LOGIC QUESTION

mmckayl

I have a logic question I was wondering if you could help me with.

So I was doing a MBF question on Test 41 section 1. Essentially, the stimulus gives you two conditionals.

I know you cannot get a contrapositive out of A (Most)-> B nor out of A B. So my question is is why can you not do this logical move, when you start with the original conditional statement
Normal

Contrapositive

A->B:

By inference we can say A (most)-> B

By inference we can say AB

The contrapositive being~B->~A:

By inference we can say ~B (most)-> ~A

By inference we can say ~AB~

While I know that we cannot get a contrapositives from “most” conditionals and “some” conditionals, from this chart it seems like we can?

I understand from ALL you can imply MOST, from MOST you can imply SOME or from ALL you can imply SOME.

So what I am missing or not realizing?

Scott Milam

@mmckayl Short answer - your chart isn't wrong, but it is misleading...

I often find it helpful to replace symbols with real-world categories when talk about things like this. It makes it so much more clear! So for this, I'll use "Dogs" for "A" and "Mammals" for "B."

We know that if something is a dog, then it is a mammal (A > B ). Since that is true, it also implies the other relationships on your chart:
- If something is not a mammal, it is not a dog (the contrapositive, ~B > ~A)
- Most dogs are mammals (most A > B )
- Most not mammals are not dogs (~B (most) > ~A)
- Some dogs are mammals (some A > B )
- Some not mammals are not dogs (~B (some) > ~A) (NOTE: I assume the original text is a typo, it flips this ~A and ~B )

However, it is important to note that these are all implied from the fact that all dogs are mammals (A > B ). If that isn't the case, we cannot make the same inferences. This becomes pretty obvious if we replace A and B with terms for which A > B is not true. For instance, replace A with "houses" and B with "one story houses."

Most houses are one story houses (A most > B ). But that doesn't mean that the "contrapositive" (its not a REAL contrapositive) is true. "Most non-one-story houses are not houses" (~B most > ~A) is clearly false.

That's why I don't like this chart - it LOOKS like everything in the left column is equivalent to the right column. But in reality, everything in the table is really all derived from the top-left cell (A > B ) and cannot be relied upon otherwise.

Hope that helps!