Compound Statements in FL with Most and Some

[bht1234]

N 0 P Q R



The inferences in the book are
N -P (crossed out P)
R -P (crossed out P)

1. Why does it automatically turn into some?
2. Why isn't the inference N R? When does the some train stop on compound statements?
3. Can we only make inferences from the "open" variables?

[Jonathan Evans]

Hi, BHT,

Welcome to the forums! Good questions!

It doesn't automatically turn into "some" so much as that's the implication of multiple statements put together. If I'm not mistaken, you are looking at the following drill question:

• Some Ns are Os.
  No Os are Ps.
  No Ps are Qs.
  All Qs are Rs.

The diagram is as follows:

1. N O
2. O P
3. P Q
4. Q R

From statements (1) and (2) we get: N O P

Now imagine that we have 10 Ns out there. What do we know about those 10 Ns? We know some of them are Os. Let's imagine that 4 of the Ns are Os.

What do we no about Os? We know that no Os are Ps. So let's see what we know so far about all 10 of our Ns.

• N, also an O, not a P
  N, also an O, not a P
  N, also an O, not a P
  N, also an O, not a P
  N, not an O, don't know whether it's a P
  N, not an O, don't know whether it's a P
  N, not an O, don't know whether it's a P
  N, not an O, don't know whether it's a P
  N, not an O, don't know whether it's a P
  N, not an O, don't know whether it's a P

In this hypothetical scenario, we know that the Ns that are Os are not Ps. For the Ns that are not Os, we don't know whether or not they are Ps. They might be; they might not be. All we know is that some Ns are not Ps. That is the implication of the two combined statements: (1) N O and (2) O P

It's similar for the latter two statements (3) P Q and (4) Q R

Imagine that we have 10 Qs. We know two things about these 10 Qs. We know they are not Ps. We also know that all Qs are Rs, so these 10 Qs are Rs. Thus, we know that these 10 Qs that are also Rs are not Ps.

Here's the thing. We don't know about any other Rs out there. Maybe there are 5 more Rs floating around. These Rs are not Qs. Since these Rs are not Qs, we don't know whether or not they are Ps. They might be. They might not be.

Therefore, we know for sure there are some Rs that are not Ps, but we don't know whether there might be some Rs that are Ps. That's how we get the second inference.

In re your second question: the train stops at the double not arrow. Once we discover no Os are Ps, we can't go any further. There is not connection between Os and Qs, so we can't make any further inferences.

In re your third question, no, it is possible to make inferences from "closed" variables; it is just typically easier to analyze open variables.

Here's an example of an inference from a closed variable:

• G H I J

Even though H is a closed variable, we can infer H J.

Thanks for the questions. I hope this helps!