Logic is the study of reasoning. It's a system for taking statements, connecting them, and figuring out what conclusions logically follow. The foundation of this system is propositional logic, which deals with simple, declarative sentences.

Propositions are the atoms of logical arguments. They are clear-cut statements without ambiguity. For example:

• "The earth is flat." (A false proposition)
• "Water boils at 100°C at sea level." (A true proposition)

Things that aren't propositions include questions ("What time is it?"), commands ("Close the door"), or opinions ("That movie was great"). In logic, we often use letters like $p$ and $q$ as shorthand for these propositions.

Propositions are interesting, but they become powerful when we connect them. We do this using logical connectives. These are the tools that let us build complex statements from simple ones.

To analyze these complex statements, we use a tool called a truth table. It's a simple chart that lists every possible combination of truth values for our propositions and shows the result for the overall statement.

In the table for AND ($p \wedge q$), you can see the statement is only true when both $p$ and $q$ are true. Now let's look at OR ($p \vee q$).

Here, the statement is true in every case except when both $p$ and $q$ are false. The implication table is often the most surprising.

The only time an implication is false is when the first part is true, but the second part is false. Think back to the promise: "If you get an A, I'll buy you ice cream." The promise is only broken if you get an A (T) and I don't buy ice cream (F). If you don't get an A (F), it doesn't matter whether I buy you ice cream or not; the promise was never broken.

When we build truth tables for more complex statements, some interesting patterns emerge. These statements fall into three special categories.

A classic example of a tautology is $p \vee \neg p$ ("p OR not p"). It's always true that a statement is either true or false. Tautologies are the bedrock of logical certainty.

The logical form of a contradiction is $p \wedge \neg p$ ("p AND not p"). A proposition and its negation can never be true at the same time. Identifying contradictions is key to finding flaws in an argument.

Most statements we deal with are contingencies. Their truth depends on the facts of the world. For example, $p \rightarrow q$ is a contingency, as its truth table has both true and false outcomes.

Propositional logic is a great starting point, but it has its limits. It treats entire statements as single, indivisible units. It can't look inside them to see how objects relate to each other.

Consider this classic argument:

1. All humans are mortal.
2. Socrates is a human.
3. Therefore, Socrates is mortal.

This argument is clearly valid. But in propositional logic, we'd have to assign a different letter to each statement:

• $p$: All humans are mortal.
• $q$: Socrates is a human.
• $r$: Socrates is mortal.

The argument becomes: $p$ and $q$, therefore $r$. Propositional logic can't see the connection between these statements. It doesn't understand that "Socrates" is an instance of "human" or that the property of being "mortal" is shared. To analyze arguments like this, we need a more powerful system that can handle objects, properties, and relationships.