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- TL;DR Summary
- Equivalence Relation fails when two propositions are not in the same sample space. Why?

It is a theorem that: two propositions implying each other, in the sense that the set of outcomes making one true is the same as the one making the other true) have the same probability. this comes from the fact that if p --> q, the P(p&q) = P(p), we have that if p <-> q, then P(p&q) = P(p)= P(q). but this is only so if p and q dwell in one sample space.

Question: what is the problem when they are not in the same sample space?

Question: what is the problem when they are not in the same sample space?