Difference between negation and contrapositive?

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• 4 Likes

  Jonathan Wang Yearly Sage ⭐

  February 2015 6867 karma

  Confusion often arises because negation is part of the process of taking a contrapositive ("flip and negate"). Just remember the following:
  
  Contrapositives return the exact equivalent of your initial statement. Saying that "All apples are delicious" is 100% equivalent to saying "If it's not delicious, it's not an apple". You can use A -> B and /B -> /A interchangeably and nothing will go wrong.
  
  Negations don't do that; they return to you an entirely different statement. If I say "It's not true that all apples are delicious", I'm not just saying "all apples are delicious" in a different way - I'm denying the statement entirely. And if you substituted the second statement for the first, the result would be radically different.
  
  SO, in summary:
  Initial statement: All apples are delicious.
  Contrapositive: All things that are not delicious are not apples. (Same as initial statement)
  Negation: It's not true that all apples are delicious. (Denying the initial statement)
• jdawg113 Alum Inactive ⭐

  February 2015 2654 karma

  negating is saying it is not the case so "If it is a dog, I like it" negated is "If it is a dog I may not like it" or "It is not the case that if it is a dog then I like it"
  
  Contrapositive would be "If I do not like it, then it is not a dog" so it is saying the same thing as the original, just if it was reversed kind of
• [Deleted User] Free Trial

  February 2015 578 karma

  All apples are fruit
  
  POSITIVE: APPLES ---> FRUIT
  
  Means if apple then fruit (exactly what our original statement said)
  
  CONTROPOSITIVE: -FRUIT ---> -APPLE
  
  Means if not a fruit then not an apple. What are apples? Fruits...right? So if your not a fruit you can't be an apple, orange, rasberry, etc.
  
  
  INCORRECT NEGATION: -APPLE ---> -FRUIT
  
  Means if not apple then not fruit. Really? Can't oranges also be fruit? You don't need apple to have a fruit. You can have other fruits too like oranges then you can say you have a fruit.
  
  INCORRECT REVERAL: FRUIT ---> APPLE
  
  Similar to incorrect negation. Means if you have a fruit then you have apple. This morning I had orange and it wasn't an apple but it's still a fruit. So it's not a must that when you have a fruit you have apple.
• jdawg113 Alum Inactive ⭐

  February 2015 2654 karma

  "not all apples are fruits" would be the negation to above (or "it is not the case that all apples are fruit")
• [Deleted User] Free Trial

  February 2015 edited February 2015 578 karma

  @jdawg113 when it comes to incorrect negation you can incorrectly negate it anyway. I did it in this format: -A then -B (if not apple then not fruit)
  
  You could also incorrect negate it your way: A then -B (not all apples are fruits, like you said)
  
  I think that's how incorrect negation works. Anyway the easiest way to remember is: for contropositive flip and negate. Do not negate the statement just by the way it already is.
• jdawg113 Alum Inactive ⭐

  February 2015 2654 karma

  "incorrect negate" where did you get that term from? just curious since a negation is a negation, so by saying incorrect negate seems like it is saying it is not a correct negation
  if it is not an apple then it is not a fruit would be a wrong negation since the original is saying ALL apples, a negation just means not all, so there can be apples that are fruits but there can be apples that are fruits
• [Deleted User] Free Trial

  February 2015 578 karma

  I learned incorrect negation and whatnot from testmasters. I find it to be easy.
• 5 Likes

  emli1000 Alum Member Inactive ⭐

  February 2015 edited February 2015 3462 karma

  SOME AND MOST RELATIONSHIPS
  EXISTENTIAL QUANTIFIERS OVERVIEW Lesson 1 of 30
  • Existential quantifiers are intersectional relationships. Some X are Y (X some Y).
  • He says that universal quantifiers talk about complete subsumpion– in other words, completely subsumed and contained. All X must be Y (X–>Y).
  *No contrapositives
  
