2.3. Standard Logical 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 a^2 + 2ab + b^2 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 indicates that two statements yield the same truth value under all possible interpretations. For example, if you compare two statements X and Y, they are logically equivalent (noted as X ≡ Y) if both are true or both are false in every scenario. This can be compared to algebraic expressions being equal when substituted with the same values, yielding the same results.

Think about two houses that look different from the outside but are identical inside. Just like how those houses provide the same experience (shelter, comfort) regardless of their different appearances, two logically equivalent statements provide the same truth value in logic, even if they are expressed differently.

We will come back to that point later but let me first define a bi conditional operator or a bi conditional statement which for which we use this notation ↔ that means an arrowhead which has an arrowhead at both ends. And this bi conditional statement is used to represent statements of the form p if and only if q or in short form p if and only if q says another way another form of representing if and only if is iff. So very often for mathematical and for various theorem statements, you must have seen conditions like prove that this is true if and only if this holds right? So wherever we are making statements of that form, we are actually making statements of the form p bi-implication q.

A bi-conditional statement asserts that both parts of the statement are true or both are false simultaneously. For instance, 'p if and only if q' means that if p is true, then q is also true, and if p is false, then q is false. This relationship can also be expressed as two implications: p implies q and q implies p.

Consider a teacher saying, "You will pass if and only if you study." This means studying is necessary to pass (if you study, you pass) and sufficient to pass (if you pass, you must have studied). It highlights the close relationship between studying and passing.

Now let us next define tautology, contradiction and contingency. So 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 sometimes true it can be sometimes false. I cannot say that it is always true or it is always false.

A tautology is a statement that remains true regardless of the truth values of its variables. An example of a tautology is 'p or not p' (p V ¬p), which is always true. Conversely, a contradiction is a statement that remains false regardless of the truth values, such as 'p and not p' (p ∧ ¬p), which is always false. A contingency, however, can be either true or false depending on the values of its variables, like 'p and q'.

Think of a light switch: if the light is on, the statement 'the light is on or off' (tautology) is always true. The phrase 'the light is both on and off' (contradiction) is always false. However, if you say ‘the light is on’ (contingency), the light can either indeed be on or not, depending on the situation.

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. We also have this distributive law this says that you can distribute the disjunction over conjunction and so on.

Standard logical identities are rules that help in simplifying and transforming logical expressions. For instance, the identity law states that combining any statement (p) with 'true' yields p (p ∧ true = p). Similarly, De Morgan's laws allow for the negation of combined statements to be distributed in a specific way, such as converting conjunctions to disjunctions when taken inside negations.

Consider following the rules of a recipe: using a teaspoon of salt (identity law) will keep the dish salty (still just salt). De Morgan's law can be thought of as portioning out ingredients: if you have a combined mixture that's mixed, managing parts in order (like separating dry and wet ingredients) keeps the essence intact but let's you see their connections clearly.

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. So for instance, if you want to verify the De Morgan’s law, so the first part of the De Morgan’s law says that the negation of conjunction of p and q is logically equivalent to negation p disjunction negation q.

To verify logical identities, you can utilize truth tables. By constructing a truth table for two logical expressions, one for the left-hand side and one for the right-hand side, you can determine if the two expressions are equivalent by checking if their truth values match in every scenario.

Think of a teacher evaluating two different methods for solving a math problem. They would create a table checking every possible scenario to see if both methods yield the same answer. If both methods give the same result for every case, they are equivalent, just like logical expressions evaluated in truth tables.

However, the truth table method of verifying logically equivalent statements 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.

While truth tables are effective for verifying logical equivalences, they become impractical with many variables. For instance, a table for three variables has 8 truth combinations, but for 20 variables, there would be over a million rows to consider, which is nearly impossible to manage.

Imagine trying to compare many different combinations of toppings on a pizza: it’s easy to manage a small number of pizzas, but as you try to add more and more toppings, the combinations become overwhelming and unmanageable, similar to truth table rows growing exponentially.

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. So for instance, these are some of the standard logical equivalent statements, which we use to simplify complex expressions and verify whether those complex expressions are logically equivalent or not.

To bypass the cumbersome nature of truth tables for complex logical statements, mathematicians utilize standard logical identities. These well-established identities allow them to simplify expressions step by step, making it easier to demonstrate logical equivalences without constructing extensive truth tables.

Consider a mechanic repairing a car: instead of checking every single component each time (like using a truth table), they rely on known shortcuts and standard parts (logical identities) to streamline the repair process efficiently.

So now let us do an example here. Suppose I want to prove that my LHS expression and RHS expression, they are logically equivalent so this is my statement X this is my statement Y. Well in this case I can use the truth table method because my expressions X and Y involve only 2 variables and I can draw truth table which will have only 4 rows, but what I want to do here, I want to demonstrate here is that without even drawing the truth table, I can show that the expression X is logically equivalent to expression Y by using logical identities.

Demonstrating logical equivalence can often be achieved without constructing a truth table. By using accepted logical identities and applying them correctly to transform one expression into another, one can reach a conclusion about equivalence. In this way, logical identities become powerful tools for simplifying statements and proving equivalences.

Imagine explaining a recipe in two different ways (like baking brownies and making chocolate cake) that ultimately yield the same dessert: rather than writing down every step for both recipes, you can show how the ingredients transform through similar methods to highlight their equivalence in the final product, paralleling how showing transformations in logic shows equivalence.

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