Struggling to grasp significance in negation

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• 1 Like

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

  February 2018 27900 karma

  The problem is that “some” can include “all.” True statement: Some cats are mammals. This does not imply that not all cats are. In more formal logic that the LSAT is not concerned with, some can be translated to “there exists. . .” There exists a cat that is also a mammal. If there existed a cat who was not a mammal—the negation in question—then it could not be true that “all cats are mammals.”
• acsimon Alum Member

  February 2018 1269 karma

  I would also add that philosophy of language has had some helpful things to say about this.

  First, you are right to think that “all A’s are B’s” does seem to be at odds with “some A’s are B’s”, but this is supposed to derive from the fact that, in most conversational situations, if you were to assert the latter you would imply “some A’s are not B’s”. Why?—Because if you knew that All A’s are B’s, why didn’t you say that instead of something weaker. The reason is, we are conditioned to infer, is that you didn’t believe that “All A’s are B’s”. You can read more about this if you google “scalar implicature”.

  Second, notice that the following (schematized) mini-discourse is entirely consistent.

  (1) Some A’s are B’s. In fact, all A’s are B’s!

  Compare this with the following inconsistent mini-discourse:

  (2) Some A’s are not B’s. In fact, all A’s are B’s!

  The fact that the pairs of sentences in (2) are inconsistent, but those of (1) are consistent, should be enough to convince you that the existentially quantified sentences in each (i.e, the initial sentences) are not equivalent to one another. Reflecting on this, you should get a better handle on the significance of negation in these contexts.

  Cheers—A.c.S
• goingfor99th Free Trial Member

  February 2018 edited February 2018 3072 karma

  There are 'polar opposites' and there are 'logical opposites.' For LSAT/logic purposes, we are primarily concerned with logical opposites.

  To illustrate logical opposites with percentages:
  All A are B
  100% of A are B

  Some A are not B (logical opposite of 'All A are B')
  1-100% of A are not B

  AND

  No A are B
  0% of A are B

  Some A are B (logical opposite of 'No A are B')
  1-100% of A are B
• goingfor99th Free Trial Member

  February 2018 edited February 2018 3072 karma

  A way I understood it: to negate a conditional statement, negate the relationship between the sufficient and necessary conditions, not the sufficient or necessary terms themselves.
• willrpayton Member

  February 2018 33 karma

  Thank you all very much for your comments, they've helped a lot, especially reading about 'scalar implicature'.
• 1 Like

  AudaciousRed Alum Member

  February 2018 2689 karma

  Yeah.. learning logical opposites was a change from polar opposites.
  The opposite of All is not None, but "Some are not". Basically, If a thing is 100%, the opposite just says "a thing is NOT 100%". In fact, most of negation seems to just be slapping a "NOT" or "it is not the case that" in front of the original idea to negate it. "Not 100%" doesn't necessarily mean 0 (none). It just means "not 100%". 50 is not 100. 99 is not 100. In any case, "SOME are not". Technically, all could be not. Why? Because you'd still be right. Say you and a friend are watching turtles. And your friend says "All turtles are blue". You say "Well, some of them are not blue". And you catch all the turtles to prove this to your friend, and all of them are brown. Not a single one was blue. You were not wrong to say "some of them are not blue" if they're all brown, right?