2.6.1. Summary of Concepts Introduced

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.

Logical equivalence means that two propositions yield the same truth value in every possible situation. This is akin to how in algebra, two expressions can be equivalent if they simplify to the same result regardless of the values assigned to their variables. For logical propositions X and Y to be equivalent, they must either both be true or both be false under all scenarios, which we illustrate with the equivalence notation ≡.

Consider two friends, Alex and Jamie, who have made a promise to meet at a café. If both say they will arrive at 3 PM, their promises are equivalent. It doesn't matter how they say it ("I will be there at 3" or "I promise to arrive at 3"), as long as they both have the same commitment, they are equivalent in their intentions.

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.

Tautology, contradiction, and contingency are fundamental concepts in logic. A tautology is a statement that will always be true regardless of the truth values of its components (like 'p or not p'). A contradiction, by contrast, is always false (like 'p and not p'). A contingency neither guarantees truth nor falsehood; it can be true in some scenarios and false in others (like 'p and q'). Understanding these terms is essential for analyzing logical expressions.

Think of a light bulb. A tautology would be, "The light is either on or off"—it's always true. A contradiction would be, "The light is on and off at the same time"—this can never happen. A contingency might be, "The light is on"; it can be true or false depending on whether the bulb is working or not.

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.

For two propositions X and Y to be logically equivalent, the statement 'X if and only if Y' must always hold true. This relationship is expressed as a bi-implication and is considered a tautology if it is true under all truth assignments. If this holds, it confirms that both propositions behave identically in all circumstances.

Imagine two recipes for the same dish—one for pizza and another for lasagna. If both recipes need exactly the same ingredients in identical proportions to be successful, we could say that following either recipe is logically equivalent to making the same dish, because both will yield the same delicious result.

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.

Logical identities, like the identity law and double negation law, provide frameworks for simplifying and manipulating logical expressions. For instance, the identity law tells us that combining a proposition with true preserves the proposition's truth. Similarly, double negation implies that negating a negated statement results in the original statement itself. These laws are fundamental tools in logical reasoning and proof.

Think about a street sign: if it says 'Do Not Enter', then saying 'Do Not Not Enter' simply means you can enter again. This simplification echoes the double negation law, demonstrating how logical identities help clarify understanding.

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.

Truth tables provide a systematic way to evaluate the truth values of logical expressions under all possible combinations of variable values. By constructing a truth table for both sides of an identity, we can check if they yield the same outcomes, thereby confirming their validity. However, as the complexity increases with more variables, constructing truth tables may become unwieldy, leading to the need for logical identities for simplification.

Consider a factory producing two types of widgets. A truth table could help you track every possible combination of production scenarios to see if the final output meets quality standards. As the number of scenarios grows (like adding more variables), using logical identities to streamline the process becomes essential for efficiency.

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