| P | Q | P∧Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
A proposition is the simplest kind of claim about the world. It is a sentence that is either true or false, never both, never neither. The sky is blue. Water boils at one hundred degrees Celsius. These are propositions because we can, at least in principle, determine their truth. Questions and commands are not propositions. They do not carry truth values the way a leaf carries veins.
The study of propositions is the foundation of all logical inquiry. Before we can reason about complex arguments, before we can construct proofs or evaluate the validity of an inference, we must first understand what it means for a single statement to be true or false. This is not as simple as it sounds. Consider the proposition "this sentence is false." It is a grammatically correct English sentence, but it cannot be assigned a truth value without contradiction.
We represent propositions with letters. P, Q, R, S. These are not abbreviations; they are abstractions. When we write P, we mean any proposition whatsoever. The letter strips away the content and leaves only the logical skeleton. This is the great power of formal logic: by removing the specifics, we reveal the structure.
The connectives join propositions together. "And" combines two propositions into a conjunction that is true only when both parts are true. "Or" creates a disjunction that is true when at least one part holds. "Not" inverts a single proposition, turning truth to falsehood and falsehood to truth. "If...then" establishes a conditional relationship, a promise that holds unless the premise is true and the conclusion false.
In the reef of formal logic, each operation takes the shape of a creature. The gate is both a mechanism and a metaphor: it opens to let truth pass through, or closes to block the false. Here are the four fundamental species of the logical reef.
| P | Q | P∧Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| P | ¬P |
|---|---|
| T | F |
| F | T |
| P | Q | P∨Q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| P | Q | P⊕Q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
A proof is a sequence of propositions, each following from the ones before it by the application of a rule of inference. It is the spine of logical reasoning, the mechanism by which we move from what we know to what we can demonstrate. A proof does not persuade; it compels. If the premises are true and the rules are valid, the conclusion is inescapable.
Consider the simplest form of deductive reasoning, known since antiquity as modus ponens. We begin with two premises: first, that if P is true then Q must also be true; second, that P is in fact true. From these two statements alone, we can conclude with absolute certainty that Q is true.
The beauty of this structure is its transparency. Every step is visible. Every justification is explicit. There are no hidden assumptions, no rhetorical flourishes, no appeals to emotion or authority. The proof stands or falls on its logical structure alone.
More complex proofs chain multiple inference rules together. We might use modus tollens to reason backward from a false conclusion, or hypothetical syllogism to link conditional statements into longer chains. We might employ reductio ad absurdum, assuming the negation of what we wish to prove and showing that this assumption leads to contradiction. Each technique is a tool in the logician’s cabinet, as precise and specialized as a naturalist’s instruments.
Logic is not a set of rules imposed from outside. It is a way of seeing what is already there, hidden in the structure of every sentence, every argument, every decision. Like the naturalist who learns to see the species in a reef by learning the shapes and patterns of life, the student of logic learns to see the architecture of thought itself. Each gate, each connective, each step of a proof is a specimen in a cabinet that has no final drawer. The collection grows as long as we continue to look carefully.