Negation Help

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• 1 Like

  Cant Get Right Yearly + Live Member Sage ⭐ 🍌 7Sage Tutor

  July 2016 27902 karma

  I think you pretty much nailed it. There is an implied “all,” and to negate an “all” statement we just need to convert it to a “some not” statement, which is exactly what you did. And I think your original “it is not the case that” statement works when you add in the implied “all” to get:
  
  It's not the case that all small animals can move more rapidly than large animals can.
  
  So here it’s clear that this is equivalent to a “some not” statement. The meaning is definitely obscured without the all though; the intended meaning becomes a very unnatural way to read the actual statement. You could probably translate it without the “all” and just add “it is not necessarily the case that,” for a more intuitive negation statement.
  
  Great analysis!
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  quinnxzhang Member

  July 2016 edited July 2016 611 karma

  @"Keane Xavier" said:
  Because the author is talking about these animals as sets - small animals and large animals - can we infer that he or she is talking about all small animals and all large animals? Can we thus read this statement as, "All small animals can move more rapidly than large animals can"?
  

  It's not so straightforward. There's an interesting class of statements called "generics", which look like universals on the surface, but are actually quite different. For example, "humans give live birth" might look like a universal, but is in fact a generic. The sentence isn't saying ALL humans give live birth; after all, over 50% of the human population can't give birth at all! There's a lot of debate about how to properly analyze generics, which you can read about here: http://plato.stanford.edu/entries/generics/. The relevant takeaway is that your sentence is probably intended to be read as a generic.
  
  Thus, your sentence, "small animals can move more rapidly than large animals", actually expresses the following proposition: "It's possible that (gen) small animals move more rapidly than large animals". The negation of this is: "It's not possible that (gen) small animals move more rapidly than large animals" (i.e. small animals canNOT move more rapidly than large animals). This is why the explanation tells you small animals must either move more slowly or at the same speed as large animals.
• Keane Xavier Alum Member

  July 2016 171 karma

  Thank you both for your input, @"Cant Get Right" and @quinnxzhang! Both of you provide great advice and thoughtful responses throughout the forums, and I appreciate your responses here.
  
  Let me ensure that I follow, @quinnxzhang. Forgive me if I don't; I don't have a background in logic, although I'm interested in learning as much as I can.
  
  From what I've gathered from your post and from Stanford's discussion of the matter, which is at times esoteric, generic statements express generalizations of individuals of a kind or predicate properties of a kind itself, but these statements aren't quantified - that is, they lack determiners or adverbs of quantification and do no thus provide specific information about the number of members of a kind that have the property or characteristic that is attributed to that kind. Although Gen is an operator that functions like an adverb of quantification, it is not considered a quantifier because it cannot convey how much or how many. In this sense, it cannot be said to share meaning with any of the other quantifiers, such as "all" or "most" or "some," and cannot be equated with them. Generic statements, furthermore, cannot be simply considered universal statements because a generic statement can be true with exceptions - unlike a universal statement.
  
  For these reasons, I cannot infer from the generic statement "small animals move more rapidly than large animals" that all small animals share that ability. The quantity of small animals that move more rapidly than large animals is uncertain. Thus, I cannot equate the original generic statement to the universal statement "all small animals move more rapidly than large animals." So in negating this generic, we simply negate it to "it's not the case that (gen) small animals move more rapidly than large animals," from which we can infer that an unspecified quantity of small animals must either move equally rapidly or less rapidly than large animals.
  
  Am I on the right track? Would you add or correct anything? Thank you for sharing your knowledge! Generic statements are quite interesting, indeed.
  
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  quinnxzhang Member

  July 2016 edited July 2016 611 karma

  @"Keane Xavier" said:
  For these reasons, I cannot infer from the generic statement "small animals move more rapidly than large animals" that all small animals share that ability. The quantity of small animals that move more rapidly than large animals is uncertain. Thus, I cannot equate the original generic statement to the universal statement "all small animals move more rapidly than large animals."
  

  Right.
  

  @"Keane Xavier" said:
  So in negating this generic, we simply negate it to "it's not the case that (gen) small animals move more rapidly than large animals," from which we can infer that an unspecified quantity of small animals must either move equally rapidly or less rapidly than large animals.
  

  This is close, but not quite there. If the original statement were simply "small animals move more rapidly than large animals", then this would be spot on. However, the original generic was a modal statement, i.e. "small animals CAN move more rapidly than large animals". The negation of this negates the possibility, i.e. "small animals CANNOT move more rapidly than large animals" or "IT'S NOT POSSIBLE that small animals move more rapidly than large animals". From this, we are able to infer that NO small animals move more rapidly than large animals. What licenses this inference isn't the (gen) operator, but the negation of the modal operator "can".
  
  It might help to think about a more intuitive analogue. Let's revisit "humans give live birth". This sentence is a generic because, while true, it's not saying that ALL humans give live birth. Now, let's modify this sentence with the modal operator "can", i.e. "humans CAN give live birth". Again, this is a generic because, while true, it's not saying that ALL humans can give live birth (males, children, seniors, infertile women, etc. can't give birth).
  
  The negation of "humans CAN give live birth" is "humans CANNOT give live birth". This negation is saying that it's impossible for humans to give live birth, from which we infer that no humans give live birth. Likewise, "small animals CANNOT move faster than large animals" allows us to infer that no small animals move faster than large animals.
  
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  Cant Get Right Yearly + Live Member Sage ⭐ 🍌 7Sage Tutor

  July 2016 27902 karma

  That’s really interesting. Totally makes sense. @quinnxzhang , you aware of any LSAT questions hinging on this distinction?
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  quinnxzhang Member

  July 2016 611 karma

  @"Cant Get Right" said:
  That’s really interesting. Totally makes sense. @quinnxzhang , you aware of any LSAT questions hinging on this distinction?
  

  I'm sure there have been generic sentences in, say, some of the RC passages, but I've never seen a question that you would miss if you didn't know what generics were. The writers are usually pretty explicit in the LR section, and when this distinction would matter, I've only ever seen the writers use universals.
  
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  Cant Get Right Yearly + Live Member Sage ⭐ 🍌 7Sage Tutor

  July 2016 27902 karma

  Right on.
• Keane Xavier Alum Member

  July 2016 171 karma

  Ah, I did miss the "can" from the original statement, didn't I? That makes sense. It seems that I have more reading to do: modal operators. Thank you, @quinnxzhang, for taking the time to respond at length. I much appreciate it.
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  Waiting For Grey Day Alum Member

  November 2017 edited November 2017 323 karma

  @quinnxzhang The distinction between modal and generics is incredibly insightful!