Formal Conditional Reasoning Question

[hnadgauda]

I am looking at Statement 5 on page 315 of the LR Strategy Guide. I do not understand how the contrapositive is derived from the statement.

Statement: S-->(-R+T) or (-T+R)

Taking the contrapositive requires negating both sides and switching sides. So we would get the following:

-(-R+T) and -(-T+R)--> -S

I don't understand how this translates to what is written in the book, which is:

(R+T) or (-R+-T)-->-S

Can you please clarify this?

[ohthatpatrick]

Before I begin, I hate this way of notating these rules.

But you're right about your contrapositive,
-(-R+T) and -(-T+R)--> -S

Those "-" in front are like a -1 outside a parenthesis.

-1(-x + y) = x - y

Of course, in our logic statements, we need to not only negate the the letters but also flip "and" to "or" (or vice versa).

So if we "distribute" the negative in
-(-R+T) and -(-T+R)--> -S
we'd get
[R or -T] and [T or -R] --> -S

What would it take to trigger the left side of that idea?

You need a representative from bracket 1 and a representative from bracket 2.

If you have (R or -T) and you have (T or -R), then you get -S.

Well you would never have R and -R. (it's a contradiction)
You'd never have -T and T.

So the rule is only going to be triggered if we have
R and T
or
-T and -R

So the math works, but that is truly awful.

Here's how I prefer to think of these rules:
S-->(-R+T) or (-T+R)

The right side is saying that "either R or T, but not both, is in". Since that means that one will be IN and one will be OUT, I symbolize the right side as a typical "enemy" rule.

In grouping games, when we have a rule that says "R and T can't be in the same group", I would put them in a circle and cross it out.

There's no way to draw that in this typing environment, but we'll make it look like -(RT).

So I would have
S --> -(RT)
and
(RT) --> -S

When I read that I think
"When S is in, then I have to split up R and T ... so one will be IN and one will be OUT"
and I read the contrapositive as
"If R and T are found in the same place, then S is OUT"

Hope this helps.