  RELATIONSHIP TO UNIVERSAL QUANTIFIERS Lesson 2 of 30
  • Conditional logic =UQ
  • UQ: learning about a relationship of complete subsumption – where 2 groups of things existed where 1 group is completely subsumed by another group (Jedi & Force users) and (watermelon subsumed by fruit-Sufficient Condition & Necessary Condition)
  • Existential quantifiers
  Relationship – intersection instead of 1 group being completely subsumed by another group we have 2 groups that happen to share some share of overlap
  Ex: Venn diagram – size of intersection between the relation of one idea and another idea (dogs & things that are cute)
  • Describes the difference between Some & Most relationships
  
  EXISTENTIAL QUANTIFIERS BRIEF Lesson 3of 30
  • There exist things such as ______ (fill in the blank- predict)
  Ex: happy people v. poor people (middle intersection)
  • Indicators: “SOME”
  • **No such thing as a contrapositive- it does NOT exist for EXISTENTIAL QUANTIFIERS
  
  “SOME” STATEMENTS MEANING Lesson 4 of 30
  • EQ
  • "Some speaks of a range”-> “All speaks of a point (100)”
  • Definition of “some” is at least 1
  • Starts at 1 to 100 (does not include 0)
  • Ex: Some dogs are playful (1-100) → all dogs are playful (100)
  • Lower bound: 1
  Upper bound: 100
  • 0 is not included in Some
  
  “SOME” STATEMENTS TRANSLATE Lesson 5 of 30
  • [Some] dogs (D) are cute (C)
  D some C
  *Some is an intersection that’s why it does not carry an arrow
  • NO SUCH THING AS A CONTRAPOSITIVE =/= /D some /C- YOU KNOW TOO LITTLE TO DETERMINE IF (NON-DOGS ARE NOT CUTE? HOW DO YOU KNOW? YOU DON’T KNOW!)
  
  “MANY” = “SOME” Lesson 6 of 30
  • On the LSAT, the word “many” just means “some”
  
  
  “MOST” STATEMENTS MEANING Lesson 7 of 30
  • Existential Quantifier
  • Intersection between two ideas
  • Definition for “Most” is half plus 1
  • Lower bound: 51
  Upper bound: 100
  • Also speaks of a range
  • Ex: Most cats are furry
  At least 51 of the cats are furry.
  
  “MOST” STATEMENTS TRANSLATION Lesson 8 of 30
  • Ex: [Most] Pandas (P) like to eat bamboo (EB)
  More than half of the pandas like to eat bamboo
  P---most-- >E
  **Direction of the arrow does matter-
  *”Most” is the subset of “some”
  • NO CONTRAPOSITIVES!
  
  “FEW” STATEMENTS MEANING AND TRANLATION Lesson 9 of 30
  • EQ
  • Definition of “few” is some are, most are not
  • Ex: [Few] dogs (D) are evil (E)
  1. Some dogs are evil – D some E (and)
  2. Most dogs are not evil- D –most-- > /E
  • On the LSAT "few" focuses on MOST ARE NOT
  
  ADVANCED: ALL IMPLIES MOST IMPLIES SOME Lesson 10 0f 30
  • All implies Most implies Some
  • All → Most → Some
  • All → Some
  • All is the most restricted – it’s just a point
  • Ex: All pencils are made from trees
  Most pencils are made from trees – P –most-- > T
  Some pencils are made from trees – P some T
  
