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Posted By Confutus

More comparison of logical theorems from Whitehead and Russel with Lukasiewicz 3-valued logic:

2.01 (P -> ~P) - >~P

This rule, one of those that is called "reductio ad absurdam", or proof by contradiction, or negation introduction in natural deduction systems, does not hold in 3VL. However, in general if a proposition holds in 2 valued logic, there is usually at least one modified or restricted version using modal concepts in 3-valued logic that does holds.   (P -> ~P) -> <>~P (If P implies not-P, then possibly not P) is valid in 3VL.  (P -> ~<>P) -> ~P (if P implies not possible P, then not P) is also valid.

2.02 Q -> (P -> Q)

Valid in 2VL and 3VL

2.03 (P -> ~Q) -> (Q -> ~P)

The contrapositive works in 3 valued logic exactly as it does in 2 valued logic; a convenience.

2.04 ((P-> (Q->R)) ->( Q -> (P -> R))

Valid in 2VL and 3VL

2.05 (Q -> R) -> ((P-> Q) ->(P -> R))

Valid in 2VL and 3VL

2.06 (P -> Q) -> ((Q -> R) -> (P - R))

Valid in 2VL and 3VL

2.07 P > (P v P)

Valid in 2VL and 3VL

2.08 P-> P

Valid in 2VL and 3VL

 
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

Lukasiewicz attempted to employ his 3-valued logic to use the modes of medieval logicians

It is possible that:P,

It is impossible that P

It is necessary that P,

It is contingent that P,

It is true that P

it is false that P.

For various reasons, this program had only limited success. In his prefix notation, he used Mp to indicate" It is possible that P", Np to indicate "Not P", and Cpq to symbolize "if  P then Q". 

Lewis was more successful in his developmment of modal ideas, and he used a diamond and box notation <>P to indicate "it is possible that P", and [] P to indicate "necessarily P". 

Neither of these notions corresponds exactly to the usual meanings of "possible" and necessary" in either a casual sense or a more sophisticated philosophical sense, but I hope to explore the differences later.  I prefer the Lewis box-and-diamond notation to the Lukasiewicz notation, and will be using it.

 

I downloaded a PDF of Russell and Whitehead's "Principia Mathematicia" so I could do more detailed comparison of the various systems. One of the the first things that stands out is the definition of the material conditional P=) Q as (~P v Q). Not p or Q.  This definition of the material conditional is standard in classical two valued logic,

 

The 3-valued Lukasiewicz  conditional cannot be defined this way. It allows doubtful conditionals, in which "if P then Q" is evaluated as having the middle truth value when P is true and Q has the middle truth value, or when P has the middle truth value and Q is false.  This may be explored further.

 

 
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.


 


 
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