Hi!
I just finished reading/working through Chapter 13: Formal Logic in the Logical Reasoning Bible. I'm relieved that I'm understanding "
most" (51-100
) of the concepts in the chapter. I did, however, eventually encounter some difficulty when it came to listing the additive inferences derived from complex Formal Logic statements.
Diagramming for me is not the issue -- I feel very confident in my efficiency/accuracy of diagramming the statements (I usually always get these right). I understand the Logic Ladder and how we can make inherent inferences "climbing down" the rungs. I also understand the concepts of recycling and checking the closed variables. What I am unsure of, however, is when to use and not use these tools. For example, when is it necessary to infer a
some relationship from an
all relationship, and when is it redundant? When is recycling advantageous, and when is it a waste of time/an unnecessary step?
For some of the drills listed on page 465 of the 2020 Logical Reasoning Bible, I found myself listing inferences that were not matched in the example key. I found all the inferences listed as correct, but I always had a couple extras. Perhaps these inferences are inherent and thus are not listed (as per the instructions of the drill), but I clearly seem to be unable to differentiate the two -- which inferences are relevant to the statements and which are not.
Let's take drill #1 for example, which reads:
Some As are Bs.
No Bs are Cs.
All Cs are Ds.
I'm able to quickly diagram this problem:
A
B
C
D
I'll talk you through my thought process and perhaps you can tell me where I'm going awry. The first thing I do is look at the ends of the chain (Principle 1). I see the
some relationship at one end between A and B, and automatically my brain thinks Some Train. I ride the Some Train and establish the relationship between A and C:
[A
B
C]
D
[A
C]
D
INF. #1: A C
And there's my first additive inference (connecting A and C through B, then removing B)! Yay, all is good so far (I hope).
So now that I've established an inference from A, B and C, I turn to the other end of the statement to see if I can infer anything from the B, C and D relationship. Well, Principle 3 says to not start with a variable involved in a double-not arrow, so I can't infer anything starting with B (moving toward D). D, on the other hand, has an arrow pointing toward it, but if I take that relationship (the relationship between C and D) and climb down the Logic Ladder, I can infer that...
A
[B
C
D]
...can be reduced to (using the Inherent Inference rule):
A
[B
C
D]
Which now allows me to take the Some Train from D to B, and I find a second inference:
A
[B
C
D]
A
[
B D]
INF. #2: B D
Now these two inferences are all that the answer key on the following page (page 466) lists. My thought process at this stage in inferring is that I haven't completely explored every avenue to see if I've found all the inferences I can find. What about recycling?
So I go ahead and throw inference #2 into the equation and see if I can infer anything.
A
[B
C
D]
A
[
B D]
Principle 9. Two consecutive
somes do not yield an inference. Nothing I can do here.
Let's throw inference #1 back into the equation.
[A
B
C]
D
[A
C]
D
This looks, to me, like I could derive a third inference from the statement. I ride the Some Train (from A to D) and find a third inference:
A
C D
A
D
INF. #3: A D
And this is the inference that is not listed in the answer key. Is this inference inherent, not additive? Have I taken an unnecessary step? Is my thought process correct in that I should (in the case of three individual relationships) be looking at the two relationships on the either end, then recycle any inferences I find into the original statement to potentially find more inferences (that's a confusing sentence)?
In some of the following drills on page 465, I found myself questioning whether or not I needed to reduce
all statements to
some statements to see if there was a possible inference I could make. Should we always be reducing to
some relationships to potentially infer? Are those additive inferences? I think I'm confusing myself even more by typing this... but I hope my questions are somewhat clear?
Thank you!
