Negating an and/or statement clarification

Comments

• Sami Yearly + Live Member Sage ⭐ 7Sage Tutor

  January 2018 10806 karma

  @"k.karmazyn" said:
  Hello 7Sagers!

  I'm going through the core curriculum a second time, and I have a question about the negation of and/or statements in conditional relationship! If, for example, we have a statement that reads "If Kay sing, Justin and Tommy sing also" we would diagram this as

  K------> J and T (with a split arrow)

  My question is, if we are doing a logical reasoning problem, and say an answer choice draws a conclusion that Kay doesn't sing...in order to prove that Kay in fact doesn't sing (i.e. deny the sufficient) we need to also deny the necessary in order for the sufficient to be denied. My question is, if ONE of the conditions in the necessary if failed (Justin not singing for example) is that enough to then contrapose back and say that Kay didn't sing? Or do both conditions (Justin and Tommy not singing) need to be failed?

  The rule of thumb is that to do the contrapositive with an "and" in neccessary you turn it into an "or". So if Justin does not sing then Kay does not sing.

  Think of it this way, if we know Justin does not sing, is it even possible that Kay can sing? Because as soon as Kay sings both Justin and Tommy have to sing. So just by knowing that one of Justin and Tommy or both cannot sing, we can conclude that Kay does not sing.

  Similarly, if we have the statement "If Kay sings, Justin or Tommy sing also" and we conclude again that Kay doesn't sing, in order to prove that would be need BOTH Justin and Tommy not to sing, or is it enough for just one of them not to sing, in order to say that Kay didn't?

  I think by knowing how we worked on the above statement, you should try to write out why it might or might need both.

  If Justin doesn't sing, is it possible that Kay can sing?
  If it is the case that Kay can sing, what must also be the case?
• thisissparta Alum Member

  January 2018 1363 karma

  1) In a statement with "and" in the necessary, each event occurs independent of the other. You don't need to contrapose both necessary conditions simultaneously in order arrive at /Kay singing.

  2) Alternatively, in the second example, you need to prove that both Justin and Tommy didn't sing. They both should not be singing for Kay to not sing.