>>11557326>>11557334To summarize, proof writing can be boiled down to showing that an implication between the premises of an argument and the conclusion is a tautology. Often the fastest way to prove this is by only looking at the cases in which the conclusion is false- since those are the only cases in which the argument could be invalid. Within the interpretations in which the conclusion is false- if there are no interpretations (truth value assignments to the variables of your statement) in which the premises are true, then the argument is valid.
So essentially, you are really just looking at interpretations in which the conclusion of an argument is false, and then trying to find a true premise. Since the premises are connected by a conjunction, this is actually quite easy because conjunctions are true only if all its constituent statements are true. So you're really only interested in interpretations in which all the premises are true and then conversely looking for a false conclusion.
So the key points are:
1. Convert all the definition of terms from calculus into atomic statements
2. Convert all theorems into implications between premises and conclusions
3. Identify interpretations (truth-value assignments of the atomic statements) in which all of the premises are false
4. Find a false conclusion among those interpretation
If you can't- the argument is valid. If you can, it's invalid.
