Self Contradictory or tautologous Biconditionals

Question

If I have a statement, i.e., (G → ∼Q) ↔ ∼(Q • G), would it be accurate to call this statement self-contradictory?

On the left side, it is not the case that Q is true given that G is true.

On the right side, it is the case that Q and G are both false.

Thus I have a contrary, correct?

However, I am being asked if this statement is consistent, self contradictory, tautologous, contingent, or logically equivalent. I am thinking that perhaps it is tautologous, since it "jives" according to the traditional square of opposition as being a contrary?

Thoughts?

Answer 1

You have mis-translated the right-hand expression (assuming the dot operator is 'and'). The expression does NOT mean 'G and Q are both false', it means 'G and Q are not both true'. So, for example, that expression will be true when G is true and Q is false - which incidentally also makes the left-hand expression true.

Answer 2

Be careful $$\lnot (G\land Q) \not\equiv (\lnot G \land \lnot Q)$$ Rather, by DeMorgan's rules, we know that $$\lnot (G \land Q) \equiv \lnot G \lor \lnot Q$$

(See DeMorgan's Law(s))

$$ $$

Now, we have that $${(G → \lnot Q)} \equiv \lnot G \lor \lnot Q \equiv \lnot Q \lor \lnot G \equiv \color{blue}{Q \rightarrow \lnot G}$$ $$ $$ And we have that $$\lnot(Q \land G) \equiv \lnot Q \lor \lnot G \equiv \color{blue}{Q\rightarrow \lnot G}$$

$$ $$

Can you make some conclusions about, $(G\rightarrow \lnot Q) \iff \lnot(Q \land G)?\,$