# 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!

# 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.

# Being revolutionary

I have decided, rather than waiting for people to discover my logic, that I need to be more aggressive, or, that is, more assertive, about promoting it. I have occasionally boasted that I am trying to start a logical revolution, or a reformation, or a renaissance, or something.

Following the suggestions that I got at LTUE on networking, I thought to look up conferences on the subject of mathematical logic. The next one remotely accessible will be in May, in Illinois, sponsored by the Association for Symbolic Logic. However, budgetary constraints are likely to put this out of reach. International conferences are even further out of reach.
In the meantime, i am trying to use social media. I renewed my participation on the Stack Exchange network, where I found a couple of questions that had easy answers since I last looked. I don’t want to pester Facebook friends or family on a subject that they have routinely found less than interesting, but I’m considering linking these posts on LinkedIn and Google+.

So, why three values in logic? In classical logic, it has been assumed that statements should be classified as true or false. This is a useful first approximation. However, we live in a world full of ambiguity, uncertainty, lack of information, and conflicting definitions of terms. Questions for the best or only reasonable answers are “I don’t know” and “i can’t tell” are excluded from logical inquiry, and “it depends” are only indirectly addressed. It seems that it ought to be possible to create a three valued symbolic logic. As it turns out, this is easier said than done. There have been several attempts made, most of them with various advantages and deficiencies.

One of the important tools of classical logic is “reductio ad absurdam”, or proof by contradiction. The presence of the third value and the failure of the law of the excluded middle mean that these cannot be directly imported from classical logic. However, it is possible that suitably modified versions can be developed and incorporated into a scheme of natural deduction.

In development of the knowledge Base, I got as far as a review of how other history applies to prehistory. Most of what is now known about prehistory has been discovered in modern times, and I am going through a review of how peoples of the world are applied to modern history. Currently, this is at cities of India. The 19th century is being connected to current events of 2018. For the 20th century, I am doing a review of how early prehistory applies. The late 20th century is being connected to weeks of 2018. The early 21st century is being connected to material culture. 2017 and the fourth quarter 2017 are being connected to weeks of 2018. I am trying to pick out notable developments in prehistory, but this requires that the weeks of 2018 be examined in more detail.

# Revolutionary

I’ve been out of town for a few days for a family funeral, so I haven’t followed up on my “Learned Professors” post.

As a result of my studies in logic, I have become convinced that the whole field of formal logic has stagnated since the 1920s. Although classical two-valued Aristotelian logic is widespread and has all kinds of applications from applications from electronics to computer science, there has been comparatively little progress in non-classical logic. This is largely because non-classical logics as they have developed have been either excessively cumbersome or seriously incomplete.
In the 19th century, George Boole pioneered the use of mathematics to represent the truth of mathematical statements. The methods he used were cumbersome and applied largely to classes of things. His logic was enormously simplified by the introduction of a fairly simple concept: The use of the inclusive “or”. Statements and arguments could be translated into mathematical symbolism, manipulated by mathematical methods, and reinterpreted as statements that were equally valid and equally correct, (or incorrect). Reasoning that was difficult or complex when expressed verbally could be simplified. This works well when applied to classical Aristotelian two valued logic. There is excellent agreement between the methods of symbolic logic and those of traditional logic.

However, it has long been recognized that classical Aristotelian logic has some severe limitations. In particular, the insistence that statements must be true or false, with no middle ground, fails to deal with the complexity of the real world, where ambiguity, uncertainty, lack of information, and conflicting definitions abound. It has worked best in mathematics, where objects can be defined without regard to whether they exist in nature. The perfect certainty of mathematics is entirely artificial.
There have been various attempts to extend logic and mathematical reasoning to the realm of the ambiguous and uncertain. These have not worked so well. In future posts, I will review a few of these.

More generally, in dealing with the knowledge base, I’ve been pushing the development of world history. This depends heavily the area I have called sociology. I’ve completed a pass through a historical review. Although I would like to work backward through weeks of 2016, I a setting this aside in favor of reviewing the roots of sociology in institutions and culture. History depends in particular on peoples of the world, and for these I am going through a summary review of history. This is mostly showing me what gaps still remain: Central Asia, Balkan and Scandinavian peoples, and Southern African peoples haven’t yet been fully treated. I am linking nations to particular weeks of 2017, to ease things when I get back to them. For Western Civilization, I have finished connecting cities as far as they have developed, reviewed the connections with other peoples, and I am now going through a review of how the area I call social mechanics applies. I am working on extending the connections of Balkan peoples to other peoples of the world, and getting notes on the history of Greece back through antiquity and prehistory. These are rather slow going.