Propositions and Truth Values: The Logic Basics Guide

A proposition is a declarative sentence that is either true or false — but not both, and not neither. "New Delhi is the capital of India" is a proposition (true); "Close the door!" is not a proposition at all, because commands are neither true nor false. Propositions are the atoms of all logical reasoning.

This matters because every proof, every program's if-statement, and every digital circuit is ultimately a structure of propositions combined by logical rules. If you cannot tell which sentences carry truth values and how those values combine, you cannot follow a mathematical argument.

Key Concepts

• Truth value: the value T (true) or F (false) assigned to a proposition. A proposition has exactly one.
• Negation (¬p): "not p" — flips the truth value.
• Conjunction (p ∧ q): "p and q" — true only when both are true.
• Disjunction (p ∨ q): "p or q" (inclusive) — false only when both are false.
• Implication (p → q): "if p then q" — false only when p is true and q is false.
• Biconditional (p ↔ q): "p if and only if q" — true when p and q match.
• Truth table: a table listing every combination of truth values and the resulting value of a compound statement.

Formulas

• Negation: ¬T = F; ¬F = T.
• Conjunction: T ∧ T = T; all other combinations give F.
• Disjunction: F ∨ F = F; all other combinations give T.
• Implication: p → q is F only for (T → F); it is T for (T → T), (F → T), and (F → F).
• Biconditional: p ↔ q is T when both match, F when they differ.
• De Morgan's laws: ¬(p ∧ q) is equivalent to (¬p ∨ ¬q); ¬(p ∨ q) is equivalent to (¬p ∧ ¬q).

Worked Example

Let p: "2 + 2 = 4" (T), q: "Delhi is in India" (T), r: "5 > 9" (F). Evaluate each compound statement.

1. p ∧ r: T ∧ F = F — "2+2=4 and 5>9" is false because r is false.
2. p ∨ r: T ∨ F = T — only one true part is enough for "or".
3. p → r: T → F = F — a true premise with a false conclusion makes the implication false.
4. r → p: F → T = T — a false premise implies anything; this surprises beginners but is the standard rule.
5. p ↔ q: T ↔ T = T — both true, so the biconditional holds.

Common Misconceptions

• Questions and commands are not propositions. "Is it raining?" has no truth value — only declarative sentences qualify.
• Implication needs no causal link. "If 2+2=4 then Delhi is in India" is true (T → T) despite the two facts being unrelated — implication is purely about truth values.
• "This statement is false" is not a proposition. It is a paradox: true makes it false, false makes it true. Genuine propositions never self-destruct like this.

Real-World Applications

• Digital circuits: AND, OR, NOT gates are physical conjunctions, disjunctions, and negations — your phone is applied propositional logic.
• Programming: every if-condition combines propositions with and/or/not; De Morgan's laws simplify tangled conditions.
• Database queries: SQL WHERE clauses filter rows using exactly these connectives.
• Mathematics: every theorem is an implication (assumptions → conclusion) whose truth table you must respect in proofs.

Propositions are sentences with truth values; connectives combine them by fixed truth-table rules. Master negation, and, or, and the tricky if-then, and you hold the keys to every proof and program you will ever meet.