2.3.3. Distributive Law

The heretofore discussed logical identities include ... we also have this distributive law this says that you can distribute the disjunction over conjunction and so on.

In logic, expressions can be manipulated in much the same way as numbers. One of these manipulations is called the Distributive Law, which states that the disjunction (logical OR) can distribute over the conjunction (logical AND). This means that for propositions P, Q, and R, the expression P AND (Q OR R) is logically equivalent to (P AND Q) OR (P AND R). This foundational rule allows the logical restructuring of statements for further analysis or simplification.

Think of the Distributive Law like distributing apples in different baskets. If you have a basket containing apples and oranges, and you want to take apples out and give them to two different friends, you can give some apples to the first friend and some to the second friend. This mirrors how you can break down a complex logical statement into simpler parts while still retaining the overall meaning.

How do we verify whether these logical identities are correct? ... hence I can say that these two are logical equivalent statements.

To verify logical identities, mathematicians often use truth tables. A truth table provides all possible truth values for a given set of propositions and shows how they result in the overall expression's truth value. By calculating the truth values for both sides of an identity, we can confirm whether they align under all possible circumstances. If they match for every value, then the two expressions are logically equivalent.

Imagine you’re a teacher grading a test. You want to know if two students’ answers to a question are effectively the same. You can create a checklist of criteria (like correctness and completeness) and evaluate each answer against it. If both answers meet all the same criteria points, you can confidently say they are equivalent in terms of fulfilling the assignment.

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

The truth table method becomes cumbersome as the number of variables increases. For more complex logical expressions with many variables, the total number of rows in the truth table increases exponentially. For example, with 3 variables, you only need 8 rows, but with 20 variables, you require 2²⁰ rows, which is impractical. This highlights a limitation when using this method for larger scale logical statements.

This is akin to organizing a large group of people into teams where everyone must be evaluated on multiple criteria. For small groups, you can easily go through each person one by one. But imagine the hassle if you had to do this for a massive conference with hundreds of attendees; the task becomes infeasible as the groups grow, much like the expansive truth table.

So we are trying to do the same thing even in the mathematical logic. ... we are using the De Morgan’s law and hence we are substituting this part with this part and so on.

In logical proofs, once we've established certain logical identities, we can use them repeatedly to transform complex expressions into simpler ones. By referring to known laws—like De Morgan's Laws or the Distributive Law—we can streamline our reasoning without verifying each individual step. This method not only saves time but also helps in focusing on the logical structure of arguments.

Consider a chef in a kitchen who knows a variety of recipes. When making a dish, the chef might use established techniques (like chopping vegetables a certain way or marinating meat) rather than reinventing each step from scratch every time they cook. This efficiency allows the chef to create delicious meals more easily, much like how using known logical identities simplifies the logical reasoning process.

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