At the heart of logic is the idea of a proposition. A proposition is a statement that can be definitively labeled as either true or false. It's a clear declaration about the state of the world.

Commands, questions, and opinions are not propositions because they can't be assigned a simple true or false value. "What's for dinner?" isn't true or false. Neither is "Go clean your room." Logic needs firm ground to stand on, and propositions provide that solid footing.

To make things simpler, we use letters like $p$, $q$, and $r$ to stand in for entire propositions. So, we might say $p$ represents "It is raining."

Rarely do we deal with single propositions in isolation. We combine them to form more complex thoughts. In logic, we use five special symbols called logical connectives to link propositions together.

We use the symbol $¬$ for negation. If $p$ is "It is sunny," then $¬p$ means "It is not sunny."

Here’s how its truth is determined:

Next, we have connectives that join two propositions.

The symbol for conjunction is $∧$. If $p$ is "It is cold" and $q$ is "It is snowing," then $p ∧ q$ means "It is cold and it is snowing." This is only true if both conditions are met.

The symbol for disjunction is $∨$. If $p$ is "She will study on Saturday" and $q$ is "She will study on Sunday," then $p ∨ q$ means "She will study on Saturday or she will study on Sunday." This is true if she studies on Saturday, or on Sunday, or on both days. This is an "inclusive or."

Now for two slightly trickier connectives.

We use the symbol $→$ for implication. The statement $p → q$ means "If $p$, then $q$." An implication is only false when the first part ($p$) is true and the second part ($q$) is false. Think of it as a broken promise. If you get an A (p is true) and I don't buy you pizza (q is false), then I broke my promise ($p → q$ is false). In all other cases, the promise is kept.

That last part can seem strange. If the first part is false, the implication is true. If you don't get an A, my promise is not broken whether I buy you pizza or not. The condition for the promise was never met.

Finally, the strongest connection.

The symbol is $↔$. The statement $p ↔ q$ means "$p$ if and only if $q$." This is like a two-way street; it's the same as saying $(p → q) ∧ (q → p)$. It's true only when $p$ and $q$ match.

The little tables we've been using are called truth tables. They are a powerful tool for analyzing complex logical statements. A truth table systematically lists every possible combination of truth values for the simple propositions and shows the resulting truth value for the compound statement.

Let's build one for a more complex statement: $(p ∧ q) → p$. We need columns for our base propositions, $p$ and $q$. Then we build up to the final expression, piece by piece.

First, we list all combinations for $p$ and $q$. Next, we figure out the truth value for the part in the parentheses, $p ∧ q$. Finally, we use the values from the '$p ∧ q$' column and the '$p$' column to determine the truth value of the final implication.

Notice the final column. It's always true! This tells us something special about the statement.

Tautology: A statement that is always true, no matter the truth values of its components. Our example, $(p ∧ q) → p$, is a tautology. Another simple one is $p ∨ ¬p$ ("It is raining or it is not raining"), which must always be true.

Contradiction: A statement that is always false. The classic example is $p ∧ ¬p$ ("It is raining and it is not raining"). This statement can never be true.

Contingency: A statement that can be either true or false, depending on the truth values of its components. Most logical statements we encounter are contingencies. For example, $p → q$ is a contingency because its truth depends on the values of $p$ and $q$.

These concepts, from simple propositions to the analysis of truth tables, are the foundation for all logical reasoning.