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?


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 '12 at 8:37
  • The second row is also confusing, I have A, shouldn't it be true? – Red Banana Jun 29 '12 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 '12 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 '12 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 '12 at 9:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.