The Heyting-algebraic definition of implication makes intuitive sense to me, or at least after you state your confusion. ‘One circle lies inside the other’ is like saying A is a subset of B, which is a statement that describes a relation between two sets, and not a statement that describes a set, so we shouldn’t expect that that mental image would correspond to a set. Furthermore, the definition of implication you’ve given is very similar to the material implication rule; that we may substitute ‘P implies Q’ with ‘not-P or Q’.
Also, I have personally been enjoying your recent posts with few prerequisites. (Seems to be a thing.)
Thanks! I’m not an amazing writer like Eliezer, but I enjoy being on LW and I want other people to enjoy it as well.
The definition of implication is actually a bit more complex, you need to take the largest open subset of “not-P or Q”. Similarly, negation isn’t just complement, but the largest open subset of the complement. That’s what makes the intuitionistic stuff work, otherwise you get classical logic as Alex said. But topology isn’t everyone’s cup of tea, so I left it out.
(Not very familiar with math.)
The Heyting-algebraic definition of implication makes intuitive sense to me, or at least after you state your confusion. ‘One circle lies inside the other’ is like saying A is a subset of B, which is a statement that describes a relation between two sets, and not a statement that describes a set, so we shouldn’t expect that that mental image would correspond to a set. Furthermore, the definition of implication you’ve given is very similar to the material implication rule; that we may substitute ‘P implies Q’ with ‘not-P or Q’.
Also, I have personally been enjoying your recent posts with few prerequisites. (Seems to be a thing.)
Thanks! I’m not an amazing writer like Eliezer, but I enjoy being on LW and I want other people to enjoy it as well.
The definition of implication is actually a bit more complex, you need to take the largest open subset of “not-P or Q”. Similarly, negation isn’t just complement, but the largest open subset of the complement. That’s what makes the intuitionistic stuff work, otherwise you get classical logic as Alex said. But topology isn’t everyone’s cup of tea, so I left it out.