What does Aristotle say?

I really don’t know. For some time, I’ve been avoiding looking closely at the origins of logic, and specifically modal logic, in the writings of Aristotle. I’ve run out of excuses and I finally downloaded a copy of Aristotle’s “Categories” (E.M. Edgehill’s translation, from the Gutenberg project), to start looking it over. I will have a little more to say on this when I’ve studied more of it.

One of my ongoing projects has to do with writing what I call historical fantasy. I finally broke down and connected two areas of particular changes and movements; namely stone age developments and the agricultural revolution, all the way back to their origins in the Knoweldge Base. This should allow me to do more with those subjects.

Switching

I’ve come close to the limit of what I currently have to say about logic. I had a comment that commended me for putting it out there for free instead of an e-book, but it appeared in my spam comments and the reference was generic. I may want to go that route anyway, including all the tables and proofs of the theorems and equivalences and various other claims. There are more ideas percolating and more extensions I can work ing, but I’ve spent the pent-up steam until I can get more questions or comments. I’ve been active on MathExchange and Google+ and have posted a few more comments and claims there, but so far nobody has really bitten.

I’ve been continuing work on the Knowledge Base. The last time I reviewed progress, I found the major subject of Institutions to be most in need of development. I’ve been going through a review of how peoples of the world apply to them and I’m currently connecting regions of China. Culture shouldn’t be too far behind it. I still need to get this caught up to current weeks, and I also want to broaden the connections to the smaller nations.

I’m also trying to use the eLearning feature on LinkedIn to pick up and renew my work in computer programming, but this isn’t the most urgent priority right now.

What’s so revolutionary?

This is a continuation of a series of posts on three-valued logic which began with “The learned professors”

There are about seven reasons this work is revolutionary.

1) First, it is a full-featured extension of classical logic into the realm of the uncertain. The arithmetic of the integers is an extension of the ordinary arithmetic of whole numbers into the negative, and the arithmetic of common factions is an extension of ordinary arithmetic into the realm of parts of objects. Both are fully compatible and include the whole numbers as a special case. In a similar fashion, all the laws of classical two-valued logic remain true in the two-valued case, but some of them must be modified in the three valued case.

2) It is truth functional and the same methods of truth truth tables and algebraic manipulation apply in this logic as in two valued logic. It follows the associative, commutative, distributive, identity, annihilator, idempotent, double negation, and De Morgan laws of Boolean algebra but does not follow the complementation laws x & ~x= F and x ∨ ~x = T. This is called a De Morgan algebra.

3) It is a truth functional system of modal logic. It differs from the Lewis systems S1-S5, because it uses a different version of the strict conditional, and because it does not include the law of the excluded middle. However, theorems of L3M that are parallel to the axioms of S5 can be formulated, therefore anything that can be proven in S5 can be proven with this logic. It has been established that S5 itself cannot be reduced to 3 values, but this is a narrow result and can probably be traced to the difference in conditionals.
It can also be established that that if the strict material conditional []( P =) Q) holds, then so does the strict Lukasewicz conditional [](P -> Q), but not conversely.

4) It is a truth functional version of intuitionism. In similar fashion to modal logic, if all the axioms of intuitionism are replaced using the strong negation ~<>P instead of the ordinary negation, and using the strict Lukasewicze conditional instead of the ordinary material conditional, the resulting axioms are theorems of this system. Therefore, anything that can be proven using these axioms is also true in this systems.

5) It clarifies long-standing controversies in logic, notably the meaning of the conditional and its relation with concepts of implication and entailment, and doubts about the universal validity of modus ponens. Along with fuzzy logic, it offers a resolution to the Sorites paradox. It addresses some of the same concerns as relevance logic. The Lukasiewicz conditional can be defined as (~Q \/ P \/ P==Q), if P== Q has been defined as P and Q having the same truth value.

6) It is connected to Fuzzy logic. It has some of the same features, although this logic includes all the values between T and F in the one value U, where Fuzzy logic gives each truth value a distinct number.

7) It shares features with paraconsistent logic. (P & ~P) => Q is not a tautology, so it is not explosive, and is not sufficient to prove any other proposition at all. However, this logic does include methods of indirect proof, which work by establishing contradictions. It is simply necessary to establish a contradiction and not merely a contrary. Expressions such as (p & ~<>P) “true and impossible”, ([]p & ~P) “certain and false”, ([]p & ~[]P) “certain and not certain”, (<>P & ~<>P) “possible and impossible”, and ([]P & ~<> P) “certain and impossible” all work.

