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.

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.

The learned professors

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.

Distracted

While I was at LTUE, I had occasion to give my son some advice about being proactive, not just waiting for things to happen. Later, I thought about it and decided that it’s a piece of advice I need to take myself. This is especially true with regard to my social life. Most of my life I have been much too passive about waiting for people to approach me, and disappointed when they did not. It’s about way past time for me to be more active about meeting people.

I also had occasion to ponder the difference between me and the hero I’m writing about in the book I’m working on. One of the reasons he is a hero is because he has habits of taking care of the little things, the daily chores, the small tasks. He doesn’t procrastinate his maintenance. Me…hah. That’s something I need to work on. It’s harder when you have to train yourself as an adult than it is to establish good habits when you’re young.

on Monday, there was a post on According to Hoyt, Nah King, Nah Queen that I spend most of the day composing and monitoring comments. I do that, sometimes, if the post and content are interesting; This is a fairly congenial bunch of commenters, and I agree with and like many of them.
Tuesday, I spent some time at , which was talking about book covers, and got a chapter written so I can take it to my writer’s group tomorrow night.
Today, I did some work on my Knowledge Base. I’m back to working in early prehistory, trying to expand and extend things useful to it.

LTUE 3

Now that I am at home and can type faster than the not-so-smart phone I bought on this trip will let me do, I can give a better report. The day before, I took my older son out to lunch, and wound up giving him some Dadly advice, that he needed to chose his own direction, decide what he wants most in life, and do what it takes to get there. Afterwards I thought I need to take my own advice, when it comes to social interaction.
The night before LTUE started, I thought to check out the hotel and get familiar with the place and parking arrangements, and ran into David Farland. Last November, I attended one of his workshops. He remembered at least my face and asked how my writing was going. I was pleased to be able to report progress.
On Thursday, one of the panels that stood out to me had the title “How to Feed an Army”, with panelists Jonathan LaForce, Kal Spriggs, Brad Torgersen, and Paul H. Smith. These are all military or former military, and they talked about logistics and supply. Water, rations, gasoline and parts, medical supplies, casualties, presenting supply information so it is most useful to busy commanders, thieves, information management, and related. Since the main character in my story is conscripted into the king’s army, this was highly valuable. This was one of the blocks in my story. I wasn’t sure what he would be doing there. Now, I have a better idea and I can proceed. Kal Spriggs is the author of several works; I’ve read his “Children of Valor” series, starting with Valor’s child Valor’s Child, and liked it. I’ve only read one of Brad Torgersen’s works, The Chaplain’s War, but I liked that one, too.
I attended a “Kaffeklatsch” with Sarah Hoyt. I’ve become a fairly regular commenter on her blog, and wanted a chance to meet her. I did and introduced myself as one of her “Huns”, but she was busy and had other friends. I didn’t want to be too much of an obnoxious pest by hanging and following her around. Perhaps another time.

Friday, besides the keynote address by Jo Walton, and a panel on “Hidden disabilities”, I wound up attending surprisingly few. Most of the ones I wanted to attend were full by the time I got there, so I wound up wandering the halls. I did have a nice conversation with Jo Walton, whose work I haven’t read, although the name is familiar to me, and she kindly recommended a couple of her works. “Effective Networking for Authors and Artists” was highly useful, and I will be changing my approach my blogging and Facebook activity based on their advice. I had a chance to speak with panelist Donna Milakovic afterwards, and she was very friendly and encouraging.

Saturday, I attended a 2-hour session on medieval weapons and armor, by C. David Belt. Since had invited those attending his session to ask him about the subject, and he was fairly mobbed, I skipped the crown and went to attend the talk by Todd McAffrey, but afterward I buttonholed him in the hallway when a panel I wanted to attend was full, and he gave me some highly useful worldbuilding advice on my hero’s probable weapons, armor, and gear. The next panel I got into was on “Writing Children”, with Sarah Hoyt among others, and another one was “From Peasant to Noble: Social Mobility in Feudal Societies”. That clarified some of the rather vague notions I had about the “Now what?” after my hero kills the dragon.

I was hoping to meet Marion G. Harmon, author of the Wearing the Cape series, which I have greatly enjoyed. I did spot him and tell him I had read and enjoyed his work. He was unnecessarily apologetic about the delay in writing the next book in his series, and appeared preoccupied about something, so after telling him his series is one of my re-reads, I left him alone.

There was more, but, hey, why try to tell all? It was one of the more productive and enjoyable events I’ve ever been to. Next time I go to one of these things, I’ll be better prepared.

LTUE 2

Still at LTUE. I got a few tips about networking, a few tips about my novel, attended some interesting panels,and had a good time. It was suggested that for networking purposes, I need to attend as many face-to-face events related to writing (and my other working interests) as possible, and become active on social media such as Twitter & Facebook. I think I need to get home & assimilate first.

LTUE

Mostly just a test of my smart phone and ability to do mobile blogging. I met a few authors, got a few hints on scenes to work on for “Dragonkiller”, made a couple of contacts for possible future editorial and artistic work, and had a few other nice conversations.

Perishing

Yes, this blog is perishing of loneliness. Partly due lack of attention from me, partly due to lack of visitors (other than spammers). Comments would be highly welcome.

I will be attending the “Life, the Universe, and Everything” symposium in Provo, Utah. There are a few authors and their fans that I’m especially hoping to meet. Since most of my writing so far has been is nonfiction and goes into the Knowledge Base, I expect I will talk about the project there.

I’m also working on some fiction stories:
One of those is a fantasy; I’m only a few thousand words in. The working title for is “Dragonkiller”.
I’m also planning what I call a historical fantasy series. I haven’t actually begun writing it yet, but I have concept I like. The working title is “Magister”.

For “research” purposes, I’ve been building a e-library of GURPS supplements. I’m expecting to use this to help me build secondary characters and situations. I’ll have to see how this goes.