How do implication and equivalence work in propositional logic?

"⇒ (“implies”)\n\nOkay, now for the toughest one. We’re going to spend significant time thinking through this one carefully, because it’s both important (in some ways, the most important of the operators) and also potentially baffling. I’ve studied this stuff for years, and I still sometimes get stuck when trying to figure out ⇒. If we say “X ⇒ Y,” we’re claiming that “if X is true, then Y is true.” Note carefully that we are not claiming that X itself is true. We’re simply asserting that if it’s true, then Y must necessarily also be true. We call the first part of a ⇒ proposition the premise, and the second part the conclusion. Here, X is the premise and Y the conclusion. So far, it seems easy. It gets a little slippery when you realize that the only claim “X ⇒ Y” is making is: “if X is true, then Y must be true.” If X is not true, then “X ⇒ Y” is making no claim at all. Confusingly enough, this means that except for the one scenario where X is true but Y is false, the statement “X ⇒ Y” itself is always true. So, besides the obviously sensible case when X and Y are both true, X ⇒ Y is true even when: (1) X is false and Y is true, and (2) X is false and Y is false. Or, to put it succinctly: X ⇒ Y is true whenever either X is false or Y is true or both.\n\nFor example, A ⇒ C is a true proposition, believe it or not. In English, it says “UMW being in Virginia implies that dogs are carnivores.” The proposition B ⇒ A is also true: “The King of England being female implies that UMW is in Virginia.” What possible sense can we make out of these nonsensical claims? The key to understanding it, for me at least, is twofold. First, remember that to a computer (or a logic system), there is no meaning to the propositions: they’re simply atomic building blocks, each of which is true or false. So the fact that to a human, the content of the propositions might have nothing to do with each other — English Kings and dogs — is irrelevant to a computer: it just thinks indifferently in terms of “X” and “Y,” and has no idea what real-world entities any of this refers to. Second, think in terms of ruling out counterexamples. When I assert X ⇒ Y, what I’m saying is “it’s impossible for X to be true and Y false, because X’s truthfulness would imply Y’s truthfulness.” Just as when I assert X ∨ Y I’m promising that either X or Y is true (or both), when I assert X ⇒ Y I’m promising that either X is false or Y is true (or both). In this way, it starts to make sense when someone says, “Iowa being in the Southern hemisphere implies that Batman’s cape is red.” That assertion is like a promise: “if it turns out that Iowa is in the Southern hemisphere, then I guarantee Batman’s cape is red.” But since Iowa isn’t in the Southern hemisphere, all bets are off. The conclusion was conditional on the premise.\n\nThe reason this operator is so important is that in artificial intelligence, the name of the game is concluding new facts from known existing facts, so that knowledge is increased. Every time a ’bot learns that X ⇒ Y is true, and then also learns that the premise (X) is true, it can conclude that the conclusion (Y) is true, even if it was never explicitly told that Y was true. This rule of logic is called modus ponens, and is the workhorse of automated knowledge bases.\n\n⇔ (“equiv”)\n\nFinally, the proposition X ⇔ Y is true whenever X and Y have the same value: they’re either both true, or both false. This can be seen as “implies in both directions,” since X ⇔ Y means “if X is true, then Y is true; and if Y is true, then X is true.” This operator is also the inverse of ⊕, since X ⊕ Y is true only if X and Y are different, and X ⇔ Y is true only if they’re the same.\n\nThese operators, which each produce another proposition (called a compound proposition) from the proposition(s) they operate on, can be combined to form complex expressions. For instance:\n\n• ¬ B is the proposition that the King of England is not female. (This is true.)\n\n• A ∧ ¬ B is the proposition that UMW is in Virginia and also the King of England is not female. (This is also true.)\n\n• C ⊕ (A ∧ ¬ B) is the proposition that either dogs are carnivores or UMW is in Virginia and the King of England is not female. (This is false, because both halves of the xor are true.)\n\n• (C ⊕ (A ∧¬ B)) ⇒ ¬ A is the proposition that if either dogs are carnivores or UMW resides in Virginia and the King of England is not female, then UMW must not reside in Virginia. (This is true, since dogs are carnivores and UMW resides in Virginia and the King of England is not female, so the left-hand side of the ⇒ is false, which means that the entire expression is true regardless of the truth value of the right-hand side (which is also false, since UMW doesn’t not reside in Virginia.)\n\n• Etc."