Equivalence

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

The strict Lukasiewicz conditional, unlike the ordinary material condition, the strict Lewis conditional, or the ordinary Lukasiewicz conditional, meets the condition for an ordering relation: It is symmetric (A => A), antisymmetric (A => B & B=> A if and only if A = B), and transitive (if A => B and B => C, then A => C). It also satisfies the requirements for a desirable logical entailment relation: if A => B is true, B is not less true than A.

In classical two-valued logic, the notion of logical equivalence is expressed by the biconditional; “if A then B, and if B then A”, or “A if and only if B”. Because of the simplicity of 2-valued logic, this also expresses the happy property of being truth functional, that is, it is possible to evaluate a logical expression algebraically, by plugging in truth values for the variables and evaluating the expressions to determine whether they are true or false. This also expresses the notion that if two expressions are logically equivalent, they have the same truth value. The laws of logic are tautologies, that is, they are expressions that are always true, whatever truth values are substituted for the logical variables. This also a mathematical equivalence, because it is reflexive, (A = A), Symmetric (if A = B then B = A, and transitive (if A = B and B = C, then A = C).

This becomes more difficult in three valued logic. Most of the “logical equivalences” used in three valued logic are biconditionals, but not mathematical equivalences, and do not reflect the notion that two expressions have the same truth value. This is particularly obnoxious with the Lukasiewicz biconditional.
<-> P
* T U F
* T T U F
Q U U T U
* F F U T

However, it is simple enough to apply the “Definite” operator, [](P <-> Q) to define a new strict Lukasiewicz biconditional
<=> P
* T U F
* T T F F
Q U F T F
* F F F T

This does have the same desirable properties of being both a logical equivalence and a mathematical equivalence.
It’s simple enough, but it hasn’t been done and does not appear in the standard references or surveys of multi-valued logic. Kleene’s 3-valued logic does have this equivalence, but it uses the standard definition of the conditional P -> Q = (~P \/ Q), which doesn’t work in three values.

Next, I will discuss some of the more revolutionary consequences and implications of this logic.

Conditionals and Implication

The elementary theory of mathematical structure distinguishes between relations, such equality and order, that may be true or false, and operations such as addition and subtraction, that yield other objects of the type. In the study of classical two-valued logic, since the objects of the type are truth values, this distinction between relations and operations becomes invisible and they can be treated the same. Thus, the relation of implication, “if P then Q” can be expressed and even defined as (P -> Q) = (~P v Q). This is known as the material conditional, and it has been much debated and challenged. However, it continues to be used in classical two-valued logic, because it works.

It works because it expresses an ordering relationship among the truth values of P and Q: in mathematical terms, the truth value of P is less than or equal to that of Q. This is more easily understood if it is reversed: The truth value of Q is greater than or equal to P; Q is no less true than P, or Q is at least at true as P. This is a fundamental criterion of valid reasoning. We want to guarantee that if we start with true premises (summarized as P), and use valid reasoning (P->Q), we can safely draw conclusion Q without introducing error. This gives a succinct understanding that often baffles elementary logic students. If a statement P is false, the conditional P->Q is true, because any other statement at all is at least as true as a known falsehood. if Q is true, the conditional ) ->Q is true, because a known truth is at least as true as any other statement. This applies strictly to the truth values and holds regardless of the content of P or Q. However, conditionals derived from this process are in practice useless. If P is known to be false, p -> Q may be true, but it tells us nothing about Q, which may be true or false. If Q is true, P -> Q is true, but we have already reached our conclusion, and P could be equally well be or false.

But this only works in the constrained world of two-valued logic. Intuition and experience suggest that these aren’t necessarily true in the wider world of reasoning which includes uncertainty. For nearly a century, the development of logic beyond classical logic has been limited by a lack of a conditional that has equal power. This is not longer the case. I consider four cases.

In two valued logic, the laws of logic are expressed in the form of tautologies, statements that are always true, regardless of the values of the variables. The law of bivalence (P \/ ~P) is always true, whether P is true or false. The law of the excluded middle ~(P & ~P) is always true. More usefully and importantly, such statements as (P v Q) <=> (Q v P) are always true.

The first is the straightforward extension of the definition of material conditional, ~P or Q. This doesn’t work in three values, because even the simplest implications, (P -> P) fails…it has the value U when P does, so that using this definition, one cannot even establish the basic principle, If P, then P, for instance “If Harvey is a giant white rabbit, then Harvey is a giant white rabbit” as a general rule. This is very bad.

C.I. Lewis, who was much concerned with the deficiencies of the material conditional when applied outside the limited context of classical two-valued logic, proposed a strict conditional, which he defined as “It is not possible for P to be true and Q to be false”, and used this as the basis for modal logic. This also doesn’t work in a three valued system. Lewis assumed it anyway, and this assumption breaks his systems. One cannot use the truth table methods in his systems to establish whether a given formula is valid. Thether a formula is a logical law or not has to be derived using the methods of deductive inference from the axioms. Although he was concerned with what he called the paradoxes of the material conditional and thought it unreasonable that the P -> Q should follow from the mere falsity of P, and proposed the strict conditional as an alternative, it is also possible to form similar paradoxes of the strict conditional.

