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tristanhaze comments on Logic as Probability - Less Wrong
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Do you know what truth tables are? The statement "A->B" can be represented on a truth table. A and B can be possible. not-A and B can be possible. Not-A and not-B can be possible. But A and not-B is impossible.
A->B and the four statements about the truth table are interchangeable. Even though when I talk about the truth table, I never need to use the "->" symbol. They contain the same content because A->B says that A and not-B is impossible, and saying that A and not-B is impossible says that A->B. For example, "it raining but not being wet outside is impossible."
In the language of probability, saying that P(B|A)=1 means that A and not-B is impossible, while leaving the other possibilities able to vary freely. The product rule says P(A and not-B) = P(A) * P(not-B | A). What's P(not-B | A) if P(B | A)=1? It's zero, because it's the negation of our assumption.
Writing out things in classical logic doesn't just mean putting P() around the same symbols. It means making things behave the same way.
'They contain the same content because A->B says that A and not-B is impossible, and saying that A and not-B is impossible says that A->B. For example, "it raining but not being wet outside is impossible."'
If you're talking about standard propositional logic here, without bringing in probabilistic stuff, then this is just wrong or at best very misleadingly put. All 'A->B' says is that it is not the case that A and not-B - nothing modal.