>P is written Mp.

It seems simple and clear enough so far, but there already pitfalls and problems that await the unwary. I will discuss these, next post.

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.