Short answers, pulled from the story.
Propositional logic is a branch of classical logic that deals with propositions, which can be true or false, and the relations between them. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or zeroth-order logic. It studies compound propositions formed by logical connectives representing conjunction, disjunction, implication, biconditional, and negation.
Chrysippus is often credited with developing a deductive system for propositional logic in the 3rd century BC, work later expanded by his successor Stoics. His logic focused on propositions rather than terms, setting it apart from traditional syllogistic logic. Most of the original writings were lost, and Stoic logic faded into oblivion between the 3rd and 6th century CE before being resurrected in the 20th century.
Propositional logic does not deal with non-logical objects, predicates about them, or quantifiers, while first-order logic does. All the machinery of propositional logic is included in first-order and higher-order logics, making propositional logic their foundation. By comparison, truth-functional propositional logic is considered zeroth-order logic.
A truth table is a semantic proof method that determines the truth value of a propositional logic expression in every possible scenario by exhaustively listing the truth values of its constituent atoms. A formula is semantically valid if and only if all the lines of its truth table come out true. Truth tables have 2 to the n lines for n variables, so they can become very long for large values of n.
The truth table's invention is of uncertain attribution. The tabular structure is generally credited to either Ludwig Wittgenstein or Emil Post, or both independently, while ideas influential to its invention appear in the works of Frege and Bertrand Russell. John Shosky concluded that it is far from clear that any one person should be given the title of inventor of truth-tables.
Classical propositional logic is decidable because each formula contains only finitely many propositional variables, so one can determine in finitely many steps whether it is satisfiable, unsatisfiable, or valid. Deciding satisfiability is an NP-complete problem, but practical methods such as the DPLL algorithm from 1962 and the Chaff algorithm from 2001 run fast on many useful cases. Recent work extended SAT solvers to arithmetic expressions, creating SMT solvers.