Logical Transformations as a Klein 4-Group

After recently studying statements and their logical transformations, I was happily surprised to discover a resemblance between these transformations and a special type of group known in group theory as a Klein 4-group.

First off, the logical transformations I was studying were the contrapositive, the converse, and the inverse. For example, if we take the statement "If x then y", we can transform it into:

  1. The contrapositive: "If not y, then not x",
  2. The converse: "If y, then x", and
  3. The inverse: "If not x, then not y"

For convenience, I will abbreviate these \(C_P\), \(C\), and \(I\), and the original statement \(S\).

In exploring these transformations further, I realized that instead of going from the original statement to one of the transformations, I could apply a transformation on top of a transformation.

For example, the contrapositive of the converse is the contrapositive of "If y, then x", which is "If not x, then not y".

However, these double transformations all became other transformations! In the example above, for instance, the resulting statement is really just the inverse of the original statement. After trying out many of these double transformations, I realized that I was actually constructing a Cayley table.

Without delving to deeply into group theory, a Cayley table is a table used to show how a collection of items behave when combined in certain ways. In my case, I realized I could use a Cayley table to show how logic transformations behaved when applied more then once on a statement.

I came up with the following table:

\(~\)\(S~\)\(C_P\)\(C~\)\(I~\)
\(S~\)\(S~\)\(C_P\)\(C~\)\(I~\)
\(C_P\)\(C_P\)\(S~\)\(I~\)\(C~\)
\(C~\)\(C~\)\(I~\)\(S~\)\(C_P\)
\(I~\)\(I~\)\(C~\)\(C_P\)\(S~\)

Which has been classified, more generally, by group theorists as the Klein 4-group. Technically, this represents a transformation in any row first being applied, and then a transformation in any column being applied second with their intersection being the resulting statement. However, as is the case with this and many other types of groups, the transformations are commutative, meaning the order they are applied in does not matter.

Maybe it is just me, but I thought it so fascinating that something as fundamental as logic could be described by something so abstract as group theory, and thus relating it to a host of other applications such as physics, chemistry, and Rubik's cubes!

I would love any feedback anyone has on this topic. I have not been able to find anything online relating logic to group theory, so any extra information or ideas would be much appreciated!

Note by David Stiff
3 weeks, 1 day ago

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I don't think this is a particularly surprising result, as every group of order 4 is either cyclic or isomorphic to the Vierergruppe... Obviously it's not cyclic (since each logical statement is idempotent); hence, the only thing to verify is closure and associativity of composition of "logical transformations" (whatever that means). There's also an application of the Vierergruppe to music composition too...

I think it would be beneficial for you to add this to the Klein 4-group Wikipedia page, as well as this page, but you may want to make sure that your writing is perfect.

Gennady Notowidigdo - 3 weeks, 1 day ago

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