predicate logic

( (P⇒Q) and (R⇒S) ) ⇒ ( (P⇒S) or (R⇒Q) )

This is an tautology. Use Proof by contradiction to prove that it is always true.

Anyone wanna help me out, I'm stuck atm :S

Use truth tables and it will become much easier to solve. Your main goal is to see if the first two sentences are true, then the next two are true.

basically, all things being equal, interchange and reverse and see if it still holds true. If it does than = law of consistency (always and forever) true
If it doesn't, where doesn't it hold true? Or possibly, is what it is true to something other than what it is saying? What is this contradiction true towards?

First you assume the left is true. You would do this for any proof (except contrapositive, but let's not worry about that).

Then to use proof by contradiction, you would assume that the negation of the right is true. That is, assume P and not S, and assume R and not Q.

You would immediately come to the conclusion that Q is true (since P is true), and R is false (since not S implies not R).

Since this is clearly a contradiction, you can say that the right side is true. This is how you do a proof by contradiction.

Simple, just use inductive proof over predicate logic. Actual proof is left to the reader as an exercise.