2.2.3. Logically Equivalent Statements

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. More formally X is logically equivalent to Y provided the X bi-implication Y is a tautology.

Logical equivalence refers to the idea that two statements or propositions are equivalent in terms of their truth values. When we say that two expressions are logically equivalent, such as X and Y, we denote this relationship with a symbol (≡). This means that for every possible assignment of truth values to the variables in these expressions, the truth values of X and Y will always match. For example, if X is true, then Y must also be true; if X is false, then Y must also be false. Additionally, a formal way to express this logical equivalence is through bi-implication, stating that the expression 'X bi-implication Y' (X ↔ Y) is a tautology, meaning it is always true. Hence, logical equivalence ensures that two propositions convey the same truth in all circumstances.

Think of logical equivalence like two different routes leading to the same destination. Imagine you have two paths to get to a friend's house: one is through the park, and the other is through a neighborhood. No matter which path you take, as long as both routes are clear, you will arrive at your friend's place. In the same way, logical equivalence means that two different statements (like two paths) will lead to the same truth value (the destination) regardless of the specific details involved.

So the statement will be true only when both the sides of this expression or the compound propositions on both the sides take the same truth value. 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. We have this De Morgan’s law which is very important which says that if you have a negation outside then you can take the negation inside and split it across the various variables and if you have conjunction inside then it becomes disjunction and vice versa.

Logical equivalence can be demonstrated through various laws and identities in propositional logic. For instance, the Identity Law states that combining a proposition (p) with 'true' using conjunction (p ∧ true) always yields 'p'. Another important law is the Double Negation Law, which asserts that negating a negation returns the original proposition (¬(¬p) ≡ p). De Morgan's Laws show the relationship between conjunctions and disjunctions when negation is applied. For example, the negation of a conjunction is equivalent to the disjunction of the negations (¬(p ∧ q) ≡ ¬p ∨ ¬q). These standard laws form the basis for understanding and proving logical equivalences in more complex statements.

Consider a light switch. A light switch in the 'on' position brings light to a room. The Identity Law, like turning the switch on with the guarantee of the light being on (switch 'on' AND it is light), always returns the truth that the room is lit just by ensuring the switch is in the 'on' position. Furthermore, if a system allows you to toggle a switch twice (Double Negation), you find yourself back where you started: flipping the switch back and forth will result in the same light being on. De Morgan's Law can be likened to scenarios where turning off two lights simultaneously (to avoid darkness) means you must ensure not both lights are off (otherwise the room remains dark). This interplay of equivalences helps visualize how logical statements hold true across various conditions.

However, the truth table method of verifying logically equivalent statement has a limitation. Namely, the limitation here is it works as long as the number of variables the number of propositional variables which are there in your identity or the statement this is small. So in all this logical identity that I have written down in this table, there are at most three propositional variables and if I try to draw the truth table of a statement having 3 variables, and there will be only 8 rows which are easy to manage. But imagine I have a logical identity which has a 20 number of variables then the number of rows and that truth table will be 2^20 and definitely you cannot draw such a huge table. So that is why it is infeasible to verify the logical equivalence of statements using the truth table method and that is why what we do here is we use some standard logical equivalent statements.

While truth tables are a foundational tool for verifying logical equivalences, they become impractical as the complexity of expressions increases. With a limited number of propositional variables, constructing a truth table is straightforward. For example, with three variables, one can quickly derive the truth values in eight rows. However, if there are 20 variables, the number of rows escalates exponentially to 1,048,576, making it nearly impossible to construct or analyze. To overcome this limitation, mathematicians employ known logical identities to simplify or rewrite expressions in a way that showcases their equivalences without needing extensive truth tables.

Imagine attempting to compare all possible routes on a massive map with hundreds of paths—it's overwhelming! Instead, navigation apps use known shortcuts and routes to simplify the journey from one point to another without needing to analyze every single road. Similarly, in logic, instead of analyzing each possible truth value for numerous variables, we apply established laws to identify equivalences. These logical identities serve as shortcuts in proof processes, allowing us to focus on deriving the necessary conclusions with ease, just like following clear routes on a map.

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