Question

I have the following three premises:

P or Q
P => R
Q => R

The symbol => represents the ‘implies’ operation. I understand that these premises constitute a dilemma, but how can they be combined into one logical expression?

Answer 1

Since each proposition in the list is true at the same time, there’s an implied and among them.

(P v Q) ^ (P => R) ^ (Q => R)

but we know that (P v Q) is True:

True ^ (P => R) ^ (Q => R)

and that leaves us with just:

(P => R) ^ (Q => R)

An implication like P => R translates to:

~(P ^ ~R)

and that can be converted to:

(~P v R)

using that, we have:

(~P v R) ^ (~Q v R)

if we factorize we get:

(~P ^ ~Q) v R

since:

(~P ^ ~Q) == ~(P v Q)

we have:

~(P v Q) v R

but we know that (P v Q) is True:

~True v R

or:

False v R

and that leads to the final answer:

R