Axiomatic Systems: Does Boolean Logic Apply?

[Manchot]

I was just thinking about something recently, and I'm now posing a question. It seems to me that in all mathematical proofs I've seen, there is an implicit assumption that the rules of Boolean logic are what are "logical." That is, if you have two mathematical statements P and Q, you assume that ~(P and Q) is the same thing as ~P or ~Q, and that P and (P implies Q) is the same thing as Q, etc. Is it possible to create other "mathematics" where these standard Boolean rules do not apply?

 

[DeadWolfe]

Yes. For example:

http://en.wikipedia.org/wiki/Constructive_mathematics

 

[tacman]

This is a very interesting question and one that has been debated among mathematicians and logicians for a long time. Axiomatic systems are the foundation of mathematics and they are based on a set of axioms and rules of inference. Boolean logic is one such system, but there are other systems that exist.

First, let's define what Boolean logic is. It is a system of logic that deals with propositions, or statements that are either true or false. It is based on three basic operations: and, or, and not. These operations follow specific rules, such as De Morgan's laws, which state that ~(P and Q) is equivalent to ~P or ~Q. In other words, Boolean logic is based on the principle of bivalence, which means that a statement is either true or false, and there is no in-between.

Now, to answer your question, it is certainly possible to create other "mathematics" where these standard Boolean rules do not apply. In fact, there are many different logics that exist, each with its own set of axioms and rules of inference. One example is fuzzy logic, which allows for statements to have degrees of truth instead of just being true or false. In this system, De Morgan's laws do not hold, and ~(P and Q) is not necessarily equivalent to ~P or ~Q.

Another example is modal logic, which deals with the notions of necessity and possibility. In this system, the rules of Boolean logic do not always apply. For instance, in modal logic, ~(P and Q) is not always equivalent to ~P or ~Q.

So, to answer your question, Boolean logic is just one of many axiomatic systems that exist, and it is not the only one that applies to mathematics. While Boolean logic is widely used and accepted, there are certainly other systems that can be used to describe and analyze mathematical concepts. It all depends on the specific axioms and rules of inference that are used in a particular system.