Logic is the study of correct reasoning. To reason correctly, we need to start with clear, unambiguous statements. In logic, these basic statements are called propositions.

For example, "Paris is the capital of France" is a true proposition. "The moon is made of green cheese" is a false proposition. Both are propositions because they have a clear truth value.

Questions like "What is your name?", commands like "Close the door," and opinions like "Chocolate is the best flavor" are not propositions. They can't be assigned a simple true or false label.

We rarely deal with single propositions in isolation. We combine and relate them to form arguments. This is done using five basic tools called logical connectives. We'll use the letters $P$ and $Q$ to stand for any two propositions.

We use the symbol $\land$ for AND. So, $P \land Q$ means "P and Q". If $P$ is "It is sunny" and $Q$ is "It is warm," the conjunction $P \land Q$ is true only if both conditions are met.

The symbol for OR is $\lor$. So, $P \lor Q$ means "P or Q". This is an "inclusive or," meaning it's fine if both are true. If someone says, "I'll bring the chips or the salsa," the statement is true if they bring chips, if they bring salsa, or if they bring both.

The symbol for NOT is $\neg$. If $P$ is "The light is on," then $\neg P$ means "The light is not on." They always have opposite truth values.

Implication (Conditional): This is an "if-then" statement, written $P \rightarrow Q$. It means "if P, then Q". Think of it as a promise. An implication is only false when the "if" part ($P$) is true, but the "then" part ($Q$) is false. In all other cases, the promise is kept.

Consider the promise: "If you get an A, then I will give you $100."

• You get an A (P is true) and I give you $100 (Q is true). My promise was kept. The implication is true.
• You get an A (P is true) but I don't give you $100 (Q is false). I broke my promise. The implication is false.
• You don't get an A (P is false), and I give you $100 anyway (Q is true). My promise is not broken. The implication is true.
• You don't get an A (P is false), and I don't give you $100 (Q is false). My promise is not broken. The implication is true.

Biconditional: This is an "if and only if" statement, written $P \leftrightarrow Q$. It's true only when $P$ and $Q$ have the same truth value (both true or both false).

For example, "You can drive if and only if you have a license." If you can drive, you must have a license. If you have a license, you can drive. The two parts are locked together.

How can we keep track of all these rules? A truth table is a simple chart that shows the output of a logical expression for every possible combination of inputs. It's a powerful tool for analyzing complex statements.

A special kind of proposition is one that is always true, no matter the truth values of its components. This is called a tautology. For example, the statement $P \lor \neg P$ ("The light is on, or the light is not on") is always true. You can prove this by creating a simple truth table.

Since the final column is all Ts, it's a tautology.

Truth tables help us check if an argument is valid. An argument is valid if the conclusion must be true whenever all the premises (the starting statements) are true. We can see this by looking for a row in the truth table where all premises are true. If the conclusion is also true in that row (and every such row), the argument is valid.

However, building truth tables for complex arguments can be tedious. Instead, we can use simple, proven patterns of reasoning called rules of inference. Here are two of the most common ones.

This pattern is incredibly intuitive. It's the 'method of affirming'.

1. Premise 1: $P \rightarrow Q$ (If it's raining, the ground is wet.)
2. Premise 2: $P$ (It is raining.)
3. Conclusion: Therefore, $Q$ (The ground is wet.)

This is the 'method of denying'. It allows us to work backward from a false outcome.

1. Premise 1: $P \rightarrow Q$ (If it's raining, the ground is wet.)
2. Premise 2: $\neg Q$ (The ground is not wet.)
3. Conclusion: Therefore, $\neg P$ (It is not raining.)

These building blocks—propositions, connectives, and basic rules—form the foundation of formal logic. They allow us to translate complex arguments into a symbolic language we can analyze with precision.

Ready to test your understanding? Let's try a few questions.

Mastering these concepts is the first step toward building and deconstructing arguments effectively.