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Task on propositional logic:

"Suppose you create a truth table for A and B, both formulas in propositional calculus, and have a look at the columns below the main connectives of A and B. When do we know for sure that A ≡ B is true?"

My understanding is that this option should be correct:

"If there is a T in every row under A and a T in every row under B."

However, the answer sheet says it's not correct. Why is that?

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  • $\begingroup$ How do you know that the answer sheet is correct? $\endgroup$
    – Somos
    Feb 21 at 20:31
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    $\begingroup$ If there's a T in every row under A then A is a tautology, no? Same for B. Two tautologies are equivalent. But that's not the only way for two formulas to be equivalent. Maybe that is what the answer sheet is saying. $\endgroup$
    – David K
    Feb 21 at 22:14
  • $\begingroup$ Does this answer your question? Do Biconditionals Have to be Logically Related? $\endgroup$ Feb 21 at 22:25
  • $\begingroup$ See Truth table for biconditional $\endgroup$ Feb 22 at 7:05
  • $\begingroup$ It is sufficent, but it is not necessary. $\endgroup$ Feb 22 at 7:52

1 Answer 1

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If $A$ and $B$ have a $T$ in each row, then both of them are tautologies, and any two tautologies are logically equivalent. So, both being always True is certainly a sufficient condition for them to be equivalent. However, it is not necessary: two statements that are False in every row are also logically equivalent. And, in general, if $A$ and $B$ have the same truth-value in every row, where for some rows they may both be True, and for other rows they may both be False, then they are equivalent.

Indeed, having the same truth-value in every row is a necessary and sufficient condition: so this is exactly when they are equivalent. My guess is that this is the answer that your problem sheet was looking for. If so, I would say the question is a bit poorly phrased, since the way it is stated it does sound like your sufficient condition should work.

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  • $\begingroup$ Thank you! I think it's just about the way they phrased it $\endgroup$ Feb 22 at 12:41
  • $\begingroup$ @РоманКирьянов You’re welcome! And since you’re new to this site: If you think my Answer sufficiently addressed your Question, you can accept it by clicking on the check mark to the left of it. $\endgroup$
    – Bram28
    Feb 22 at 13:14

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