Natural language is powerful but often ambiguous. A statement like "You can have cake or ice cream" is unclear. Does it mean you can have one, the other, but not both? Or can you have both? To reason with precision, we need to translate these statements into a formal system. This is the role of propositional logic.

In this system, we don't care about the specific meaning or content of a sentence. We only care about its truth value: is it true or false? We represent simple, declarative sentences as variables, typically lowercase letters like $p$, $q$, and $r$. These are called atomic propositions. For example, $p$ could represent the statement "It is raining," and $q$ could represent "The ground is wet."

By stripping away the linguistic nuance, we can focus entirely on the logical structure of an argument. Our goal is to connect these atomic propositions in a way that is precise and unambiguous.

We link propositions together using logical connectives, also known as truth-functional operators. Each connective has a strict rule that determines the truth value of the complex proposition it forms, based solely on the truth values of the propositions it connects. You're likely familiar with the basic ones: conjunction (and), disjunction (or), and negation (not). Let's formalise these and introduce some more powerful ones.

The inclusive disjunction ($\,\lor\,$) is true if at least one of the propositions is true. This is the default "or" in logic. In contrast, the exclusive disjunction ($\,\oplus\,$) is true only when exactly one of the propositions is true. It captures the "one or the other, but not both" sense of "or".

The most important and often misunderstood connective is material implication. It's crucial to separate its logical definition from our everyday idea of cause and effect.

This leads to some counter-intuitive results. For example, the statement "If Paris is in England, then the sky is green" is logically true. Why? Because the premise ($p$: "Paris is in England") is false. Since the premise is false, the implication holds true regardless of the conclusion's truth value. The only thing the implication forbids is moving from a truth to a falsehood.

A truth table is our primary tool for analysing complex propositions. It systematically lists all possible combinations of truth values for the atomic propositions and shows the resulting truth value for the entire formula.

To build one, start with columns for each atomic variable. If there are $n$ variables, there will be $2^n$ rows. Then, add columns for each sub-formula, building up from the simplest parts to the full expression. Let's analyse the formula for logical equivalence, $(p \rightarrow q) \land (q \rightarrow p)$.

We need columns for $p$ and $q$, then for the two implications $p \rightarrow q$ and $q \rightarrow p$, and finally one for the conjunction that connects them.

By comparing the final column with the truth table for $p \leftrightarrow q$, you can confirm they are indeed logically equivalent. This is a powerful method for proving equivalences.

The final column of a truth table reveals the nature of a proposition. We can classify any formula into one of three categories.

Tautologies are also known as valid formulae. They represent logical truths that hold regardless of the facts of the world.

Contradictions are unsatisfiable. They represent logical falsehoods.

Most statements we deal with in arguments are contingencies. Their truth or falsity depends on the state of the world. By formalising them, we can rigorously test the validity of the structures that connect them.

This formalisation provides the syntax and semantics needed to build and analyse complex logical arguments with clarity.