Here he says that:

If you have A is sufficient for B it means that every time you have A you will have B, without exception: A -> B

If you have A is necessary for B it means that every time you have B you will have A, without exception A <- B

But I'm in doubt on what truth tables does each one holds. Can someone help me?

1 Answer 1


A -> B is just implication:

A B A -> B
t t   t
t f   f
f t   t
f f   t

A <- B is the same as B -> A, so the above truth table holds, but with different variable names.

  • I don't understand how the last two rows are possible: "Everytime you have A, you'll have B" - In the last two rows, I don't have A - considering that "have A = A is true".
    – Red Banana
    Jun 29, 2012 at 8:37
  • The second row is also confusing, I have A, shouldn't it be true?
    – Red Banana
    Jun 29, 2012 at 8:46
  • 1
    A -> B means: "When A is true, then B is also true." That is not the same as "If, and only if, A is true, then B is true." When A is false, B may be either true or false, and the implication is still true. For example, take the implication: "When it rains, the street is wet." Now, what do we know about the street when it doesn't rain? Nothing! The street could be either dry or wet – maybe because someone is washing his car. In the second row, A is true but B is false, but we want B to be true, whenever A is true. Thus the implication is false.
    – danlei
    Jun 29, 2012 at 9:14
  • Also note, that implications don't imply causality. A -> B doesn't mean "B because A" or something like that. It's really just "If A, then B", nothing more, nothing less. So, whenever A is false, B's value doesn't matter, the implication is true. If B is true, A's value doesn't matter, because the only case in which the implication could possibly become false needs a false B. That means A -> B is the same as -A v B ("not A or B"). Search for a nice introductory logic tutorial, if you're interested in this. It's too much to cover thoroughly in a few comments.
    – danlei
    Jun 29, 2012 at 9:19
  • If you want to learn formal logic properly, that is. I still hope that I answered your question understandably. Or is there something else unclear about your example?
    – danlei
    Jun 29, 2012 at 9:27

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