Monthly Archives: April 2018

More connections

Since history depends so heavily on sociology, I have been reviewing and rewriting the connections of sociology to institutions. This is not producing much significant progress, because I haven’t been working in the details. A review of the history of peoples of the world has also produced fairly minimal progress. Nations are now connected to weeks of 2017, which is a bit further back than I have analyzed events. I am now free to begin connecting nations to biographies. For Western Civilization, I have felt a need to examine historical roots, which has meant connecting Greece to early antiquity and prehistory. I’ve finally accomplished this, and extended about a dozen other nations to the same periods, so there will be a little bit of catching up involved. I’ve also done a review of social mechanics and religion to Western Civilization, and it’s now time to finish connecting elements of government. I’m taking Asiatic peoples through a review of connections to Western Civilization. Oriental peoples, and India, are still being connected to particular nations. Communities are being connected to biographies, and so are social mechanics as. For Institutions in general, I’ve done a review of peoples, social mechanics, culture, and anthropology. These are now being connected to biographies. Connecting more things to biographies has long been a part of the overall plan, and I’m pleased to finally be making more progress with it.

Aristotle’s “Categories” relates to linguistics and logic, which is not quite my area of focus. I’ve now downloaded his next book; “On Interpretation”which has more of the content I am looking for.

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.