2.5.1. Using Logical Identities for Simplification

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. 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.

Logical equivalence refers to two propositions that have the same truth values under all circumstances. This means if one proposition is true, the other must also be true, and vice versa. The notation ≡ indicates that these two statements are equivalent, not just equal in value like regular numbers. For example, if X is true (say the sky is blue), then Y (the sky is not dark) must also be true. If X is false, Y must also be false.

Think of two friends who both agree on a topic, say the best flavor of ice cream. If one says chocolate is the best (X) and the other agrees (Y), they are equating their preferences, just like logically equivalent propositions share truth values.

A tautology is a proposition which is always true, irrespective of what truth value you assigned to the underlying variables. An example of a tautology is the disjunction of p and ¬p.

A contradiction is a proposition that is always false, regardless of the truth value assigned to the underlying variables. An example of a contradiction is p conjunction ¬p.

A contingency is a proposition that can be either true or false, depending on the truth values assigned to its variables. For example, p conjunction q can be true or false depending on the specific truth values of p and q.

A tautology is a statement that is universally true. Think of it like saying, "It is either raining or not raining"; this statement will always hold true. A contradiction, on the other hand, is always false, like saying, "It is raining and not raining at the same time"; this is logically impossible. A contingency is a statement that can be sometimes true and sometimes false, such as "It is raining" which can be true if the weather is bad and false otherwise.

Consider a light switch. In the case of the tautology, the light switch is either on or off, which will always hold true. For a contradiction, if I say, "The switch is both on and off at the same time," that makes no sense. For a contingency, if I make the statement, "The light is on;" it could actually be off, depending on the scenario.

There are various standard logical equivalent statements which are available and they are also called by various names. For instance, the conjunction of p and true is always p (identity law). The double negation law states that the negation of the negation of p is p. De Morgan’s laws allow us to interchange conjunctions and disjunctions under negation. The distributive law, similar to arithmetic, allows distributing conjunction over disjunction.

Logical identities are key rules that simplify logical expressions just like formulas simplify math problems. For example, the identity law means that if you add 'true' to any proposition, it's equivalent to just that proposition (because true does not change its reality). Similarly, double negation states that if you deny something twice, you end up back at the original statement. De Morgan's laws allow you to switch between 'and' and 'or' when negating statements, while the distributive law helps manage more complex logical expressions by breaking them down into simpler parts.

Think of baking a cake. You know that if you add sugar (true) to flour (p), you still have flour. If you take away sugar and then add it back, you haven’t changed anything about the flour; similarly in logic, these laws allow us to manipulate statements without altering their basic truth.

To verify whether these logical identities are correct, we can use the truth table method. We can draw a truth table for the left-hand side of the expression and another for the right-hand side, then check if they are the same. However, if a logical identity involves too many variables, constructing truth tables becomes impractical.

Truth tables visually represent the truth values of propositions for every possible scenario, allowing us to prove logical identities. However, as the number of variables increases, the tables grow exponentially, making them unwieldy and impractical for complex expressions. In such cases, known logical identities help; rather than proving from scratch, we can leverage established identities to simplify and verify equivalence.

Imagine trying to verify all possible pizza toppings with a chart - with just a few toppings, it’s manageable, but with dozens, it becomes overwhelming. Instead, if you know a popular topping combinations (like pizza in its classic form), you take those as given and adjust as needed, similar to using logical identities.

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