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MUCH WILL BE ASKED

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Quantifiers, basic.
« on: January 02, 2008, 06:22:30 PM »
Scored 159 on DEC 1, I will be retaking and shooting for the upper 160's, yet I lack some basic on quantifiers, my goal is too master these ideas and I hope that will boost my LR.

1) Some A's are B's
   Some B's are C's
 -------------------
   No conclusion?

2) Some A's are B's
   All B's are C's
-------------------
     No conclusion?

3) All A's are B's
   Some B's are C's
--------------------
  Some A's are C's??

4) Some A's are B'c
   Most B's are C's
-------------------
  No Conclusion?, why?

Most and Some cannot derive conclusions? Anyone have examples of this?

5) Most A's are B's
   Most B's are C's
--------------------
  No Conclusion?

6) All A's are B's
   Most B's are C's
-------------------
  Some B'c are C's?

If I left someout place enlighten me.
Is MANY the same as MOST, or like SOME, can someone explain.
Anyone have any good resources to get these downpacked?
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gollymolly

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Re: Quantifiers, basic.
« Reply #1 on: January 02, 2008, 07:15:05 PM »
Draw venn diagrams! That always helps me.

lp1283

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Re: Quantifiers, basic.
« Reply #2 on: January 02, 2008, 08:20:38 PM »
Powerscore LRB does a good job of explaining some of the formal logic needed for the exam. 

Many = Some
_____________________________ ___________________

Some = At least one, possibly all.

Most = A majority, possibly all.
_____________________________ ___________________
Some = 1 to 100 (at least one)

Most = 51 to 100 (a majority)

All = 100

philneuro

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Re: Quantifiers, basic.
« Reply #3 on: January 02, 2008, 08:25:48 PM »
Scored 159 on DEC 1, I will be retaking and shooting for the upper 160's, yet I lack some basic on quantifiers, my goal is too master these ideas and I hope that will boost my LR.

1) Some A's are B's
   Some B's are C's
 -------------------
   No conclusion?

2) Some A's are B's
   All B's are C's
-------------------
     No conclusion?

3) All A's are B's
   Some B's are C's
--------------------
  Some A's are C's??

4) Some A's are B'c
   Most B's are C's
-------------------
  No Conclusion?, why?

Most and Some cannot derive conclusions? Anyone have examples of this?

5) Most A's are B's
   Most B's are C's
--------------------
  No Conclusion?

6) All A's are B's
   Most B's are C's
-------------------
  Some B'c are C's?

If I left someout place enlighten me.
Is MANY the same as MOST, or like SOME, can someone explain.
Anyone have any good resources to get these downpacked?

MUCH WIIL BE ASKED, here are some examples utilizing the quantifiers "all", "some", and "no". Much of the content is repetitive and nonsensical, but I intentionally made the content repetitive and nonsensical in order to highlight the concepts of inherent and additive inferences. For instance, I reuse premises and/or conclusions from one or more arguments to create new arguments. I also use inherent inferences of premises and/or conclusions from one or more arguments to create new arguments. Example: If we know that All As are Bs, then we know that Some As are Bs. Hopefully, this will be evident below.

I don't discuss inherent and additive inferences here, as I'm guessing you are already familiar with them. In the LRB, there's a great discussion of those two concepts and much more. Anyways, here are the examples. (disclaimer: these examples were created on the fly. so there may be mistakes present. if there are any mistakes, please let me know. thanks!)

1. All humans are mammals.
All American citizens are humans.
-----------------------------------------
All American citizens are mammals.

2. No humans are aliens.
All American citizens are humans.
-----------------------------------------
No American citizens are aliens.

3. All humans are mammals.
Some American citizens are humans.
---------------------------------------------
Some American citizens are mammals.

4. No humans are aliens.
Some United States workers are humans.
-------------------------------------------------
Some United States workers are not aliens.

5. All humans are mammals.
No trees are mammals.
-------------------------------
No trees are humans.

6. No trees are mammals.
All humans are mammals.
-------------------------------
No humans are trees.

7. All humans are mammals.
Some trees are not mammals.
----------------------------------
Some trees are not humans.

8. No trees are humans.
Some American citizens are humans.
--------------------------------------------
Some American citizens are not trees.

9. All humans are mammals.
Some humans are smart beings.
--------------------------------------
Some smart beings are mammals.

10. Some humans are smart beings.
All humans are mammals.
--------------------------------------
Some mammals are smart beings.

11. No trees are (naturally composed of) steel.
Some trees are perennials.
-------------------------------
Some perennials are not (naturally composed of) steel.

12. Some mammals are not smart beings.
All mammals are warm-blooded.
---------------------------------------------------
Some (things that are) warm-blooded are not smart beings.

13. All mammals are warm-blooded.
No (things that are) warm-blooded are (things that) lack a spinal column.
-------------------------------------------------------------------
No (things that) lack a spinal column are mammals.

14. Some humans are smart beings.
All smart beings are (things that) at least can count to ten.
---------------------------------------------------------------------
Some (things that) at least can count to ten are humans.

15. No humans are reptiles.
Some reptiles are (things that) lay eggs.
-----------------------------------------------
Some (things that) lay eggs are not humans.