  INTERSECTION RELATIONSHIPS Lesson 11 0f 30
  • WHAT’S AN EXISTENTIAL QUANTIFIER? First, understand that just like the conditional relationships, this idea also expresses a relationship. Whereas the conditional relationship expresses the idea that one group is being completely subsumed by another group, here, with the existential quantifier, we’re talking about a different kind of relationship. We’re talking about the relationship of intersections. There are two types of intersection relationships.
  • The three logical indicators here are “some” which means at least 1 up to all; “most” which means half plus 1 up to all; and “few” which usually means some are and most are not.
  INTERSECTION TRANSLATIONS Lesson 12 0f 30
  TRANSLATIONS TO LAWGIC FOR INTERSECTION STATEMENTS:
  • The “Some” Group: some, many, several, at least one, lots, not none
  • The “Most” Group: most, a majority, more than half
  • The “Few” Group: few
  **Remember that there are no contrapositives for these intersection relationships. They don’t exist. You can try to get a “contrapositive” for these statements and you’ll see that they don’t express the same idea at all. For each of these statements, there is just one way in Lawgic to express them.
  
  ADVANCED: NEGATE SOME STATEMENTS Lesson 13 0f 30
  • Some – None
  • Ex: Some dogs are brave
  D some B
  • Negate [No] dogs are brave
  D→ /B
  B→ /D
  • All- Some Not
  
  ADVANCED: NEGATE ALL STATEMENTS Lesson 14 0f 30
  • All- Some… Not… (0-99)
  • Ex: All cats(C) are pretentious (P)
  C→P
  • Negate: [Some] cats are [not] pretentious
  C some /P
  
  DENY THE RELATIONSHIP Lesson 15 of 30
  • A→ B (all statement)
  To deny this statement the arrow cannot happen
  A is the S.C & B is the N.C
  Then you must deny the arrow
  • A some /B
  • A and /B
  
  HOW TO NEGATE STATEMENTS IN ENGLISH Lesson 16 of 30
  • Negating the Conditional Sentence
  To negate a conditional statement, you should realize that what you’re negating is the conditional relationship. In other words, where the original statement says that A and B exist in a conditional relationship, you’re saying that no, A and B do not exist in a conditional relationship. That A is not sufficient for B and B is not necessary for A. For example, negate this statement:
  o All Jedi use the Force.
  Did you say “No Jedi use the Force?” That’s not right. To negate this statement, you’re denying the conditional relationship between the categories Jedi and Force users. Whereas the original statement is stipulating that the categories of J and F exist in a conditional relationship, you’re saying J is not sufficient for F (and F is not necessary for J). So, in English, it becomes an intersection statement.
  o Some Jedi do not use the Force.
  See how that’s different from “No Jedi use the Force?” That’s how it works for all conditional statements. You can formulate the contradiction (i.e., the negations) by saying “Some [of the sufficient] is not [the necessary].” Or, if grammatically it doesn’t make sense to use “some,” you can fall back on the more general rule of “One can be [the sufficient] and not be [the necessary].” For example, “If the President endorses this bill, then it will pass.” Here, it clearly doesn’t make sense to use “some.” So, we fall back on the more general rule and say “The President can endorse the bill and it could not pass.”
  • Negating the Intersection Sentence
  You should have some idea of how to contradict intersection statements by now. It’s just the reverse of the rule for contradicting conditional statements. You are denying the fact that an intersection relationship exists between the two groups. For example:
  Some cats are furry.
  
  The contradiction is that there is no intersection between cats and furry things. “No cats are furry” or “All cats are not furry.” You replace the “some” with a “no.” Alternatively, you can replace the “some” with an “all… not.” Let’s look at another example:
  
  Sometimes, when I eat too much ice cream, I get a stomachache.
  
  The contradiction is “I never get a stomachache from eating too much ice cream.” Or, “Whenever I eat too much ice cream, I do not get a stomachache.”
  For now, if you need to rely on these translation mechanisms, that’s okay. You’re training your intuition so that one day, you can tag the clause “it’s not the case that…” in front of any sentence and understand what the negation of that sentence is.
  
  
  Logical Opposites Range Construction Example
  All 100 All... are... All cats are pretentious.
  Not all 0-99 Some... are not... Some cats are not pretentious.
  
  Logical Opposites Range Construction Example
  Some 1-100 Some... are... Some dogs are brave.
  None 0 No... are... No dogs are brave.