# logical equivalence between statements

I get confused when i think about logical equivalence between conditional statements. For example saying that

∼(P⇒Q)=P∧∼Q.

If there is variables involved then the statement on the left says that there exists some value of that variable for which P does not imply Q. The statement on the right P and not Q is true for all values of that variable, This is what i understand but i think im wrong because these statements are logically equivalent and are meant to say the same thing about P and Q.

I can understand if they did not include variables but not if they do can someone help explain, thanks.

• Equivalence here simply means that the two statements have the same truth table, entry by entry. For any assignment of truth values for $P,Q$, the two statements are either both true ir both false. – AnyAD Dec 1 '18 at 23:59

The way you are trying to understand equivalence is slightly correct, but mostly not: first one is saying $$P$$ does not imply $$Q$$. In a truth-table, if you notice, a conditional is true in two and only two cases: either $$P$$ is false or $$P$$ is true and $$Q$$ is true. Therefore, to say $$P$$ does not imply $$Q$$ is to say: $$P$$ is true and $$Q$$ is false.
Now notice, $$P$$ and not $$Q$$ is true exactly when both $$P$$ is true and $$Q$$ is false. Did you notice anything similar between this sentence and the last sentence of the first paragraph?
This similarity means $$\lnot (P \implies Q)$$ and $$P \land \lnot Q$$ have the same truth-table. Therefore, they are uquivalent.
• @CarlosBacca I think you misunderstand a logical implication. To say $P$ implies $Q$ is to say if $P$ is true Then $Q$ must be, by definition true. In order for the conditional to makes sense. Therefore, you can not isolate propositions involved in a conditional like you have done. $P\implies Q$ is true when $P$ and $Q$ are semantically and syntactically related. Not isolated. – Bertrand Wittgenstein's Ghost Dec 2 '18 at 0:06