MUCH WILL BE ASKED

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Re: Quantifiers, basic.
« Reply #4 on: January 02, 2008, 09:35:47 PM »
philneuro, thanks for all the examples, by the way are the examples I put right, as far as the conclusion because im curious if my thinking is right as far as like a MOST and SOME not having a conclusion, etc. Thank you in advance.


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philneuro

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Re: Quantifiers, basic.
« Reply #5 on: January 02, 2008, 11:04:43 PM »
philneuro, thanks for all the examples, by the way are the examples I put right, as far as the conclusion because im curious if my thinking is right as far as like a MOST and SOME not having a conclusion, etc. Thank you in advance.




Here's my take. Hope it helps.

1) Some A's are B's
   Some B's are C's
 -------------------
   No conclusion?

Correct. Two “some’s” never produce any inferences (or conclusions).

2) Some A's are B's
   All B's are C's
-------------------
     No conclusion?

There is a conclusion here. It is” Some As are Cs.” The argument can be diagrammed as follows:  A s B  C; yields As C.  Notice that “B” is common to both premises. Thus, “B” is only written once in the previous diagram.

However, if the second premise stated “All Cs are Bs”, there would be no conclusion. As B  C; yields no conclusion.

See LRB, Chapter 11: Formal Logic for a detailed discussion of the nature of the arrow diagrams used above and the relationships that hold between the arrow diagrams used above. But for present purposes, s stands for “Some”,  stands for “All”, and –m stands for “Most”.

3) All A's are B's
   Some B's are C's
--------------------
  Some A's are C's??

Correct. If all of the entities belonging to the class of As are Bs and some of the entities belonging to the class of Bs are Cs, then it must be true that some of the entities belonging to the class of As are Cs.

4) Some A's are B'c
   Most B's are C's
-------------------
  No Conclusion?, why?

Correct. A s B –m C yields no conclusion because there is no guarantee that the entities belonging to the class of As that are Bs are also entities that belong to the class of  Cs.

 For instance, what if it is true that both some cats are dogs and that most dogs are horses.

4’) Some Cats are Dogs.
     Most Dogs are Horses.
-------------------------------
???
 Can we correctly assume that either some or most cats are horses? No. At least not from the two premises just mentioned. This is partly (maybe wholly) due to the fact that in the premise “some Cats are Dogs”, no assertion is made about all cats and no assertion is made about all dogs.  So, once again, there is no guarantee that the entities belonging to the class of As (cats) that are Bs (dogs) are also entities that belong to the class of Cs (horses).
   

Most and Some cannot derive conclusions? Anyone have examples of this?

See 4’ above and the explanation that follows.

5) Most A's are B's
   Most B's are C's
--------------------
  No Conclusion?

See 4’ above and the explanation that follows. Some of what is written there applies here and helps explain why no conclusion can ever be drawn from the two premises involving “most”.

However, there is a relationship involving the usage of two “Most’s” that yields a conclusion. Here it is.

Most As are Bs.
Most As are Cs.
-------------------
Some As are Cs.

A m—B –m C

“if there are five B’s, and three of the B’s are A’s, and three of the B’s are C’s, then there must be an overlap of at least one A and C, and therefore we can conclude that some A’s are C’s” (LRB, p. 330).

6) All A's are B's
   Most B's are C's
-------------------
  Some B'c are C's?

Correct. But there is another conclusion to be drawn here. It is “Some As are Cs.” The conclusion you have above is an “inherent inference” from the statement “Most Bs are Cs.” Consider this, 1. “All dogs are mammals.” If that is true, then it is uncontroversially true that 2. “Most dogs are mammals.” And it is also uncontroversially true that 3. “Some dogs are mammals.” Although 2 and 3 are not very informative when we know 1 is true, 2 and 3 are nevertheless uncontroversially true.

My attempt to represent the "all", "most" and "some" arrows failed above when trying to diagram the arguments. Sorry.

upgrade

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Re: Quantifiers, basic.
« Reply #6 on: January 02, 2008, 11:21:10 PM »
I just reviewed this material tonight, in the PowerScore Logical Reasoning Bible.  What philneuro posted is a good rundown of the way the LRB describes this subject. I have found that Venn diagrams are sometimes useful, but in many cases I have found them to be too difficult to be sure that my drawing fits all the criteria.  With the arrows for all, most, some, and none you don't have to worry about the drawing matching every scenario.

lp1283

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Re: Quantifiers, basic.
« Reply #7 on: January 03, 2008, 04:53:15 PM »

Here is something I wrote up about it in the context of a pretty hard problem from the June 2004 test.

http://www.lsatdiscussion.com/index.php/topic,1198.msg4398.html#msg4398

Whoa and this was suppose to be done in less than 5 mins (a stretch)? Yikes.

theo

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Re: Quantifiers, basic.
« Reply #8 on: January 06, 2008, 06:18:12 PM »
Powerscore LRB does a good job of explaining some of the formal logic needed for the exam. 

Many = Some



I've heard it explained it even better:

'Many' is a word that only leaves you guessing
Guessing 'bout a thing you really ought to know, ooh!
You really ought to know...
I really ought to know...

http://www.youtube.com/watch?v=_swFHp-0_sY
quid leges sine moribus vanae proficiunt?

MiamiLaw008

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Re: Quantifiers, basic.
« Reply #9 on: January 06, 2008, 06:25:01 PM »
In almost every  situation I can remember that has  both many and few in the same stimulus, many is GREATER than few.