How Does Scaled Boolean Algebra Map to Numerical Operations in Measure Theory?

[friend]

Does this define "measure"?

I've been reading the following paper:

Scaled Boolean Algebra, by Michael Hardy, arXiv:math/0203249v1.

and I'm wondering how much his efforts prove. He seems to prove that Boolean Algebras (such as exist in set theory and Propositional logic) can be mapped to math if the variables used (sets in set theory and propositions in logic) are part of a partially ordered set. If this partial ordering prescribes that some variables are larger or have more weight than others, then this weight can be mapped to relative increasing numerical values. He then goes on to show that this also allows set theoretic union or logical disjunction to be mapped to addition and set theoretic intersection or logical conjunction to be mapped to multiplication. And all this seems to be a way of defining a measure on sets in what seems to be the usual way. Is this the only way to map set theory and/or Propositional logic to numerical math operations? Are his efforts necessary to define a measure? Thanks.

 

[SW VandeCarr]

I've been reading the following paper:

Scaled Boolean Algebra, by Michael Hardy, arXiv:math/0203249v1.

and I'm wondering how much his efforts prove. He seems to prove that Boolean Algebras (such as exist in set theory and Propositional logic) can be mapped to math if the variables used (sets in set theory and propositions in logic) are part of a partially ordered set. If this partial ordering prescribes that some variables are larger or have more weight than others, then this weight can be mapped to relative increasing numerical values. He then goes on to show that this also allows set theoretic union or logical disjunction to be mapped to addition and set theoretic intersection or logical conjunction to be mapped to multiplication. And all this seems to be a way of defining a measure on sets in what seems to be the usual way. Is this the only way to map set theory and/or Propositional logic to numerical math operations? Are his efforts necessary to define a measure? Thanks.

As far as I know, the usual definition of a measurable space (endowed with a non negative measure) does not rely on the definition of partially ordered sets. A non empty set, the collection of all its subsets and the cardinality of these subsets define a non negative measure space. The measure of this space is a real valued function defined on the subsets of this set that is additive (typically countably additive) for disjoint subsets and zero for the empty set.

The partial ordering of elements in a set does not rely on cardinality (greater than one). It is based on an ordinal representation of elements in a set

 

[Stephen Tashi]

Is this the only way to map set theory and/or Propositional logic to numerical math operations?

One hesitates to answer any question about Boolean algebra that has an "and/or" in it!

Also, to say that someone maps "set theory" to "math operations" is open to a wide interpretation. The garden variety of measure theory that is used to define integration on the real numbers does provide a system for assigning numerical values to sets that are in an algebra of sets and does give formula for computing the measure of intersections, unions and complements of these sets.

Are his efforts necessary to define a measure?

From glancing at that paper, his main effort isn't to prove the possibility of assigning a measure (in the technical sense of the word "measure", as in "measure theory"). So, no, not all his efforts are necessary to define a measure on a boolean algebra. He defines something called a "scale". I think his efforts are mainly directed to showing that algebras that have a scale behave like algebras that have a probability measure - but I haven't read the paper.

 

[friend]

Perhaps I'm confusing scale with measure. In measure theory, the bigger the set the larger the measure it has. But then again the size of a set could also be a means of partial ordering, right, so that it also gets assigned a larger scaling number.

What I'm really curious about is how one goes from set theory operations or from propositional logic operations to math operations. And I was thinking that this paper provided some means of doing that, perhaps the only means. Is it a "map" that gets us to math, or is there something more basic like simply counting elements in the set after set operations are performed? That doesn't seem to work for converting logic operations to math because true and false can only be converted to 0 or 1. But scaling the decree of evidence for a proposition from any value between 0 and 1 does seem to allow consistent math operations. Any guidance in these matter would be very much appreciated.

 

[Stephen Tashi]

Perhaps I'm confusing scale with measure. In measure theory, the bigger the set the larger the measure it has. But then again the size of a set could also be a means of partial ordering, right, so that it also gets assigned a larger scaling number.

In "measure theory", a "measure" has a technical and abstract definition. When you say "bigger" it isn't clear what measure you have in mind. To talk about the "size" of a set might mean to talk about it's "cardinality". That can lead to discussions of the "sizes" of infinite sets in terms of tranfinite cardinal numbers. Or the "size"of a set of number might mean something like it's length. Then we get into technical questons about assigning length to sets like "the set of all rational numbers between 0 and 1". If you formed a partial order of sets by length the the interva [2,4] would be bigger than the interval [0,1]. If you partially order sets by inclusion ( [itex] \subset [/itex]) then the two intervals are not comprable since neither is a subset of the other.

The definitions of "measure" and "cardinality" are fairly standard in mathematical literature. The definition of "scale" is (as far as I know) unique to that paper.

What I'm really curious about is how one goes from set theory operations or from propositional logic operations to math operations.

And I was thinking that this paper provided some means of doing that, perhaps the only means. Is it a "map" that gets us to math, or is there something more basic like simply counting elements in the set after set operations are performed?

There is no claim in that paper that you can develop a theory of evidence based on simply counting the elements in a set. Nor is there any theory of probability that says you can always assign probabilities to sets by counting their elements. The cases when you can do that have special properties, such as the assumption that all elements of the set have equal probability. There information that all elements have equal probability must be, in a manner of speaking, added to the situation. It isn't an axiomof set set theory. Likewise, when people create measures on structures, they usually begin by assuming that there is a function that assigns a number to some of the things in that structure. Then they deduce that numbers can be computed for the rest of the structure. To applying the theory to real life problems, the function that assigns the numbers depends on particular details of the problem.

That doesn't seem to work for converting logic operations to math because true and false can only be converted to 0 or 1. But scaling the decree of evidence for a proposition from any value between 0 and 1 does seem to allow consistent math operations.

You aren't stating a clear mathematical question. You apparently see some paradox, but I can't interpret what you mean.

 

[SW VandeCarr]

The word "scale" generally applies to "measures" like temperature scales (Celsius and Fahrenheit). These are not true measures in that they to not refer to non negative additive quantities. To solve most equations in physics involving temperature, the scale value must be converted to an absolute measure such as Kelvin.

As far as partially ordered sets go, a set of cardinality n can have more than one unique ordering if n is greater than one, so any such measure would be ambiguous relative to the size of the set. So the answer to your question would be no.