How do we know that basic logic is real?

Question

For example, with the equation; 1+1=x. People claim that simple logic dictates the value of x, which they say is "2". How can we be sure that the logic cited is real and correct?

Answer 1

The example you gave is probably a definition (the number "2" is defined as the sum of 1 and 1), so it does not require logic. I say "probably" because you could chose to define it in some other way and subsequently prove that 1+1=2. There are different ways of building up number theory starting with different sets of assumptions and definitions.

Basic logic sets the rules for manipulating with symbols to which we conventionally assign meanings such as "false", "true", "and", "or" and "not". The below document gives a good introduction.

In fact we do not know if number theory is consistent. It could be that our basic knowledge about numbers (such as "if y>x and z>y then z>x") is inconsistent. At some very deep level, the consistency of all math is probably improvable.

Answer 2

Any proof that logic is valid would, itself, be logical in nature, making it inadmissable due to begging the question. Basically, we have to accept that logic is valid, or we'd have no way in which to advance beyond knowledge in the taxonomy of rational thought. If you reject logic, you're left with nihilism. Nothing can be known, and progress is impossible. That's not the case, so we must have been right about logic and our ability to draw conclusions. Logically speaking, of course.

Answer 3

we cant.

Logic and every other math-system is based on axiomas

changing the axiomas will change your logic and still you will have valid reasoning within the choosen set of axiomas.