The material conditional, P→Q defined as ￢P∨Q, is usually thought not to match the usual linguistics, as seen by many paradoxes. The Wikipedia article gives few good examples. I tried to resolve them by giving an alternate logic system.
It is a many-valued logic that incorporates multiple presence of worlds. Every statement may hold in some worlds, and may not hold in the other worlds. ￢P queries for worlds where P doesn't hold, P∧Q queries for worlds where both P and Q hold, and P∨Q queries for worlds where either P or Q holds. A tautology is the statement holding in every world, and a contradiction is the statement holding in no worlds.
The most distinctive feature of this logic is its definition of implication. P→Q queries whether the worlds where P holds is included by the worlds where Q holds. If the inclusion satisfies, the implication is a tautology. Otherwise, the implication is a contradiction. As a consequence, every implication is bivalent, unlike usual statements.
This seems to solve many paradoxes of material implication. To demonstrate:
P→￢P might be true in classical logic, but unless P is a contradiction, it's always a contradiction in my new logic.
Let P and Q be completely unrelated statements. P be "Dannyu NDos is in his office," and Q be "Dannyu NDos will go swimming today." Then P→Q is a contradiction. Blatantly false, unlike ￢P∨Q.
Q→⊥ is a tautology if and only if Q is a contradiction.
Long story short, my logic goes along with Plato's idea, but insists that there are multiple ideas.
However, I also see some problems:
The main question is, how be so sure that implications are bivalent, when we all are living in one reality? If there really were alternate worlds, how be so sure that everyone in every world will agree whether the implication holds?
Here, law of excluded middle works, double negations can be eliminated, and proof by contradiction works because (P→⊥)→￢P is a tautology. Does this mean constructivists will refuse to accept my new logic system?
How did my new logic system succeeded being a relevance logic without being paraconsistent?