Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic can be also considered an advancement from the earlier propositional logic. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan-completely independent of Leibniz. Although his work was the first of its kind, it was unknown to the larger logical community. The 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Propositional logic was eventually refined using symbolic logic. Consequently, the system was essentially reinvented by Peter Abelard in the 12th century. However, most of the original writings were lost and the propositional logic developed by the Stoics was no longer understood later in antiquity. This advancement was different from the traditional syllogistic logic, which was focused on terms. For more, see Other logical calculi below.Īlthough propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic ( Stoic logic) by Chrysippus in the 3rd century BC and expanded by his successor Stoics. However, alternative propositional logics are also possible. Truth-functional propositional logic defined as such and systems isomorphic to it are considered to be zeroth-order logic. The principle of bivalence and the law of excluded middle are upheld. In classical truth-functional propositional logic, formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself. Premise 1: P → Q ) are represented directly. The premises are taken for granted, and with the application of modus ponens (an inference rule), the conclusion follows.Īs propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed any more by logical connectives, this inference can be restated replacing those atomic statements with statement letters, which are interpreted as variables representing statements: Conclusion: It's cloudy.īoth premises and the conclusion are propositions. Premise 1: If it's raining then it's cloudy. The following is an example of a very simple inference within the scope of propositional logic: In English for example, some examples are "and" ( conjunction), "or" ( disjunction), "not" ( negation) and "if" (but only when used to denote material conditional). Logical connectives are found in natural languages. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. Propositions that contain no logical connectives are called atomic propositions. Compound propositions are formed by connecting propositions by logical connectives. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Propositional calculus is a branch of logic. Existential generalization / instantiation.Universal generalization / instantiation.
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