Negating an (And/Or in Conditionals) relationship

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• 3 Likes

  Heart Shaped Box Alum Member ⭐

  May 2017 2426 karma

  The difference between negation and contrapositive is that the former is to negate a logical idea while the latter is just a restatement of the same idea. The contrapositive is the exact same original statement in the contrapositive format.

  I believe you are correct in the Tom statement negation: T→NOT(J and S)

  When you negate a joint relationship, "and" becomes "or". So I would negate it as T ---> J or S. It's not the case when Tom plays, J and S both play. It could be J doesn't play, or else S doesn't, or neither of them do. "Or" is always inclusive when not specified.
• 1 Like

  batniki1 Alum Member

  May 2017 edited May 2017 226 karma

  That was incredibly helpful, @"Heart Shaped Box" . Thank you
  "Or" is always inclusive when not specified ... if a cheatsheet was worth it, this would be mine.
• Heart Shaped Box Alum Member ⭐

  May 2017 2426 karma

  Your welcome, glad to be of assistance!
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  jknauf Alum Member

  May 2017 edited June 2017 1741 karma

  So a conditional statement is a logically valid statement which means: If X happens then Y must happen. (If it rains, the ground must be wet) Well, given that we accept our conditional statement as completely true, we must also be able to also determine the opposite of our statement. (If the ground is not wet, then it did not rain)
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  akistotle Member ⭐ 🍌🍌

  May 2017 edited May 2017 9377 karma

  @therealnas said:

  "If Tom plays, then Jerome and Simmi play too"
  Translated into lawgic that would be: T→(J and S) (which could be split)

  T→(J and S)

  　　 J
  　↗︎
  T
  　↘︎
  　　S

  Yes, this is correct as you can see in this video.
  https://7sage.com/lesson/contrapositives-demorgans-law/

  Now, if we negate the statement altogether, what happens then? "It could be the case that if Tom plays, neither Jerome nor Simmi play" am I right? ....T→NOT(J and S)

  The negation isn't "T→NOT(J and S)," which would mean "if T plays, neither Jerome nor Simmi play."

  The negation of the conditional statement would be /[ T→(J and S) ] (=NOT[ T→(J and S) ]), which can be represented as T and /(J and S).

  J.Y.'s lesson on denying the relationship:
  https://7sage.com/lesson/deny-the-relationship/

  T and /(J and S)
  /(J and S) = /J or /S

  T and /J or /S

  Thus, the negation of this conditional statement "If Tom plays, then Jerome and Simmi play too" is as follows:

  Tom plays but Jerome doesn't play.
  Tom plays but Simmi doesn't play.
  Tom plays but both Jerome and Simmi don't play.

  I hope this helps
• batniki1 Alum Member

  May 2017 edited May 2017 226 karma

  Yes, @akistotle, now I understand it very well. That was a very helpful and explicit explanation! Much appreciated! I have another question, and I think you might be able to answer that...
  Why does the relationship drop the arrow and get an "and"? Could it not be represented as T→/J or /S.... Or did you write it with an "and" to avoid confusion?

  Moreover,
  You said that the negation becomes... It could be the case that if Tom plays, Jerome doesn't play or Simmi doesn't play. (here the "or" is inclusive because it's not stated otherwise...) and it translates as follows:

  "/[ T→(J and S) ] (=NOT[ T→(J and S) ]), which can be represented as T and /(J and S)"-your translation

  Why, then, do we not cross T when we negate the relationship?? Is there a possibility to have /T? In other words, could we say "It could be the case that if Tom doesn't play then something happens to Jerome and Simmi"... Or is this entirely wrong because it would get into the territory of failing the sufficient condition, which we cannot do? I know we can have /T if we take the contrapositive of the now negated statement and thus write J and S→/T but I don't understand why negating a relationship automatically keeps the S.C. the way it is and only negates the N.C.
• 1 Like

  akistotle Member ⭐ 🍌🍌

  May 2017 edited May 2017 9377 karma

  Why does the relationship drop the arrow and get an "and"? Could it not be represented as T→/J or /S.... Or did you write it with an "and" to avoid confusion?

  T→/J or /S represents a conditional statement. When you negate the original conditional statement, T→(J and S), it will no longer be a conditional statement because you have to deny the relationship that exists between T and (J and S).

  That arrow is what represents the relationship between the two ideas. T is the sufficient condition for (J and S), and (J and S) are the necessary condition for T. When you want to negate the conditional statement, you have to deny that such a relationship exists.

  So the negation can also be like:

  T <--some--> /(J and S)

  Sometimes Tom plays without Jerome or Simmi.

  You said that the negation becomes... It could be the case that if Tom plays, Jerome doesn't play or Simmi doesn't play. (here the "or" is inclusive because it's not stated otherwise...) and it translates as follows:

  "/[ T→(J and S) ] (=NOT[ T→(J and S) ]), which can be represented as T and /(J and S)"-your translation

  Why, then, do we not cross T when we negate the relationship??

  Because the original statement doesn't tell us anything about cases in which Tom does not play. [/T and (J and S)] is telling us about situations where Tom doesn't play, and the original statement does not contemplate such a situation.

  The logical negation of [T→(J and S)] is [T and /(J and S)].

  The original statement tells us the situations in which Tom plays (T) are completely within the situations in which J plays (J) and the situations in which S plays (S).

  When you logically negate of this conditional statement, you say that there are some cases Tom plays (T) but J or S doesn't play.

  The contrapositive of [T→(J and S)] is [/(J and S)]→/T], and the negation of the contrapositive is also [/(J and S) and T].

  Hope this helps (and I would appreciate if someone could also confirm the above because I'm still learning too....)