Pattern Proof - Logic

Question

Symbolize and test for the validity of the following arguement using pattern proof

If u send me a message then I will come to meet you. If you do not send me a message then I will go to China. Therefore either I will come to meet you or I will go to China

Answer 1

If p = you send me a message, q = I will come meet you , and r = I will go to China

You have the statements p --> q and. ~p --> r

From here there are a few things you could do:

~p --> r is equivalent to ~r --> p by contrapositive

So, by the law of syllogism, you can get ~r --> p --> q so

~r --> q which is equivalent to r V q (r or q)

The last equivalency can be shown with a truth table.

However, you said "either" which implies exclusive or. This means you also need to show that r ∧ q (r and q) is false.

Actually, exclusive or is equivalent to

( r∨ q) ∧ ~(r ∧q)

Since the statements are only implications, you cannot be sure that I will not go to China and come meet you. It was not stated that I will only come meet you if you message me, nor was it stated that I will only go to China if you don't message me. So, if you really meant "either " , then it would not be a true statement. However, if you just meant "or" without the "either", then it is a true statement.