Now, we want to define what we call as logically equivalent statement. So before trying to understand what are logical equivalent statements? Remember in algebra and in mathematics, you often come across expressions of this form. We say for instance that a² + 2ab + b² is equal to (a + b)². That means these two expressions are the same expression. What do I mean by same expression? Well, by that I mean that whatever value you assign to a and b, the left-hand side and right-hand side will give you the same answer. That is why these two expressions are the same expression. In the same way in mathematical logic if we have a compound proposition X and a compound proposition Y then I say that they are logically equivalent and I use this notation ≡. This is not an 'equal to' notation, this is representation of equivalence, this is also called as an equivalence notation. So I say that X and Y are logically equivalent if they have the same truth values. What I mean by that is I mean that if X is true then Y is true if X is false then Y is false that means it never happens that X and Y takes different truth values.

Now let us next define tautology, contradiction and contingency. A tautology is a proposition which is always true, irrespective of what truth value you assigned to the underlying variables. So, for example, if I consider this compound statement namely the disjunction of p and ¬p, then this will be always true; that means if p is true, then this is true and even for p equal to false this statement is again true. That means it does not matter whether your p is true or false; this statement is disjunction of p and ¬p will always be true and hence this is a tautology. Whereas a proposition is called a contradiction if it is always false irrespective of what truth value I assign to the underlying variables. So an example of contradiction is p conjunction ¬p. So you can verify that if p is false then this statement is false. And even for p equal to true this overall statement is false that means this statement is always false for every possible truth assignment of p and hence it is a contradiction. Whereas a contingency is a proposition, which is neither a tautology nor a contradiction that means it can be sometime true it can be sometimes false.

More formally X is logically equivalent to Y provided the X bi-implication Y is a tautology, right? Because if X bi-implication Y is a tautology, then it means that whenever X is false Y has to be false whenever X is true Y has to be true.

So there are various standard logical equivalent statements which are available which are very commonly used in mathematical logic and they are also called by various names. So for instance, the conjunction of p and true is always p that is called this law is called as the identity law. In the same way we have this double negation law which says that if you take the negation of negation of p then that is logically equivalent to p.

How do we verify whether these logical identities are correct? Well, we can verify using the truth table method namely we can draw, we can construct a truth table of the left hand side of the expression, we draw the truth table of the right hand side of the expression and verify whether the truth tables are the same.