Lukasiewicz took a different approach. He defined the conditional with a truth table:
Q
T U F
T T U F
P U T T U
F T T T
This has the same table as the material conditional, except that the central entry, corresponding to U -> U, is regarded as T. There is no obvious reason why this should be so, except that it seems to work. With this conditional, it is now possible to formulate some basic laws, such as “P -> P” as tautologies. The problem is that it fails as an implication. One of the basic rules of logical inference, modus ponens, which can be expressed as “(P & (P -> Q)) -> Q”, is almost a tautology, but fails in one case. It has the value of U in the case where P has the value of U and Q of F, although it is T in every other case. When Lewis and Langford discussed Lukasiewicz’ three valued logic, they observed other failures of this type, and speculated on laws that would work, but ultimately dismissed the logic as practically unworkable. They missed the truth by a whisker. Modus ponens fails because it should fail, and it should fail because Lukasiewicz allows doubtful conditionals. If P is doubtful, and P -> Q is also doubtful, using unrestricted modus ponens could lead to reaching false conclusions from a combination of doubtful premises and doubtful conditionals. This is not and should not be valid logic.

What went unobserved is that this deficiency could be repaired. What is truly remarkable is that this solution has gone unnoticed, unpublished, and unpublicized for nearly a century. It is simply to apply the “definite” modal operator to his conditional and define a strict Lukasiewicz conditional as [](P ->Q). This is where, to borrow and adapt Mark Twain’s phrase, “The difference between the right conditional and the almost right conditional is like the difference between lightning and lightning bug”. Shazam!

Conjunction and Disjunction

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.

Negation and uncertainty

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.

Pitfalls and problems in 3-valued logic

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.

Truth values and modes

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 Brief Logical History

I have just connected this blog to Google+ and LinkedIn. In case this works and you are just now seeing this and joining me, I began this particular series last week with The Learned Professors I continue my series of blog posts on my developments in logic. Welcome to the Revolution.

Mathematical or symbolic logic was developed by George Boole, although not it its present form. It was at first quite cumbersome. When the “inclusive or” (A or B or both) was adopted instead of the “exclusive or” (A or B but not both), it became possible to substantially simplify its expressions. Two major branches developed: Propositional logic, which dealt with the truth of simple sentences, and predicate logic, which dealt with classes and sets of objects.

Beginning in 1910, Bertrand Russell and Alfred North Whitehead attempted to develop logic as a formal system with a few axioms and deduce the rest of it using logical methods. This led to the development of alternate logics in which some of the rules of inference of traditional logic were modified and their consequences explored. Two of the principal varieties were intuitionistic logic, which was formalized by Arend Heyting in the 1930s, following the thinking of Earl Brouwer; and the Lewis Systems of modal logic, which is the logic of possibility and necessity. These systems were developed in the 1920s.

Also, in the 1920s, Jan Lukasiewicz, a Polish logician, developed a three valued propositional logic, and following his lead, other multi-valued logics were developed. Notable among the three valued versions are the Kleene and Priest logics, the Bochvar logic, and the Post logics. Of these, the Kleene and Priest logics are most closely related, and I will skip over the Bochvar logic and the Post logics.

None of these is quite adequate as an extension of classical logic. On the one hand, intuitionistic logic and the Lewis systems lack the important, powerful tool of truth tables in order to evaluate propositions and proposed theorems. On the other hand, the multi-valued logics lack good deductive rules of inference and are forced to use elaborate circumlocutions in order to prove or establish useful theorems and results. They are unbearably cumbersome and their explanations are turgid and full of symbolism and notation that only the initiate can read or comprehend.

An alternate approach, developed by Lotfi Zadeh, is “fuzzy logic”, which is similar to the logics developed by Lukasiewicz, but using numeric degrees of truth.

My approach is revolutionary because by providing a sound basis of inference in Lukasiewicz 3-value logic, it offers the possibility of unifying intuitionism, modal logic, the most important three valued logics, and concepts of fuzzy logic with the same power and ease of that classical logic. I mean to unify and simplify a good part of logic.

My biggest difficulty is getting anyone to pay attention. My “Hey Lookee Here What I Found!” has met with a resounding “Ho Hum”. However, lack of interest is not disproof. Since I haven’t been granted entry to the sacred halls of upper academia and have no mentor to guide me in the secrets of successfully getting published, and since I don’t know how to phrase or market my ideas to make them interesting, I’m doing this here in small chunks.