— Ch. 1 · Foundations And History —
Propositional logic.
~4 min read · Ch. 1 of 6
Chrysippus developed a deductive system for propositional logic in the 3rd century BC. His work focused on propositions rather than terms, distinguishing it from traditional syllogistic logic. Most of his original writings were lost between the 3rd and 6th century CE. Stoic logic faded into oblivion until its resurrection in the 20th century. Gottfried Leibniz created symbolic logic in the 17th or 18th century with his calculus ratiocinator. This early development remained unknown to the larger logical community. George Boole and Augustus De Morgan recreated many advances independently. Gottlob Frege published Begriffsschrift in 1879, combining features of syllogistic and propositional logic. Gerhard Gentzen and Stanisław Jaśkowski invented natural deduction methods later. Evert Willem Beth developed truth trees as another proof technique. The invention of truth tables remains uncertain, with Ludwig Wittgenstein and Emil Post both credited.
Syntax And Formal Language
Propositional variables serve as atomic formulas within formal languages. These variables typically appear as capital roman letters like P, Q, and R. Definitions establish that atomic propositional variables are themselves formulas. If A is a propositional connective, applying it to sequences of formulas creates new formulas. Definition three states nothing else qualifies as a formula. This closure clause excludes infinitely long expressions from well-formed status. Colin Howson calls the composition principle responsible for building complex formulas. Composite formulas containing one or more connectives become molecular sentences. Backus-Naur form provides context-free grammar specifications common in computer science. Schematic letters often use Greek symbols like alpha, beta, and gamma. Propositional constants represent specific propositions while variables range over all atomic possibilities. Special symbols T and F denote truth and falsity respectively. Some authors treat these as zero-place truth-functors or nullary connectives.