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  • An example of why this is incorrrect.

    If a card is the ace of spades, it is black.

    A card is black if and only if it is the ace of spades.

    There are other conditions under which B (a card is black) can happen, so the second statement is not true.

    A conclusion that would be correct is "If a card is not black, it is not the ace of spades.". The condition is that if A is true B will also always be true, so if B is false we can be sure that A is false as well - i.e. "If not B, not A".

  • No.

    B iff A is defined as "If B then A and if A then B".

    If that doesn't make it clear enough for you, then try writing out the truth table for both statements.

    • I think I understand now, but what has left me scratching my nose (metaphorically):

      Why is it called "B if and only if A", if what it really means is "B only if A and vice versa"? (Am I correct in thinking that's what it means?)

      I just don't understand how that translates grammatically. To me, "B if and only if A" sounds the same as "B only if A". I can accept that they mean different things in the context of logic, just like I can assign any meaning to any label, like I could say that "dog" now means "kite" in a certain context. But it seems unintuitive and doesn't really make sense to me. Does that make sense?

      • "B if and only if A" is a shorthand way to write "B if A and B only if A". It's like how "He is young and thin" means "He is young and he is thin". We could write it the second way without trouble, but the first way is shorter, we agree that they mean the same thing, and we prefer to conserve energy when writing.

        The form "if and only if" is merely a convenient shorthand. Shorthand is usually more convenient for the writer than for the reader. 🤷‍♂️

        Imagine these natural language sentences and analyze how they are different:

        • I'll go outside if it's not snowing
        • I'll go outside only if it's not snowing

        Hint: what do you do in each case when it's snowing?

      • "If A then B" means if A is true, then B is guaranteed to be true. Note that if B is true and A is false, "if A then B" is still true.

        "B only if A" means the only way for B to be true is for A to be true. It's weird, but it has the inverse truth table as "(not A) and B".

  • The first statement only tells you when B is true. It says nothing about when it is false. The second statement both tells you when B is true (if A) and when it is not (only if A). Therefore, the two statements cannot be equal.

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