Formal Logic Additive Inferences Drill

[akanshalsat]

Hello!

As I'm reviewing formal logic, I understand most of the rules but I'm confused about the inferences made, all of which I got wrong. These come from page 436 from the Logic Reasoning Bible, chapter 13. For example, number 1:


Some As are Bs
No Bs are Cs
All Cs are Ds

I diagrammed this correctly to be: A B C D

however, when I went to simplify it, I saw the some train and reduced the work to: A (not)C D

From there, I used the some train again, and reduced it to: A D,

but from the answer key I see 2 separate inferneces made of A (not) C & . D (not) B

This happened with me throughout the drill, so I'm confused.. are we not trying to reduce it down? How are we to know to stop making inferences from A to C and then go from D to B?

[akanshalsat]

The same goes for question number 3 on that page, I'm confused on how to approach this

[Adam Tyson]

You do indeed want to make those links when you can, akashalsat, and "reduce things down" if you can, but not every link can be made, and you've made one error here that I see.

We know that at least some A are B, and that those A that are B are not C (because there is no overlap between B and C). So your first inference is correct - at least some A are not C. The problem is with the next link you made, where you appear to be connecting A to D. There is no way to make that link! How do we know that something is a D? We only know that it is a D if we know that it is a C. Do we have a link that proves any A IS a C? Nope - we only know that some A are NOT C, but that "some" could mean "all" (because the concept of "some" includes everything other than "none").

Looked at a slightly different way, the original diagram proves some A are not C, but "not C" tells us nothing about D. We know that C is sufficient for D, but the absence of C proves nothing about the necessary condition D. Since A only tells us about the absence of C, it cannot tell us anything about D!

To help you out try assigning some meaning to these abstractions. I am going to make A=Angels, B=Beautiful, C=Cynical, and D=Deceptive. Our original claim chain, then, is this:

Some Angels are Beautiful
Nothing Beautiful is Cynical
Everything Cynical is Deceptive

Based on this, I know there is at least one Angel that is Beautiful, and that Angel therefore cannot be Cynical. Could that Angel be Deceptive? Could there be a Beautiful, non-Cynical, Deceptive Angel? Sure there could - why not? Just because that Angel isn't Cynical tells me nothing about whether they are Deceptive, right? Don't make the Mistaken Negation there!

Now, can I prove that there are any Angels that are Cynical, and thus not Beautiful? No, because the "some" that are Beautiful might be all of them! Sure, there COULD be a non-Beautiful, Cynical (and thus Deceptive) Angel, but I cannot prove that such a heartbreaking creature exists. I can only prove that they are not all that way, because some (at least one) IS Beautiful.

I hope that has shed a little light on the situation. Keep asking, we'll keep answering!

[akanshalsat]

Ahhh, this makes so much more sense!! Thanks so much Adam!

Will definitely keep asking