Conditional Logic 'some A's are B's'

[samuelfbaron]

A --> B; and
A --> C.

We infer that some B's will be C's. If all A's are B's and C's, then those that are A's will be people that are both B's and C's. That means that in at least one instance, someone out there is both a B and a C.

Is it possible to explain this reasoning or break it down a bit? How is it guaranteed that some B's are C's?

[endless_sekai]

I will try and help you out. You have the correct reasoning, but sometimes it is hard to understand it when you have it in abstract form. So to that end, lets put it into more understandable English. I am going to run through 2 examples.

First a very simple example
All apples are fruits
All apples had stems at one point.
Therefore, some fruits had stems at one point.

From general knowledge:
We know an apple is a fruit
We also know that apples had stems at one point.
Thus it is easy to see some fruits had stems at one point.

Above we assumed from general knowledge that there are such a thing as apples (duh) in the next example it gets a little more tricky.


A second more complicated example is where general knowledge doesn't help us. That is we must know something further to guarantee some Bs are Cs

All people named Alex (As) are bald(Bs).
All people named Alex (As) are chubby(Cs).

What this essentially means if there is anyone at all named Alex then that person is both bald and chubby.

So lets assume there is a Alex, then that means there is SOME person who is both bald and chubby. Consequently, this means that there AT LEAST ONE person who is a bald person and who must be chubby (Some Bs are Cs) and that there is at least 1 person who is a chubby person who must be bald (Some Cs are Bs). Thus, if there is an A then it will be that case that some Bs are Cs and some Cs are Bs.

However the important thing to note here is that there is ABSOLUTELY NO WAY to guarantee that Some Bs are Cs without first assuming that there is an A (in this case). That is, to draw the inference some Bs are Cs it must be the case that some A exists.

After all, if some people are bald and some people are chubby, it can be the case that there is never a single bald and chubby person. However, given the relationship we know between Alex and being bald and the relationship between Alex and being chubby, having Alex forces some overlap between bald and chubby.

Hopefully this didn't confuse you too badly.

[tommywallach]

Hey Guys,

Endless' explanation here is spot-on perfect. Indeed, the only exception to the original rule you stated ("How do we know there is some instance where a B is C?") is if there is no "A-thing" in the first place. For example:

(A) Any unicorn would have a horn.
(B) Any unicorn would have four legs.

Are there definitively some four-legged, horned creatures? Not necessarily, because there aren't necessarily unicorns. Of course, this is a stupid example, but you get the point. In the first example that Endless gave, we know that fruit exists, so the rest follows naturally. We don't necessarily know that Alex's exist, so the second example wouldn't be definitive until that was made clear. Of course, the LSAT wouldn't play with that, because it's too silly. : )

-t