Polarity (proof theory)
Polarity in proof theory is like playing with magnets. You know how you have magnets and they have a North pole and a South pole? Well, things in proof theory can also have poles, but instead of magnets, these things are called formulas.
A formula is like a statement that can either be true or false. Just like how a magnet has two different ends, a formula can have two different poles, and they are called polarities. These polarities can be positive or negative.
Positive formulas are like the North pole of a magnet, they attract other positive formulas and repel negative ones. Negative formulas are like the South pole of a magnet, they attract negative formulas and repel positive ones.
So in proof theory, when we want to prove something, we need to make sure that all our formulas have the right polarity. We need to make sure that all the positive formulas are true and all the negative formulas are false. If we get our polarities wrong, then our proof won't work and we won't be able to show that something is true or false.
Think of it like a puzzle where we need to fit all the puzzle pieces in the right way. If we try to fit a piece in the wrong way, it won't fit and the puzzle won't be complete. In proof theory, we need to make sure that all our formulas fit together in a way that makes sense and that our proof is complete.
So, to sum it up, polarity in proof theory is like playing with magnets where formulas have positive and negative poles that attract or repel each other, and we need to make sure all our formulas fit together in a way that makes sense to prove something is true or false.
A formula is like a statement that can either be true or false. Just like how a magnet has two different ends, a formula can have two different poles, and they are called polarities. These polarities can be positive or negative.
Positive formulas are like the North pole of a magnet, they attract other positive formulas and repel negative ones. Negative formulas are like the South pole of a magnet, they attract negative formulas and repel positive ones.
So in proof theory, when we want to prove something, we need to make sure that all our formulas have the right polarity. We need to make sure that all the positive formulas are true and all the negative formulas are false. If we get our polarities wrong, then our proof won't work and we won't be able to show that something is true or false.
Think of it like a puzzle where we need to fit all the puzzle pieces in the right way. If we try to fit a piece in the wrong way, it won't fit and the puzzle won't be complete. In proof theory, we need to make sure that all our formulas fit together in a way that makes sense and that our proof is complete.
So, to sum it up, polarity in proof theory is like playing with magnets where formulas have positive and negative poles that attract or repel each other, and we need to make sure all our formulas fit together in a way that makes sense to prove something is true or false.