Formal Logic Practice questions (pg. 435 in LRB)

[akanshalsat]

Hello!

I just have a few questions with some of the drill questions on page 435

for number 5, I know that the diagram is: I J K L
-The inference that they went from J to L being: J L, why wasn't it J L?
B/c if it had been written J K L, we would have retained the and have written: J L

For number 3, I know the diagram is E F G H

why can't we also infer E G and E H?

[Jonathan Evans]

Hi, Akan,

Great questions!

for number 5, I know that the diagram is: I J K L
-The inference that they went from J to L being: J L, why wasn't it J L?
B/c if it had been written J K L, we would have retained the and have written: J L

The reason is because and have different implications.

implies that there is overlap between two groups. For example:

• "Some dogs are pets."
  Dogs Pets
  Pets Dogs

Now lets add a conditional statement into the mix.

• "All dogs are mammals."
  Dogs Mammals

We know that there is overlaps between pets and mammals:

• Pets Mammals

In contrast is a material equivalence, a bidirectional conditional. It implies that two sets are equivalent to each other. For example:

• "Someone is a college graduate if and only if he or she has been awarded an undergraduate degree from an academic institution."
  
  UD CG

However, if you connect a one-directional conditional to this bidirectional conditional, it remains one-directional:

• "All college graduates must repay their student loans.
  
  CG RSL

The implication is still UD Student loans. We do not know that everyone who must repay his or her student loans is also a recipient of an undergraduate degree (*RSL UD). Thus, we cannot infer *UD RSL.

For number 3, I know the diagram is E F G H

why can't we also infer E G and E H?

This situation is similar to the one discussed above. One-directional conditional statements do not imply their converse. In other words, we don't want to make a Mistaken Reversal™ error. Here's an example:

• "No cats are dogs."
  
  C D
  
  "All dogs are mammals."
  
  D M

It does not follow that "no cats are mammals." We cannot infer *C M.

Does this make sense? Thanks for the good questions!