Identity Entails Logic

the inventors of the three laws of classical logic, never stated the laws relationships to each other.

 
Logical implication (or entailment) is a well understood logical function.  I represent the entailment in the diagram with the arrow labeled "implies".  The three boxes contain the three classical laws of logic.  I used the classical expressions for them. 

The first law is the law of identity.  It merely says that every time you talk about A, you mean the same identical thing. 

The other two formulas are variations on the afeard Law of the excluded middle.   Don’t try to use logic in a situation where the Law of the excluded middle does not obtain ... in other words where it is not ... err entailed.   Identity entails it.  That’s what the diagram says. For example If A might not be A, then not (not A) might not be A either.  Here is another example of where identity is failing.

In other words:  My diagram lays down a precise formula for when you can, and can not, apply the classical laws of logic.  If my diagram does not do so, then i surely want to know.
 
My diagram is the three laws of binary logic (see Barbara Cubed) expressed in second order logic notation showing (for the first time?) the precise relationship between them.

Comments


Perhaps another way to say this is …
 

  “if you are inside a classical logic  box, where those three axioms hold, then there is no need to even be aware of the relationship between the axioms … they are all of equal value. But what if you do not even know whether you are in a classical logic box or not ?    That is where the law of identity comes into play first … if it holds, then you know the other two laws will hold as well … if identity does not hold, then you are outside of a classical logic box.”


One research question might be:  Assume that you do not know whether identity holds in a context … but you find that LEM holds … if you assume that LEM holds, can you then prove that identity holds as well ?  ← i am betting not.

But i seem to remember that you need all three axioms to prove anything.  They are three independent axioms.

these are the three laws of logical thought.   not my coinage … it was coined in our culture when logic was developing.   so @nathan note! ... we have two quite different sets of 3 laws going in this room.


Yep. I agree. Reality is not binary and this nicely proves it.

Your A is never exactly my A no matter how similar they may seem by all the communication we can use to compare them. There will always be some difference, and usually more individually aware difference than not. Binary logic only works inside one individual, or a computer, and sometimes not even either of those.  

So then, what is the real logic of the verses?