How far do I need to go if I want to know if a proposition is a tautology, contingency or a contradiction?

Question

I'm just learning about this three possibilities in a logical proposition. But now I am simplifying propositions and then I have to determinate if it is a contradiction, tautology or contingency.

I have simplified this proposition, but I don't know if I have to continue simplifying more for answer if it is a tau, contr, or conti. Or at least is there something that I am missing? let me know!

Proposition

$$[(P\land Q)\to\lnot R]\leftrightarrow[\lnot (P\land Q)\lor \lnot R]$$

Conditional [~(PʌQ)v~R]↔[~(PʌQ)~R]

De Morgan   [(~Pv~Q)v~R]↔[~(PʌQ)~R]
   
De Morgan   [(~Pv~Q)v~R]↔[(~Pv~Q)~R]

Answer 1

Let $S=P\land Q$. Then the formula is $$( S\to \lnot R)\leftrightarrow (\lnot S\lor \lnot R).$$ Now recall that $(X\to Y)\equiv (\lnot X\lor Y)$.

Answer 2

To answer the question in your question title (rather than the one specific case):

You should simplify it all the way to $ \top $ (or $ \mathrm T $, $ \operatorname { True } $, or however you want to write this) or $ \bot $ (or $ \mathrm F $, $ \operatorname { False } $, etc). If you can do this, then you know that it's a tautology or a contradiction.

If you can't simplify it that far, then either it's contingent or you have failed to simplify it fully. If it's simplified to something that you recognize and know is contingent, then you're done. Or you can simplify to a normal form (the conjunctive normal form, or the disjunctive normal form), which is a step-by-step process using the operations that appear in the expression; and then you'll know that it's simplified as far as it can go and you'll know what it is. Or if you suspect that it's contingent but aren't quite sure, then you can find one way to assign truth values to each propositional variable that makes the proposition evaluate to $ \operatorname { True } $ and one way to assign truth values that makes it evaluate to $ \operatorname { False } $, and then you'll know that it's contingent.

Alternatively, you can skip all simplification and make a truth table, looking at every possible way to assign truth values to the propositional variables and seeing how the proposition evaluates in each case. If this is always $ \operatorname { True } $, then it's a tautology; if it's always $ \operatorname { False } $, then it's a contradiction; and if it's $ \operatorname { True } $ at least once and $ \operatorname { False } $ at least once, then it's contingent.

Both the truth tables and the normal forms are usually slower than simplifying in an ad-hoc way, but they're guaranteed to give you the answer. Although since you're still learning, you probably want to try simplifying when this is not faster, to give yourself some practice with it (and to make it faster). Once you've learnt enough, however, then it's up to you which approach you want to take (simplifying following your gut, simplifying step-by-step to a normal form, or making a truth table).