The on-line semi-intellectual journal of an oddball generalist.

Posted By Confutus

The weekend was a busy one. I took my mother to a High School reunion in northeastern Arizona on Saturday, to church Sunday, and a funeral on Monday. I've been doing a little work on Family history, and I got results from some medical tests.

 

I'm doing a comparison of propositions of classical two-valued logic as developed in Whitehead and Russel's Principia Mathematica. 

1.01 P -> Q defined as  ~P v Q

As already discussed, this is not true in Lukasiewicz 3-valued logic.  

1.2 (P v P) -> P

True in 2VL and 3VL

1.3 Q .-> (P v Q)

True in 2VL and 3VL

1.4 (P v Q) -> (Q v P)

True in 2VL and 3VL

1.5 P v (Q v R) -> Q v (P v R)

True in 2VL and 3VL

1.6 (Q -> R) -> ((P v Q) -> (P v R))

True in 2VL and 3VL

 


 
Posted By Confutus

George Boole invented a system of algebra which dealt with classes of objects and logic.  A few years ago, I read through his work and attempted to make sense of it, and found it difficult and cumbersome. It was greatly improved and simplifed  a few years later when logicians began using an inclusive "or" (A or B or both), rather than the exclusive sense that Boole had used and  which is more common in ordinary discourse. This took off in three directions: 1) The algebra of sets, assisted by John Venn.  2) Propositional logic; the algebra of simple, unanalyzed statements, and 3) Predicate logic, the algebra of propositional functions and quantifiers (some, and all), which is has something of an intermediate status, and which has mostly replaced the old Aristotelian syllogistic logic. 

A major development in logic was the "Principia Mathematica", which introduced an axiomatic development of logic, and attempted to base all of mathematics on logic. This wasn't entirely successful, but it had an important influence on later works.

After developing the 3-valued logic, Lukasiewicz went on to test to the propositions of Principia Mathematica to see how many of them continued to hold. Most did, many did not, and a few were contradictions in3 valued logic. I intend to follow this line and examine which ones hold up, and which do not. Lukasiewicz went on to develop a many-valued logic, which could use any number between 0 and 1 for a truth value.

At about the same time, the American logician C. I. Lewis took a different approach, and attempted to improve on propositional logic using the axiomatic approach.

Both Lukasiewicz and Lewis attemped to develop Modal logic, including notions of possibility and necessity.  These two attempts proved to be incompatible, but the reasons for the incompatibility have not been fully explored by logicians. Their works are somewhat difficult to follow, since Lukasiewicz developed a form of notation, prefix or "Polish" notation, in which an operator precedes the expressions it operates on, while Lewis used the same style of expressions used in Principia Mathematica. This has fallen out of favor and is not much used today.  Nevertheless, some comparison of the various systems is still possible.


 
Posted By Confutus

Continuing on the subject of 3-valued Lukasiewicz logic, Lukasiewicz initially proposed an interpretation of "possible" for his third truth value, but its behavior doesn't really sustain such an intepretation, and he channged it later.

In common with classical logic, one of the basic functions is negation. In common with classical logic, The negation of a true statement is false, and the negation of a false statement is true. The negation of a statment with the third truth value has the third truth value.  This makes sense; if I don't know whether "Today is Friday" is true or false, I also don't know whether "Today is not Friday" is true or false, either.

There are two other possibilities for negation.  A strong negation of a statement with the third truth value would be false. A weak negation of a statement with the third truth value would be true.  I don't expect to make much use of the strong or the weak negation, because these can be expressed in other ways.

 

There is a larger problem with his notion of equivalence. He uses logical equivalence and defines it as a biconditional; if A is true, then B is true, and if B is true, then A is true. However, One of the elementary concepts of matthematical structure is an equivalence relation, which is defined as a relation that is reflexive (A = A), symmetric (if A = B, then B = A), and transitive (if A = B and B = C, then A = C). The relation Lukasiewicz defined fails the third property. His logical equivalence is not a mathematical equivalence. Most of the other three-valued logics that have been devised also fail this test.  In contrast, the classical two-valued biconditional is a mathematical equivalence.

 

This deficiency can be repaired using concepts Lukasiewicz defines later, but this an apparenly unrecognized and unremarked  major reason three-valued logic hasn't been as useful as classical two-valued logic.


 
Posted By Confutus

My attempted change to more political blogging didn't go well. I also switched from working directly on my knowledge base to more computer programming. Then I decided to move from West Virginia to Phoenix, Arizona, and getting moved and established took higher priority than blogging.

 

One of the things I've been wanting to do for a while was to resume my studies in logic. I e-mailed a professor at ASU, who didn't sound very interested. That..peeves me, every time. So I decided to begin with reviewing my research.  I went out to ASU to the library to look up some old references, and take some notes.

My work is a new extension of the 3-valued logic of Jan Lukasiewicz, which he presented in 1921, so I looked up where I could find it. There are a couple of sources for an English translation, the better one was:

Jan Lukasiewicz: Selected works

Ed. Ludwig Borkowski

North Holland, Amsterdam, 1970

 

The paper "On Three Valued Logic"  (p 87-88 in Selected Works) provides conventional definitions for the conditional, logical equivalence, conjunction (and) and disjunction (or), and negation, and then three-valued versions of the same. I have determined that these definitions are inadequate, but they can be supplemented based on concepts he describes later. 

One thing he notes immediately is that some of the laws of classical two-valued logic do not hold in the three-valued version, notably principles of bivalence and the excluded middle, and makes the claim that "in three valued logic there are no antimonies", (or paradoxes). This is not quite the case; there are numerous problems with statements that are true and useful in classical logic that fail in the three valued version. Why they fail is work examination. For instance, he notes that  the common rule "If A implies B and B implies C then A implies C, which applies in classical logic,  fails in the three valued logic. In order for this logic to be useful for logical purposes, common rules of reasoning need to be at least explored and explained.


 

 

 
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