Propositional Logic
Study formulas built from propositions and truth-functional connectives. Truth tables define how formulas evaluate under assignments, while equivalence compares formulas across every possible assignment in the table.
A proposition is treated as true or false.
In classical propositional logic, an atomic proposition is a statement represented by a symbol such as P or Q and assigned one of two truth values. The formal system then studies how compound formulas depend on those assigned values.
Combine proposition symbols with connectives and parentheses.
Give each atomic proposition a truth value.
Apply connective definitions from the inside outward.
Hold the four input states fixed. Change the connective.
With two propositional variables there are four possible truth-value assignments. A connective is defined by what output it assigns to each of those rows.
| P | Q | IMPLIES |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Material implication is false only in the row where P is true and Q is false.
This is the one false row for material implication: the antecedent is true while the consequent is false.
Same truth column, different syntax.
Two propositional formulas are logically equivalent when they have the same truth value under every assignment to their variables. Matching words or symbols are not required; matching evaluations are.
Negating a conjunction is equivalent to disjoining the negated components.
Definitions first, intuition second.
Everyday language often carries implication, exclusivity, emphasis, timing, and context beyond these truth-functional definitions.
reverses the truth value of P
true only when both P and Q are true
true when at least one of P or Q is true
false only when P is true and Q is false
true when P and Q have matching truth values
true when exactly one of P and Q is true