Recently I come across the following question :
What can be said about the truth value of $Q$ when: $P$ is true and $P\Leftrightarrow Q$ is false?
The answer to this question is "clearly" that Q is false.
But for some time using only my intuition without using the true/false truth values table, I was convinced that we can't decide whether $Q$ is true or false. I was articulating in my mind that $P \Leftrightarrow Q$ is wrong. Hence the relation between P and Q can be anything but equivalence, Then even knowing that P is true we can't decide anything about Q.
I want to identify this maths logical/intuition fallacy. Is this type of "limitations" imposed by the vocabulary of the formal logic we study early in our mathematic courses, and which does not always reflect the human language, is a part of the study in advanced mathematics logic?
Most of the answers went in the direction of giving the definitions for the equivalence operator Propositions and open statements. I am aware of the mistake concerning these definitions. It is my fault for not stating my background. I was preparing for an instruction session given to some first-year bachelor students. Having an analysis background, I never give this kind of sloppy intuition/ logic fallacies a deep thought. My question can be roughly formulated: " are the study of these sloppy intuition/ logic fallacies part of mathematics"