Propositional logic
Branch of classical logic dealing with propositions and logical connectives.
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Propositional logic is a branch of classical logic, also called statement logic, sentential calculus, propositional calculus, sentential logic, or zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them, and is considered the foundation of first-order logic and higher-order logic.
- field
- Logic
- known_for
- Foundation of first-order logic; truth-functional propositional logic; use of propositional variables and logical connectives
- key_figures
- Chrysippus, Gottfried Leibniz, George Boole, Augustus De Morgan, Gottlob Frege, Gerhard Gentzen, Stanisław Jaśkowski, Evert Willem Beth, Ludwig Wittgenstein, Emil Post
Lore & Background
Although propositional logic had been hinted by earlier philosophers, Chrysippus is often credited with development of a deductive system for propositional logic as his main achievement in the 3rd century BC, which was expanded by his successor Stoics. The logic was focused on propositions, different from traditional syllogistic logic focused on terms. However, most original writings were lost, and between the 3rd and 6th century CE, Stoic logic faded into oblivion, to be resurrected only in the 20th century.
Symbolic logic, important to refine propositional logic, was first developed by the 17th/18th-century mathematician Gottfried Leibniz, whose calculus ratiocinator was unknown to the larger logical community. Many advances were recreated by logicians like George Boole and Augustus De Morgan, independent of Leibniz. Gottlob Frege's predicate logic builds upon propositional logic, combining features of syllogistic logic and propositional logic.
Advances after Frege included natural deduction (invented by Gerhard Gentzen and Stanisław Jaśkowski), truth trees (invented by Evert Willem Beth), and truth tables. The invention of truth tables is of uncertain attribution, with ideas preceding them from Philo, Boole, Charles Sanders Peirce, Ernst Schröder, Frege, and Bertrand Russell. The tabular structure is generally credited to either Ludwig Wittgenstein or Emil Post, with others including Jan Łukasiewicz, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis.
Reader's Guide
Propositional logic is significant as the most thoroughly researched branch of classical truth-functional propositional logic, in which formulas have precisely one of two truth values (true or false), upholding the principle of bivalence and the law of excluded middle. It is considered zeroth-order logic compared to first-order logic. All machinery of propositional logic is included in first-order and higher-order logics, making it foundational.
Its legacy includes the formal language using propositional variables and logical connectives (conjunction, disjunction, implication, biconditional, negation) to form compound propositions. It specifies a standard of logical consequence based solely on meanings of propositional connectives. Arguments are defined as premises and a conclusion, with validity requiring that if all premises are true, the conclusion must be true; soundness requires validity and all premises true.
Propositional logic deals only with declarative sentences having truth value, not questions or imperatives. It remains a core subject in university logic curricula, providing the basis for understanding more complex logical systems.
Did You Know?
- Propositional logic is also called zeroth-order logic, in contrast to first-order logic.
- Chrysippus is often credited with developing a deductive system for propositional logic in the 3rd century BC.
- The invention of truth tables is of uncertain attribution, with the tabular structure generally credited to either Ludwig Wittgenstein or Emil Post.
- Natural deduction was invented by Gerhard Gentzen and Stanisław Jaśkowski.
Frequently Asked Questions
What is Propositional logic?
Propositional logic is a branch of classical logic that studies whole statements (propositions) as truth-valued units and examines how they combine through connectives such as AND, OR, and NOT. It goes by several other names, including statement logic, sentential calculus, and zeroth-order logic.
Who are the key figures behind Propositional logic?
Its history stretches from the Stoic Chrysippus through Gottfried Leibniz, George Boole, and Augustus De Morgan, with later formal work by Gottlob Frege, Gerhard Gentzen, Stanisław Jaśkowski, Evert Willem Beth, Ludwig Wittgenstein, and Emil Post.
What role does Propositional logic play in the broader logic landscape?
It provides the truth-functional, zeroth-order machinery—propositional variables plus logical connectives—on which first-order and higher-order logics are constructed. Without this foundational layer, the more expressive quantified systems would lack their basic structural base.
How does Propositional logic differ from first-order logic?
Propositional logic treats entire sentences as indivisible truth-bearers and reasons only about their combinations, whereas first-order logic adds quantifiers and predicates so you can reason about objects and their properties within a domain. Think of propositional logic as the outer shell of logical structure; first-order logic peels inside the sentence.
Why should a set-theory fan care about Propositional logic?
Every proof system, model, and consistency result in set theory ultimately depends on the truth-functional connectives and inference rules that propositional logic formalizes. It is the minimal logical engine that makes the richer language of set-theoretic axioms interpretable and manipulable.
More in Set Theory & Logic 1-19
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